Schlumberger_-_Well_Performance_Manual (1)

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Well Performance Manual
TABLE OF CONTENTS 
AUTHOR’S PREVIEW ............................................................................................. 
PREFACE ......................................................................................................... 
CHAPTER 1 INTRODUCTION .......................................................................... 
NODAL Analysis ..................................................................... 
CHAPTER 2 RESERVOIR SYSTEM.. ................................................................ 
INFLOW PERFORMANCE RELATIONSHIP ....................... 
Single-Phase Flow.. ............................................................. 
Productivity Index.. ............................................................. 
Productivity Ratio ............................................................... 
Sources of Information.. ...................................................... 
Important Definitions.. ........................................................ 
Boundary Effects.. ............................................................... 
Two-Phase Flow.. ................................................................ 
Phase Behavior of Hydrocarbon Fluids.. ............................ 
Vogel’s IPR.. ....................................................................... 
Composite IPR.. .................................................................. 
Standing’s Extension of Vogel’s IPR.. ............................... 
Fetkovich Method ............................................................... 
Multipoint or Backpressure Testing.. .................................. 
Isochronal Tests.. ................................................................. 
Horizontal Well.. ................................................................. 
Tight Formations.. ............................................................... 
Type Curve.. ........................................................................ 
Homogeneous Reservoir Type Curve.. ............................... 
Transient IPR.. .................................................................... 
CHAPTER 3 COMPLETION SYSTEM ............................................................. 
Pressure Loss in Perforations.. .................................................. 
Pressure Loss in Gravel Pack ................................................... 
CHAPTER 4 FLOW THROUGH TUBING AND FLOWLINES.. ...................... 
Single-Phase Gas Flow in Pipes.. .............................................. 
Multiphase Flow.. ...................................................................... 
Liquid Holdup.. ......................................................................... 
No-Slip Liquid Holdup.. ............................................................ 
Superficial Velocity.. ................................................................. 
Mixture Velocity.. ..................................................................... 
Slip Velocity.. ............................................................................ 
Liquid Density.. ......................................................................... 
Two-Phase Density ................................................................... 
Viscosity.. .................................................................................. 
Two-Phase Viscosity.. ............................................................... 
Surface Tension.. ....................................................................... 
Multiphase-Flow Pressure Gradient Equations.. ....................... 
111 
iv 
l-l 
l-2 
2-l 
2-l 
2-l 
2-2 
2-2 
2-5 
2-6 
2-11 
2-16 
2-16 
2-16 
2-17 
2-17 
2-18 
2-19 
2-20 
2-22 
2-24 
2-25 
2-25 
2-27 
3-l 
3-3 
3-13 
4-1 
4-3 
4-6 
4-6 
4-7 
4-7 
4-8 
4-8 
4-8 
4-8 
4-9 
4-9 
4-9 
4-9 
i 
Two-Phase Friction.. ................................................................. 
Hydrostatic Component.. .......................................................... 
Friction Component.. ................................................................ 
Acceleration Component.. ......................................................... 
Flow Patterns.. ........................................................................... 
Calculation of Pressure Traverses.. ........................................... 
Gradient Curves ......................................................................... 
CHAPTER 5 WELL PERFORMANCE EVALUATION 
OF STIMULATED WELLS.. ......................................................... 
Pumping Wells.. ........................................................................ 
Gas Lift Wells ........................................................................... 
Pre-Acid Test Computation Sheet ............................................. 
Post-Acid Test Computation Sheet ........................................... 
APPENDICES 
4-10 
4-12 
4-12 
4-12 
4-13 
4-16 
4-17 
5-l 
5-l 
5-2 
5-7 
5-15 
A PRESSURE LOSS EQUATIONS .................................................. A-l 
B 
C 
D 
E 
F 
G 
FLUID PHYSICAL PROPERTIES CORRELATIONS.. ............... 
Oil Properties.. ........................................................................... 
Gas Solubility (Rs). ................................................................... 
Formation Volume Factor (Bo) of Oil.. .................................... 
Oil Viscosity.. ............................................................................ 
Gas Physical Properties.. .......... ..- ............................................. 
Real Gas Deviation Factor (2). .................................................. 
Gas Viscosity (u g). ................................................................... 
Rock and Fluid Compressibility.. .............................................. 
Oil Compressibility (co). ........................................................... 
Gas Compressibility (cg). .......................................................... 
Rock Pore Volume Compressibility (cf) ................................... 
GRADIENT CURVES.. .................................................................. 
CALCULATION OF GAS VBLOClTY ........................................ 
PARTIAL. PENETRATION ........................................................... 
PRAT’S CORRELATION.. ............................................................ 
SLIDES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
B-l 
B-l 
B-l 
B-2 
B-5 
B-5 
B-8 
B-10 
B-12 
B-12 
B-13 
B-14 
C-l 
D-l 
E-l 
F-l 
G-l 
ii 
CHAPTER 1 - INTRODUCTION 
A well can be defined as an inter- 
phasing conduit between the oil 
and gas reservoir and the surface 
needed to produce reservoir fluid 
handling facility. This interface is 
ble asset. The physical descrip- 
to the surface, making it a tangi- 
For optimal production, a well 
tion of a well is quite involved. 
engineering considerations. The 
design requires some complex 
optimal production refers to a 
maximum return on investment. 
The physical description of a typ- 
ical oil or gas well is shown in 
Fig. 1.1 . 
" 
-TOTAL u)sS IN TUBINQ 
I,, the performance of a well the Fig. 1 .I Possible pressure losses in the producing system 
drainage volume of the reservoir for aflowing well. 
draining to the well plays an im- 
portant role. A well combined with the reservoir draining into it is normally called an oil 
a br gas production system. A production system is thus composed of the fdlowing major components - 
porous medium, 
completion (stimulation, perforations, and gravel pack), 
vertical conduit with safety valves and chokes, 
artificial lift system such as pumps, gas lift valves, etc, 
horizontal flowlines with chokes, and other piping components e.g. valves, 
elbows, etc. 
In an oil or gas production system,the fluids flow from the drainage in the reservoir to 
the separator at the surface. The average pressure within the drainage boundary is often 
production system and is assumed to remain constant over a fixed time interval during 
called the average reservoir pressure. This pressure controls the flow through a 
depletion. When this pressure changes, the well's performance changes and thus the well 
needs to be re-evaluated. The average reservoir pressure changes because of normal 
reservoir depletion or artificial pressure maintenance with water, gas, or other chemical 
injection. 
The separator pressure at the surface is designed to optimize production and to retain 
lighter hydrocarbon components in the liquid phase. This pressure is maintained by using 
mechanical devices such as pressure regulators. As the well produces or injects, there is a 
continuous pressure gradient from the reservoir to the separator. It is common to use a 
1-1 
wellhead pressure for the separator pressure in NODALt calculations assuming that the 
separator is at the wellhead or very near it. Such assumptions imply negligible pressure 
loss in the flowline. 
NODAL Analysis 
A node is any point in the production 
system (Fig. 1.2) between the drainage 
boundary and the separator, where the 
pressure can be calculated as a function 
nodes in the complex production 
of the flow rates. The two extreme 
boundary (8) and the separator (1). The 
system are the reservoir drainage 
pressures at these nodes are called the 
separator pressure psep, respectively. 
average reservoir pressure j j r and the 
The other two important nodes are the 
bottomhole (6) , where the bottomhole 
flowing pressure pwf is measured by a 
downhole gauge, and the wellhead (3) 
where the wellhead pressure Pwh is 
measured by a gauge attached to the Fig. 1.2 Location of Various Nodes 
(Mach et al. 1981) a pressures are measured or calculated at each node, then the pressure loss between the Christmas tiee Or tKe flow arm. If the nodes can be calculated as a function of the flow rates. Some of the nodes (2.4, and 5 in 
Fig. 1.2) where a pressure drop occurs across the node due to the presence of a choke, 
restrictions (safety valves), and other piping components are called the functional nodes. 
For each component in the production system such as the porous medium, completion, 
tubulars, chokes, etc., the flow rate (4) is functionally related to the pressure differential 
(Ap) across the component, Le., 
q = f@P) (1.1) 
The following chapters of this manual establish such mathematical relationships for 
different component segments of the production system. Based on these relationships, the 
parameters that are important for the optimization of production through these 
components are discussed. NODAL systems analysis is used as a method of combining 
all these component system design procedures to help design and optimize the total 
system. 
Reference 
1. Mach, J.M., Proano, E.A.. and Brown, K.E.: “Application of Production Systems Analysis to 
Determine Completion Sensitivity on Gas Well Production,” paper 81-pet-13 presented at 
the Energy Resources Technology Conference and Exhibition, Houston, Texas, Jan. 18-22, 
1981. 
f Mark of Schlumberger 
1-2 
e CHAPTER 2 - RESERVOIR SYSTEM 
INFLOW PERFORMANCE 
RELATIONSHIP 
The Inflow Performance Relationship or 
IPR is defined as the functional relation- 
ship between the production rate and the 
bottomhole flowing pressure. Gilbert 
(1954) first proposed well analysis using 
this relationship. IPR is defined in the 1 
pressure range between the average reser- 2 
The flow rate corresponding to the atmo- 
voir pressure and atmospheric pressure. 
spheric bottomhole flowing pressure is 
defined as the absolute open flow potential , of the well, whereas the flow rate at the 
average reservoir pressure bottomhole is 
always zero. A typical inflow performance - 
relationship is presented in Fig. 2.1. Fig. 2.1 A Typical IPR Curve. 
Single-phase Flow 
For single-phase oil or liquids, the inflow performance relationship shown in Fig. 2.1 is 
stated by Darcy’s law for radial flow, as follows - 
- 
4 
(SLOPE(-PROOUCTlVlNlNOEX(PI] 
0 q, (slb/D) - 
qHM- ABSOLUTE OPEN FLOW POTENTIAL (AOFP) 
7.08 x l e 3 k,, h Br - pwf) 
~ o B , [ l n ( ~ ) - 0 . 7 J + s t + D q o 
qo = 1 (2.1) 
oil flow rate into the well, stb/D, 
formation volume factor of oil, bblhtb, (defined in Appendix B), 
viscosity of oil, cp, (Appendix B), 
permeability of the formation to oil, , md, 
net thickness of the formation, ft, 
average reservoir pressure, psia, 
bottomhole flowing pressure, psia, 
radius of drainage, ft, 
fl - where A is area of circular drainage in sq ft , 
wellbore radius (ft) , 
total skin, 
2-1 
Dqo = pseudo skin due to turbulence. In oil wells, this term is insignificant 
especially for low permeability reservoirs. 
It can be shown that, for re = 1,466 ft, rw = 0.583 ft, st = 0 and no turbulence, Darcy’s law 
simplifies to - 
This simple equation is often used for the estimation of flow rates from oil wells. 
Productivity Index 
An inflow performance relationship based on Darcy’s law is a straight line relationship as 
well can flow with atmospheric pressure at the bottomhole. The Productivity Index (PI) is 
shown in Fig. 2.1. Absolute open flow potential (AOFP) is the maximum flow rate the 
the absolute value of the slope of the IPR straight line. Thus, 
PI - -9 
@r - Pwf ) 
Based on Darcy’s law, 
PIail = 
7.08 X 10-3 h - - 90 bbl 
po Bo. [In (z )- 0.75 + s , ] (Fr-pwf) ’ (psi-D) 
The PI concept is not used for gas wells, as the IPR for a gas well is not a straight line but 
a curve. 
Productivity Ratio 
The Productivity ratio is defined as the ratio of the actual Productivity Index to the ideal 
Productivity Index (s,= 0). 
PI (actual) Productivity ratio = (ideal, = o) 
where, 
= 0.87 ( kh )s t 162.6 qpB 
m = slope of semilog straight line (Homer or MDH). 
The productivity ratio is also called the flow efficiency, completion factor, or condition 
ratio. 
Example Prblem No. 2-1 
Darcy's Law is perhaps the most important relationship in petroleum reservoir 
engineering. It relates rate with the pressure drawdown and is often used to decide on an 
appropriate stimulation treatment. The following exercises illustrate such uses: 
Oil Well 
kh (Pe - Pwf) 
h (reservoir thickness) = 50 ft, 
pe (initial reservoir pressure) I 3,000 psi, 
pwf (flowing bottomhole pressure) = 1,000 psi, 
B (formation volume factor) = 1.1 res bbl/stb, 
p (viscosity) = 0.7 cp; re (drainage radius) 
rw (well radius) = 0.328 ft (7-7/8" well) 
I) Impact of drainage area 
40 
8.42 1,489 160 
8.07 1,053 80 
7.73 745 
4% 
9% 
I 640 I 2,980 I 9.11 1 16% 
Increasing the drainage area by a factor of 16 results in a maximum rate 
decrease by 16%. In other words, the drainage area for a steady state well does 
not have a major impact on the rate. However, the drainage radius may have a 
profound effect on the cumulative recovery per well. 
2-3 
2) Impact of permeability and skin. 
For the given variables q = 7.73 + 920 k 
I s = o I s = 10 I 
10.0 
0.5 0.01 1.2 0.01 
5 0.1 12 0.1 
52 1 .o 119 1 .o 
519 10.0 1.190 
If k = 10 md, elimination of skin from 10 to 0 would result in more than 600 stb/d 
increase (i.e., candidate for matrix acidizing). 
If k = 0.1 md, elimination of skin would lead to a maximum increase of 7 stb/d. 
Example Problem No. 2-2 
For the following oil-well data, calculate 
(a) the absolute open flow potential 
relationship curve, 
and draw the inflow performance 
(b) Calculate the Productivity Index. 
Data: 
Permeability, = 30 md 
Pay thickness, h = 40 ft 
q w -ABSOLUTE OPEN FLOW POTENTIAL (AOFP) - 3.672 STBPD 
Avg. reservoir pressure, Fig 2.2 IPR Curve for the Example ProblemReservoir temperature, T = 200°F 
Well spacing, A = 160 acres (43,560 ft2/acre) 
Drilled hole size, D = 12-1/4 in. (open hole) 
Formation volume factor, Bo = 1.2 (bbl/stb) 
Oil viscosity, po = 0.8 cp 
(assume skin = 0 and no turbulence) 
jjr = 3,000 psig 
2-4 
0 Solution 
(a) Drainage radius, r, = d-ft 
= 1,490 ft 
Wellbore radius, r, = 0.51 ft 
Applying Darcy's law for radial flow, 
Absolute open-flow potential, 
q o = 
7.08 x 10-3 (30 x 40) (3,000 - 0) 
(0.8 X 1.2) [ In (z:)-o.75] - 
qo = 7.23 26'550 = 3,672 stb/D 
(b) Productivity Index = 9- 7.08 X 10-3 k h - 
@I - Pwf) ~ ~ B ~ [ ~ ( ~ ) - 0 . 7 5 ] 
Sources of Information 
1. A Transient Well Test Interpretation, e.g., buildup, drawdown, and interference, 
gives - 
Note: In injection wells, the buildup test is called a falloff test, and the 
drawdown test is called an injectivity test. 
2. Special well tests, i.e., extended drawdown or reservoir limit tests, are used to 
determine the drainage shape and re. 
3. Well logs and cores are also used to determine k and h. 
Quite often, if properly conducted and interpreted, well test interpretation methods yield 
2-5 
the most representative values of the reservoir parameters such as (kh/p),&, etc. These 
values are normally the volumetric average values in the radius of investigation, whereas 
logs and cores determine the value of k at discrete points around the wellbore. 
Important Definitions 
Permeability (k): It is a rock property that 
measures the transmissivity of fluids 
through the rock. In the simplest form, 
Darcy’s law when applied to a rectangular 
slab of rock is 
W P , - Pz) 
PL 
q = 
where, 
q = volumetric flow rate, cc/sec, 
1.1 = viscosity of fluid, centipoise , 
k = permeability of the rock, 
L = length of the rock, cm, 
A = area of cross section of flow, 
p, - pz = pressure difference, atmosphere. 
darcy, 
Fig 2.3 Darcy’s Law for Linear Flow. 
cmz, 
From this equation, a permeability of one darcy of a porous medium is defined when a 
medium will flow through it, under viscous flow at a rate of one cubic centimeter per 
single-phase fluid of one centipoise viscosity that completely fills the voids of the 
second per square centimeter cross-sectional area under a pressure gradient of one 
atmosphere per centimeter. This definition applies mainly to matrix permeability. In 
carbonates, some sands, coals, or other formations that often contain solution channels 
and natural or induced fractures, these channels or fractures do change the effective 
permeability of the total rock mass. It can be shown that in a low permeability matrix, a 
few cracks or fractures can make an order of magnitude change in the effective 
permeability of the rock. It can also be shown that the permeability (in darcies) of a 
fracture of width “w” (in inches) per unit height is given by - 
k=54.4~106~*. 
Consequently, a fracture of 0.01 in. width in a piece of rock will be equivalent to the rock 
permeability matrix may substantially increase the effective permeability of the bulk 
having a permeability of 5,440 darcies. Note that a few such cracks in a very low 
rock. 
Thickness (h): The net pay thickness is the average thickness of the formation in the 
drainage area through which the fluid flows into the well. It is not just the perforated 
interval or the formation thickness encountered by the well. 
2-6 
Average Reservoir Pressure (&): If all the wells in the reservoir are shut in, the 
stabilized reservoir pressure is called the average reservoir pressure. The best method of 
obtaining an estimate of this pressure is by conducting a buildup test. 
Skin Factor (8,): During drilling and com- 
pletion, the permeability of the formation 
near the wellbore may often be altered. 
This altered zone of permeability is called 
the damage zone. The invasion by drilling 
mudcake and cement, and presence of high 
fluids, dispersion of clays, presence of 
saturation of gas around the wellbore are 
some of the factors responsible for reduc- 
tion in the permeability. However, a suc- 
cessful stimulation treatment such as 
tive improvement of permeability near the 
acidizing or fracturing results in an effec- 
damage. The skin factor determined from a 
wellbore, thus reducing the skin due to 
bore mechanical or physical phenomena 
well test analysis reflects any near well- 
that restrict flow into the wellbore. Most 
common causes of such restrictions in ad- 
- 
I 
Fig 2.4 Positive Skin =Damaged Welibore or 
Reduced Wellbore Radius. 
dition to damage are due to the partial penetration of the well into the formation, limited 
perforation, plugging of perforation, or turbulence (Dq). These non-damage related skins 
are commonly known as the pseudo-skin. It is important to note that the total skin includ- 
ing turbulence can be as high as 100 or even more in a poorly completed well. However, 
the minimum skin in a highly stimulated well is about -5. 
The skin factor (st) is a constant which relates the pressure drop in the skin to the flow 
rate and transmissivity of the formation (Fig. 2.4). Thus, 
St = r41.50BQ, 
Lipski 
Applk,,, = 0.87 m s, 
= (p',,,f - pwf) in Fig 2.4 
where, 
m 
St 
= slope of semi-log straight line from Homer or Miller, Dyes and 
Hutchinson obtained from a buildup or drawdown test, respectively, 
(psibog cycle). 
= Sd + sp + spp + SI"* + so + s, + ....., 
2-1 
where, 
St = total skin effect, 
sd = skin effect due to formation damage (+ ve), 
spp = skin effect due to partial penetration (+ ve), 
sp = skin effect due to perforation (+ ve) (Appendix E), 
SNrb = Dq, skin effect due to turbulence or rate dependent skin (+ ve), 
so = skin effect due to slanting of well (- ve), 
SS = skin effect due to stimulation (mostly - ve). 
Only positive skin can be treated in this 
manner. It is very important to note that sd 
can be at best reduced to zero by acidizing. 
However, induced fractures can impose a 
negative skin (s,) in addition to rendering 
the damaged skin (sd) to zero. 
Using the concept of skin as an annular 
area of altered permeability around a 
wellbore, Hawkins showed the well model 
in Fig. 2.5. 
where, 
k, = reservoir permeability, 
= permeability of altered or 
damaged zone, 
r, = radius of altered or 
damaged zone, 
r, = wellbore radius. 
Fig 2.5 Well and Zone of Damaged or Altered 
Permeabilify. 
This formula also suggests that when skin s, is zero, Le., the well is not damaged, the 
positive skin indicates a damaged well, whereas a negative skin implies stimulation. 
permeability of the altered zone kd equals the reservoir permeability k, or r, equals r,. A 
Darcy's law for a single-phase gas is - 
7.03 X 10-1 kg h ( F: - pif ) 
qg = jIg.Tr[ln('C)-0.75+s,+Dqg] 
rw 
2-8 
where, 
9s = gas flow rate, Mscf/D, 
kg = permeability to gas, md, 
- 
z = gas deviation factor determined at average temperature and average 
pressure, fraction (Appendix A),' 
- 
T = average reservoir temperature, (deg. Rankine), 
pg = gas viscosity, cp (Appendix A), calculated at average pressure and - 
average temperature. 
All other parameters are defined in Eq. 2.1. Note that the skin is reduced by a stimulation 
treatment only; the turbulence is reduced by increasing the shot density or perforation 
interval or a combination of these two. 
Darcy's law for gas flow can be simplified by substituting - 
z = 1, pg = 0.02 cp, t = 200"F, or 660"R 
e as, ln@)-0.75 = 7.03 
qg = 77 X 10-7 kh 6; - pif), 
where, 
qg = gas flow rate, Mscf/D, 
k = permeability, md, 
h = reservoir thickness, ft. 
This equation is used for a quick estimation of the gas flow rate from the well. 
Turbulence Dq, in Eq. 2.2 is known as the skin due to turbulence. In gas wells this may 
be quite substantial and may need evaluation to decide the means to reduce it. In very 
highly productive oil wells, this term may also be of significance.To evaluate the skin 
due to turbulence, Darcy's law can be rewritten as - 
(iir-pWd = aoqo + bo% 2 
2-9 
where, 
k & C = a,+ boqo oil 
90 
Fig-. 2 . 6 ~ Plots Based on Four Point Test. Fig. 2.6b Evaluation of Four Point Test Data Aper 
Jones, Blount, and Glaze. 
Based on a four point test where the bottomhole flowing pressure pwr is calculated for four 
stabilized flow rates (q), the following plots can be made on Cartesian graph paper shown 
in Fig. 2.6a. The intercept and the slope of the straight line shown in Fig. 2.6a, give the 
values of the constant a and b defining the straight line. The turbulence factor can be cal- 
2-10 
culated from b. Diagnostics from a four point plot are presented in Fig. 2.6b. It is impor- 
tant to note that the well data presented in Case 1 does not show any turbulence because 
the slope of this line is zero, resulting in a value of zero for b. However, the turbulence or 
the skin due to it increases as the slopes increase as shown in Cases 2 and 3. 
Jones, Blount, and Glaze4 modified Darcy’s law for radial flow with an analytical ex- 
pression of the turbulence factor “D’ as a function of the perforated interval and gas or oil 
turbulence coefficient in the rock (0). These equations are presented in Appendix A 
(Sections ID and IIB). 
Boundary Effects 
Most reservoir engineering calculations as- 
sume radial flow geometry. Radial geome- 
try implies that the drainage area of a well 
is circular and the well is located at the 
center of the drainage circle. In many 
cases, the drainage area of a well is rectan- 
Applications of equations based on radial 
gular, or of some other noncircular shape. 
could lead to substantial error. Darcy’s law 
geometry to a noncircular drainage area 
can be modified for a bounded drainage 
radius of different shapes of boundary as 
follows - 
7.08 X l e 3 kh ( pr - pwf) 
‘lo = B o b [ In (x) - 0.75 +SI for oil 
Table 2.1 Factors for Different Shapes and Well 
Area of System Shown and A1I2lre is Dimensionless. 
Positions in a Drainage Area Where A = Drainage 
for gas 
and 
PI =+= 7.08 X 1e3 kh 
Pr - Pwf B o b [ In (x) -0.75 +s I 
for oil 
where X is given in Table 2.1 for various drainage areas and well locations. 
Example Problem No. 2-3 
(a) A buildup test in a well after a constant rate production, q, = 100 BPD indicates 
St = 2 
Calculate the pressure loss due to skin for Bo = 1. 
2-1 1 
e Solution: 
ApPskh = 141.2 X - q ~ o ~ s , = 1 4 1 . 2 ~ - ~ 2 = 1 4 1 . 2 x 1 0 = 1 , 4 1 2 p s i 1 0 0 20 
b) Draw IPRs for the given well data and make a tabular presentation of skin vs 
absolute open flow potentials (AOFP). 
Given - 
oil well 
k = 5 m d 
h = 2 0 f t 
po = 1 .1 cp 
spacing = 80 acres 
PI = 2,500 psig 
s =-5,-1,0, 1,5, 10,50 
Bo = 1.2 RB/STB 
r, = 0.365 ft 
Solution 
Drainage radius, re = = 1,053 ft. 
AOFP = q = 7.08 X l e 3 k h FI 
~ o B ~ [ I n ( ~ ) - 0 . 7 5 + s ] rW 
- - 7.08 x 10-3 x 5 x 20 x 2,500 
- 1,341 - 7.97 - 0.75 + s 
2-12 
skin, (s) 
- 5 
- 1 
0 
1 
5 
10 
50 
(STB/D) 
AOFP 
604 
216 
186 
163 
110 
78 
23 
For oil wells, since the PI is a straight line, the fir and AOFF’ will uniquely define the IPR. 
(c) Draw the IPR curve for the following gas well data. Calculate the AOFF’. 
k = 1md h = 200ft 
Reservoir temperature = 200” F 
p = 0.019cp z = 1.1 
pr = 3,500 psig spacing = 80 acres 
r, = 0.365 ft skin, s = 1 
Solution 
From Section (b), 
lnk -0.75 = 7.22 
rw 
From Darcy’s law, 
7 .03x l@kh( FT-p$) 
qg (MscfD) = (neglecting turbulence) 
j i s Z T [ l n ( ~ ) - 0 . 7 5 + s ] rw 
7.03 x lo” x 1 X 200 ( 3,508- p i f ) 
0.019 x 1.1 x 660 (7.22 + 1) 
- - 
= 1.24 X 10-3 (3,502 - p $) 
2-13 
PWdPSik!) 
3,500 
3,000 
2,500 
2,000 
1,500 
1.000 
500 
0 
Flow rate 
(Msfcrn) 
0 
4,030 
7,440 
10,230 
12,400 
13,950 
14,880 
15,190 
(d) Solve problem (b) for a square drainage instead of a circular drainage and 
skin = 0. 
Solution: 
1,341 1,341 
= In x - 0.75 + s - In x - 0.75 - 
From Table 2.1, 
0.571 4- 1 066 
x = 
rw 
"- - 0.365 - 2,920 
= In 2,920 - 0.75 =7.98 - 0.75 = 
1,341 17341 185 (stb/D) 
1,000 2,175 0.825 
1,340 1,606 1 .O403 
1.2000 
A straight line is drawn through these points and the slope and intercept are determined. 
The equation of this straight line is 
i!r.Z&f = 0.1997 + 0.000625 q, 
q 
Intercept, a = 0.1997, 
2-14 
Slope, b = 0.000625, 
AOFP = q for pwt = 14.7 psia. 
Solve the quadratic equation in q - 
0.000625 q2 + 0.1997 q - 2985.3 = 0 
q = 
-0.1997 f 4(0.1997)2 + (4 x 0.000625 x 2,985.3) = 2,032 stb/D 
2 x 0.000625 
Example Problem No. 2-4 
The following data is from a four point (flow after flow) test conducted in an oil well. 
1,000 2,175 
1,340 1,606 
1,600 1,080 
Fr = 3,000 psia 
Using Jones, Blount, and Glaze method, calculate 
i) a and b, and 
ii) AOFP. 
Solution 
ded. Plot q vs 
- 
on Cartesian graph paper, based on the data provic 
9 
Therefore, 4 = 
-0.1997 rt 
1.25 X lo3 
The positive root of this equation is 
The absolute open flow potential of this well is 2,031 (stbD). 
2-15 
Two-Phase Flow 
Darcy's law is only applicable in single-phase flow within the reservoir. In the case of an 
oil reservoir, single-phase flow occurs when the bottomhole flowing pressure is above the 
depletion of a reservoir, the reservoir pressure continues to drop unless maintained by 
bubblepoint pressure of the reservoir fluid at the reservoir temperature. During the 
pressure falls below the bubblepoint pressure which results in the combination of single- 
fluid injection or flooding. Consequently, during depletion the bottomhole flowing 
phase and twophase flow within the reservoir. This requires a composite IPR. Before 
discussing the composite IPR, a brief review of phase behavior is presented. 
Phase Behavior of Hydrocarbon Fluids 
Reservoir fluid samples taken at the bot- 
tomhole pressure, when analyzed in Pres- 
sure-Volume-Temperature (P-V-T) cells 
Temperature (P-T) diagram. A typical 
generate phase envelopes in the Pressure- 
black oil P-T diagram showing the physi- 
cal state of fluid is presented in Fig. 2.7. 
Based on the average reservoir pressure, 
bottomhole flowing pressures, and the cor- 
responding temperatures on this diagram, 
one can decide on the type of reservoir 
fluid; Le., single phase, two phase, or a 
combination. Such information is used to 
determine the type of IPR equation to be 
used. 
e 
Vogel's IPR 
In the case of two-phase flow in the reser- 
voir where the jjr is below the bubblepoint 
relationship is recommended (Fig. 2.8). 
pressure, the Vogel's inflow performance 
This IPR equation is - 
A BUCK OIL 
8 COMPOSITE 
C WET O M 
E BOLN. (115 DRNE OIL 
D onv QAS 
OIL 
TEMPERATURE I'R) - 
Fig. 2.7 Typical Phase Diagram for Black Oil. 
This IPR curve can be generated if either 
the absolute open flow potential qomax and 
the reservoir pressure jjr are known or the 
reservoir pressure jjr and a flow rate and q (STBPD) 
pressure are known. For either case, a 
buildup test for pr and a flow test with a 
bottomhole gauge are needed. 
the corresponding bottomhole flowing 
Fig 2.8 Diferent Forms of Inflow Performance 
Relationships IPR. 
2-16 
Composite IPR 
The composite IPR is a combination of the 
Productivity Index based on Darcy's law 
above the bubblepoint pressure and Vogel's 
IPR below the bubblepoint pressure. This 
IPR is particularly used when the reservoir 
pressure pr is above the bubblepoint pressure 
is below the bubblepoint pressure (Fig. 2.9). 
(Pb) and the bottomhole flowing pressure pwf 
Thus, 
and 
j. , 
q- q 0 M M 
Fig. 2.9 Vogel's Composite IPR. 
where, 
Note that Vogel's IPR13 is independent of the skin factor and thus, applicable to 
stimulated wells. 
undamaged wells only. Standingll extended Vogel's IPR curves to damaged or 
Standing's Extension of Vogel's IPRStanding extended the effect of skin on Vogel's IPR equation and came up with the 
concept of a flow efficiency factor or I%. If &. (Fig. 2.4) is defined as the bottomhole 
flowing pressure for an undamaged well, 
then, 
= 1 
- 
- - - PI - P'wf 
Pr-Pwfil,<o 
Damaged Well 
Undamaged well 
Stimulated Well 
2-1 7 
Thus FE can be calculated using well testing methods. Vogel's IPR curves for different 
values of FE are presented in Fig. 2.10. 
where, 
These PD's and tD's are obtained from appropriate type curves or well test information 
such as- and s, and other available well reservoir parameters. kh 
I.I 
From the definition of flow efficiency, 
P L '6 - FE @r - pwa) 
So, Vogel's IPR can be written as - 
For a large negative skin or high FE 
predict a lower rate with lower bottomhole 
(FE >1) and low pressures, these IPRs 
pressure contrary to reality. Clearly, this 
method cannot be recommended for such 
cases. Thus, in the case of stimulated 
wells, alternative methods to calculate an 
inflow performance relationship must be 
used. 
Fetkovich Method 
Multipoint backpressure testing of gas 
wells is a common procedure to establish 
the performance curve of gas wells or de- 
liverability. Fetkovichl* applied these tests 
on oil wells with reservoir pressures above 
and below the bubbleuoint messure. The A ~ ~~ ~ 
general conclusion fr&n these backpres- Fig. 2.10 Standing's Correlationfor Wells with FE 
sure tests is that as in gas wells, the rate 
pressure relationship in oil wells or the oil 
Values not Equal to I. 
well IPR is of the form - 
2-18 
This equation is also referred to as the oil and gas deliverability equation. The exponent 
“n” was found to lie between 0.5 and 1.000 for both oil and gas wells. An “n” less than 
used. The coefficient “C” represents the Productivity Index of the reservoir. 
1.0 is often due to nondarcy flow effects. In such cases, a nondarcy flow term can be 
Consequently, this coefficient increases as k and h increase and decreases as the skin 
increases. 
The Fetkovichl2 IPR is a customized IPR for the well, and is obtained by multipoint 
backpressure testing, e.g., flow after flow or isochronal testing. 
Multipoint or Backpressure Testing 
Such tests are performed in a shut-in well which has achieved a stabilized shut-in 
pressure throughout the drainage area. These tests are also called deliverability tests 
because they are used to predict the deliverability of a well or flow rate against any 
backpressure (pd) imposed on the reservoir. Typically, these backpressure tests consist of 
a series of at least three stabilized flow rates and the measurement of bottomhole flowing 
pressures as a function of time during these flow intervals. The results of backpressure 
tests are presented in log-log graph papers as log (F , - pZw, ) vs log q. Typical flow after 
flow test sequences are presented in Fig. 2.1 1 and Fig. 2.12. 
2 
Fig. 2.11 FlowAfier Flow, NormalSequence Fig. 2.12 Flow Afier Flow, Reverse Sequence 
(Afier Fetkovich). (Afier Ferkovich). 
2-19 
Problem on Flow-After-Flow Test: 
The purpose of this exercise is to calculate C, n, and AOF using the conventional and LIT 
equations. 
Given 
ijr = 201 psia 
I Duration I PWi I m(p) I Flow Rate 1 
(hr) 
0.00 3.56 20 1 0 
(psia) I (MMpsiaYcp) 1 (MMcf/D) 
3 
5.50 3.18 190 4 
4.44 3.28 193 2 
3.97 3.35 195 2 
2.73 3.38 196 
After the deliverability test is conducted, the coefficients of the deliverability equation 
points are plotted, the best fit straight line is drawn through them. The straight line then 
such as C and n are computed from the log-log plot of the - p f f ) vs q. After these 
obtained is called the deliverability curve. 
where, 
and 
calculated for any q and the corresponding pwf obtained from the deliverability curve. 
Isochronal Tests 
Isochronal tests are performed in low permeability reservoirs where it takes a prohibitive 
oil or gas reservoirs that normally need stimulation. Typical isochronal tests involve 
amount of time to yield stabilized backpressure. This happens in very low permeability 
periods are the same and the shut-in times are maintained long enough for the pressure in 
flowing the well at several rates with shut-in periods in between. The duration of the flow 
extended drawdown (Fig. 2.13). 
the drainage to stabilize to the average reservoir pressure. These tests are ended with an 
2-20 
Fig. 2.13 Isochronal Tesl, Flow Rate and Pressure Diagram. 
q Mud 
- 2 2 Fig. 2.14 p - p wfvs. q for Isochronal Test. Fig. 2.15 Graph of Log C vs. Log t for Isochronal Test. 
2-21 
q 4 
qse 
93 
EXTENDED FLOW RATE 
q 2 
91 
t- 
- PC- 
P 
Fig. 2.16 Modified Isochronal Test. Flow Rate and Pressure Diagrams. 
2 
To analyze these tests, log @ I - P:~.) vs log 
q are plotted as shown in the next figure 
for each of the flow periods. The best 
straight lines are drawn through these 
points, one for each of the flow periods. 
The slopes of these straight lines should be 
the same, and these lines should be paral- 
lel. These lines should get closer with in- 
creasing time. The slope n is calculated for 
any of these straight lines by the equation 
used for the flow-after-flow test. The co- 
efficient C is calculated from the straight 
line with slope n plotted through the point 
corresponding to the last stabilized rate in 
the extended drawdown test period. 
Horizontal Well Test. 
Fig. 2.17 5 - p $vs. q for Modified Isochronal 2 
Darcy's law suggests that the net thickness 
or the productive length of vertical wells is 
directly proportional to the productivity of 
tal well can be considerably greater. In 
a well. The productive length of a horizon- = W L " I I 
horizontal wells, the productivity does not 
directly increase with its length. The pro- Fig. 2.18 Horizontal Well Drainage Model. 
ductivity increase in horizontal wells with the length of the well is much slower. How- 
ever, horizontal wells can be very long unless economically limited. In heterogeneous 
2-22 
reservoirs or naturally fractured reservoirs, these wells can be drilled perpendicular to the 
natural fracture planes to substantially improve productivity. In the Rospo Mare field in 
Italy, a horizontal well is reported to produce 10 times its vertical neighbors. In very thin 
low permeability reservoirs, such increases are also possible. 
The inflow performance relationship for a horizontal well at the mid-thickness of a 
reservoir (Fig. 2.18) is 
7.08 X kh h Gr - p i ) 
Qo = 
P O P 0 {h[+d-]+F1n[ Ph )} 6) CP + 1) rw 
for L > Rh and (L/2) < 0.9 rch , 
where, 
a = one-half the major axis of a drainage ellipse in a horizontal plane (Fig. 2.18). 
= !j [ 0.5 + 4- Os 
k =- 
where subscripts h and v refer to horizontal and vertical, respectively. This equation has 
Darcy's law for gas. 
all the variables in oilfield units and can be easily converted for a gas well IPR as in 
Example Problem No. 2.4 
Gas Well: Calculate flow rate through this horizontal well. 
Spacing 160 acres 
Horizontal permeability, kh 0.06 md 
Vertical permeability, k, 0.06 md 
Average reservoir pressure, pr 800 psia 
Bottomhole flowing pressure, pd 400 psia 
Wellbore radius, r, 5 in. 
Reservoir temperature, T, 80" F 
Specific gravity of gas, y e 0.65 
Net pay thickness, h 1,600 ft 
Horizontal well length, L 2,770 ft 
2-23 
Calculations 
z - - 
qsc = 
0.9 (from Standing's correlation) 
0.0123 cp (from Carr et al.) 
a Tight Formations 
703 x 10-6 h - 703 x 10-6x 1,600 
540 x 0.0123 x 0.9 
- 
T Fg-i = 0.19 
~ @ , - p ~ . ) = 0 . 1 9 ( 8 0 0 ~ - 4 0 0 ~ ) = 9 1 , 2 0 0 
2 
91,200 k (+)= 1 , 0 5 7 ~ Mscf 
0.79 + 0.58 p In (1,920 p ) 
It is very difficult to design a meaningful well test in a very low permeability (less than 
0.1 md) reservoir. The main problem in such cases is the time required to reach infinite-acting radial flow which is very large, making such tests impractical. Consequently, it 
becomes very difficult to obtain the reservoir parameters such as (kh/p), S, PI, etc., to 
establish the Productivity Index in such cases. Multipoint tests to determine the IPR also 
rate if these wells flow. Unfortunately, these low permeability wells are normally 
become quite difficult due to the long time taken by such wells to stabilize at any flow 
fractured and once they are fractured their effective wellbore radii increases substantially. 
In such cases, it becomes even more difficult to obtain radial flow permitting a Homer 
yield the fracture properties like fracture conductivity, fracture half-length, etc. Very 
type analysis from a postfracture well test. These post-fracture tests in such cases only 
often, where conventional analysis (e.g., Hornedsemilog analysis) fails to interpret well 
test data, a type-curve matching technique is used to determine the required reservoir data 
such as (Wp), S, etc. 
Once the reservoir parameters are determined, type curves can also be used to generate 
the transient inflow performance relationships. The tight formations normally remain 
transient for a long time after the production resumes. During this transient time, type 
curves can be used to generate the transient IPR. Transient IPRs allow the calculation of 
cumulative production during the transient time, in addition to the flow rates normally 
obtained using a steady-state PR. 
2-24 
Type Curve 
Type curves are graphical representations of the solution of the diffusivity equation for 
constant rate drawdown under different boundary conditions. The diffusivity equation is a 
mathematical description of the fluid flow phenomena through the reservoir into the 
wellbore. Each type curve assumes the following reservoir and well types - 
homogeneous reservoir with or without wellbore storage and skin; 
homogeneous reservoir with or without induced fractures in the wellbore; 
* dual porosity or naturally fractured reservoirs; 
layered reservoir, etc. 
The three variables in x, y, and z dimensions in the type curve are dimensionless pressure, 
dimensionless time, and a variable representing the near wellbore condition or the 
boundary shape, respectively. Depending on the wellbore (completion) condition, the z 
variable may be - 
wellbore storage (c) and skin (s) in the case of homogeneous reservoirs, 
fracture conductivity (FcD) in the case of wells with an induced fracture. 
All type curves are plotted on log-log graph paper, so the shape of the curves strictly 
depends on the pressure and time data obtained from transient well tests. The effects of 
other parameters such as kh, k, q, +, etc., are strictly translational. This can be explained 
with a real type curve for a homogeneous reservoir with wellbore storage and skin. 
Homogeneous Reservoir Type Curve 
The type curve for a homogeneous reservoir with wellbore storage and skin presents pD as 
a function of tD/CD for different CDea in Fig. 2.19 (Gringarten et all8 nopetrol Johnston 
The wellbore storage dominated periods for all values of C,ek fit on one unit slope line. 
Schlumberger). This type curve is commonly known as a Flopetrol Johnston type curve. 
The end of a unit slope line for different values of CDez8 is marked on the type curve. The 
start of the infinite-acting radial flow is also clearly marked. This curve does not show the 
effect of boundaries to make it applicable strictly to an infinite reservoir. In reality, as the 
well sees the effect of a boundary, these type curves bend upward. The dimensionless 
time when these boundary effects are seen depends on how far the boundary is from the 
wellbore. A typical type curve for a well with no flow boundary is presented later for 
fractured wells. 
The advantage of the Flopetrol Johnston type curve is the ease of matching and clear 
definition of the flow regimes, such as end of wellbore storage, beginning of infinite-act- 
ing radial flow, etc. In this type curve, the dimensionless variables are defined as follows- 
pD = dimensionless pressure 
kh 
141.2 qBok 
- - AP 
tD/cD = 0.000295 - kh At 
k 
2-25 
102 SKIN FACTOR, s 
10 
9 n 
1 
10" 
10" 1 2.4 10 102 io3 104 
b / C D 
APPROXIMATE START OF SEMILOG STRAIGHT LINE 
Fig. 2.19 Homogeneous Reservoir Type Curve 
These dimensionless groups represent universal pressure and time scales. The type curves 
production or injection rates. The presentation of these dimensionless variables in log-log 
actually represent a global description of the pressure response with time for different 
coordinates makes possible a match of the pressure vs time data obtained from a well test. 
The rationale for that is - 
logp, = log Ap+log kh 
141.2 qBp 
= log Ap + log y 
where y = constant for a particular reservoir. 
Similarly, 
= log At + log x 
where x = f (k,h,p,C) = constant for the well reservoir system. 
Consequently, log pD and log tl, are actually log Ap and log At, respectively, translated by 
2-26 
e some constants defined by reservoir parameters. Therefore, if the proper type curve representing the reservoir model is used, real and theoretical pressure vs time curves are 
identical in shape but are translated in scale when plotted on the same log-log graph. 
Transient IPR 
Infinite Homogeneous Reservoir 
Transient IPR curves for homogeneous reservoirs can be generated using the Flopetrol 
Johnston type curve as follows. 
Example Problem No. 2-5 
Given 
k = l m d 
h = 20 ft 
Po = 1 CP 
Bo = l.ORB/STB 
pr = 2,000 psia 
r, = 0.5 ft 
- 
Calculation 
$ = 0.2 
ct = 10-5 psi-1 
C = 0.001 bbl/psi 
time = 0.1, 1, 10, 1 O O h r 
S - - . 1.21 
tD = 0.000295 - - = 5.9 At kh At 
CD P C 
Absolute Open 
At (hr) ( B W 
Flow Potential pD (from TC) tdCo 
0.1 506 0.56 0.59 
1 
39 1.2 590 100 
48 5.9 59 10 
90 3.15 5.9 
For homogeneous reservoirs, the well known infinite-acting semilog approximation for a 
2-21 
0 well with a skin s, producing at a constant rate q after the wellbore storage effects subside is given by - 
PO = 1/2 ( In to + 0.80907 + 2s ) . 
This equation represents the homogeneous reservoir type curve until the pressure 
transients start seeing the boundary. This time depends on the boundary radius and can be 
calculated based on some of the reservoir properties as follows. 
The equation for po can be simplified as - 
90 = @I - Pwf) 
162.6 !&BO[ { log ( k )- 3.23 + 0.87,) +log (t) ] 
$PCt r , 
The transient IPR equation for gas reservoirs is presented in Appendix A, Section IV. 
Example Problem No. 2-6 
Same as previous problem. r, = 2,000 ft 
time to end of infinite-acting radial flow, t (hr) 
= 948 0.2 x 105 x 2,0002 = 7,584 hr 1 
qOlAOFP - - 20 x 2,000 
162.6 log - 3.23 + 0.87s +log (t) [I (0.2 x x OS* 1 1 
- 246 - log t +4.3 
I t (hr) I a. (Absolute Oaen Flow Potential) I 
I 0.1 I 74.55 I 
10 
100 
1,000 
46.42 
39.05 
33.70 
7,584 30.07 
Note the discrepancy in the absolute open-flow potentials at early times less than 100 
hours. This is due to the wellbore storage effects not considered in the semilog approxi- 
mation. 
2-28 
Homogeneous Reservoir With Induced Vertical Fracture 
Meng and Brown8 presented type curves for wells with an induced vertical fracture at the 
center of the reservoir for different closed rectangu1ar.drainage areas. The fluid is 
considered slightly compressible with constant viscosity p. For gas flow, the real gas 
pseudopressure function (Al-Hussainy et al., 1966) is used where gas properties are 
evaluated at the initial reservoir pressure. The dimensionless variables used in these type 
curves are defined as follows. 
pD = dimensionless wellbore pressure drop 
tDXf = dimensionless time 
F,, = dimensionless fracture conductivity 
A few of these type curves are presented in Fig. 2.20. It is important to note that theearly time pressure behavior depends on the FcD; whereas the late time or after 
depletion starts, the pressure response is influenced by the shape and the size of 
drainage. 
As in the homogeneous reservoir case, transient IPR curves can be generated for fractured 
reservoirs using appropriate type curves presented in Fig. 2.20. These type curves can be 
used for both single-phase oil or single-phase gas. In the case of gas, m (p) is used instead 
of pressure, For oil wells below the bubblepoint pressure, Vogel’s IPR is used. A step-by- 
step procedure to calculate transient IPR is presented below. 
1. Calculate the dimensionless fracture conductivity defined earlier. 
2-29 
Fig. 2.20 Constant rate type curve forfinite-conducrivityfiacture-closed square system 
@elye = 1). 
2. A drainage geometry xJy. is assumed for a closed reservoir; calculate the 
fracture penetration ratio x,/&. 
3. Calculate dimensionless time tolr for any assumed time and for known 
parameters as k, 4, C,, xrr etc. 
4. From the type curves, determine the dimensionless pressure pD (tmh Fco, xf/x., 
X J Y ~ 
5. Calculate qb and PI at the bubblepoint pressure using the following - 
and 
Ibblepc where pb is the bubblepoint pressure and qb is the rate at the bu 
pressure. 
6. Calculate qVogel, where, 
)int 
2-30 
7. Calculate pwf vs q below the bubblepoint pressure using Vogel's equation - 
For gas wells, Steps 1 to 5 are followed to generate IPR curves. 
Exercises 
1. Given data (oil well) - 
Reservoir permeability, k 
Formation thickness, h 
Reservoir porosity, (I 
Total system compressibility, c, 
Oil formation volume factor, Bo 
Oil viscosity, ~o 
Initial reservoir pressure, pi 
Bubblepoint pressure, pb 
Square drainage area, A 
Design fracture half-length, x, 
Design fracture conductivity, kfw 
Simulated producing time, tp 
0.2 md 
30 ft 
0.15 
1.14 x lo5 psiL 
1.44' Rb/stb 
0.35 cp 
5,200 psi 
3,500 psi 
40 acres 
500 ft 
2,000 md-ft 
5, io, 30,365 days 
Hint: Generate transient IPR curves for different producing times. 
2. For the following gas well data, 
calculate the absolute open flow 
potential of the well. 
k, = 0.1 md 
pr = 3,000 psia 
h-300ft 
Reservoir temperature = 200'F 
y8 = 0.7 
Spacing = 160 acre 
Drilled hole size = 12 1/4 in. 
skin = 1 
Neglect turbulence. 
- 
I 
1 
:ig. 2.21 Transient IPR for afractured well in a 
closed square reservoir. 
2-3 1 
3. Calculate In (r&) for rw = 7 in. and for drainage areas of 20,40, 80, 160, and 320 
acres. 
Hint: (make a table) 
Drainage Acres In ( r h d r, re 
20 
40 
80 
160 
320 
4. Draw the IPR for a well with the following data. 
k = 5 0 m d Depth = 5,000 ft 
h = 100 ft 
pr = 2,000 psia (Producing all oil) - 
Determine the absolute open flow potential. 
2-32 
0 References 
1. Evinger, H.H. and Muskat, M.: “Calculation of Theoretical Productivity Factor,” 
Trans., AIME (1942) 146, 126-139. 
2. Gilbert, W.E.: “Flowing and Gas-Lift Well Performance,” Drill. and Prod. Prac., 
M I (1954) 126. 
3. Govier, G.W., et al.: ‘Theory and Practice of the Testing of Gas Wells,” Energy 
Resources Conservation Board, Alberta, Canada, 3rd Ed, 1975. 
4. Jones, L.G., Blount, E.M. and Glaze, D.H.: “Use of Short Term Multiple Rate Flow 
Test to Predict Performance of Wells Having Turbulence,” paper SPE 6133, 1976. 
5. Cinco-Ley, H. and Samaniego-V, F.: “Transient Pressure Analysis for Fractured 
Wells,” JPT (Sept. 1981) 1749-1766. 
6. Giger, F.M., Reiss, L.H., and Jourdan, A.P.: “The Reservoir Engineering Aspect of 
Horizontal Drilling,” paper SPE 13024, 1984. 
7. Joshi, S.D.: “Augmentation of Well Productivity With Slant and Horizontal Wells,’’ 
Paper SPE 15375 (1986); JPT, (June 1988) 729-739. 
8. Meng, H-Z. and Brown, K.E.: “Coupling of Production Forecasting, Fracture 
Geometry Requirements and Treatment Scheduling in the Optimum Hydraulic 
Fracture Design, paper SPE 16435, 1987. 
9. Muskat, M.: “The Flow of Homogeneous Fluids Through Porous Media,” MRDC, 
Boston, 1982. 
10. Prats, M.: “Effect of Vertical Fractures on Reservoir Behavior - Incompressible 
Fluid Case,” SPEJ (June 1961) 105-1 18. 
11. Standing, M.B.: “Inflow Performance Relationships for Damaged Wells Producing 
by Solution Gas Drive Reservoirs,” JPT (Nov. 1970) 1399-1400. 
12. Fetkovich, M.J.: “The Isochronal Testing of Oil Wells,” paper SPE/AIME 4529, 
1973. 
13. Vogel, J.V.: “Inflow Performance Relationships for Solution Gas Drive Wells,” JPT 
(Jan. 1968), 83-93. 
14. Homer, D.R.: “Pressure Buildup in Wells,” Third World Petroleum Congress, Sec. 
E, 503-52 1. 
15. Agarwal, R.G., Al-Hussainy, R., and Ramey, Jr., H.J.: “An Investigation of 
Wellbore Storage and Skin Effect in Unsteady Liquid Flow: 1. Analytical 
Treatment,” SPE Trans. (Sept. 1970) 279-290. 
16. Al-Hussainy, R. and Ramey, H.J.: “Application of Real Gas Flow Theory to Well 
Testing and Deliverability Forecasting,” JPT (May 1966) 637-642. 
2-33 
17. Brown, K.E.: “Technology of Artificial Lift Methods” Vol. 1, Tulsa, OK, PennWell 
Publishing. 
18. Mukherjee, H. and Economides, M.J.: “A Parametric Comparison of Horizontal and 
Vertical Well Performance,” paper SPE 18303, SPEFE, June 1991. 
2-34 
CHAPTER 3 - COMPLETION SYSTEM 
Most of the oil and gas wells are com- 
pleted with casing. The annulus behind the 
casing is normally cemented. Once the 
casing is cemented, it is hermetically insu- 
lated from the formation. To produce any 
fluid from the formation, the casing is per- 
forated using perforation guns. Such perfo- 
rated completions provide a high degree of Fig, ;7,1 Shnped Charge, 
control over the pay zone. because selected 
intervals can be-pdrforated and stimulated 
and/or tested as desired. It is also believed 
that hydraulic fracturing and sand control 
tions. However, the perforations impose 
are more successful in perforated comple- 
restrictions to flow from the formation to 
the wellbore in the form of extra pressure 
losses. Consequently, if not adequately 
designed and understood, perforations may 
substantially reduce the flow rates from a 
well. 
Detonation Front 
Outer Uner 
Stagnatlon Polnt 
Inner Liner 
I “I\\\ 
Shaped charge perforating is the most 
common and popular method of perforat- 
ing. A typical cross section of a shaped 
charge is shown in Fig. 3.1. As the shaped 
charge is detonated, the various stages in Fig 3,2a Jet,slug Formation, 
the jet development are shown in Fig. 3.2. 
The velocity of the jet tip is in excess of 
30,000 ft/sec, which causes the jet to exert 
an impact of some 4 million psi on target. 
Every shaped charge manufacturer pro- 
vides a specification sheet for charges as to 
the length of penetration and diameter of 
the entrance hole in addition to other API 
required specifications. A Schlumberger 
Table 3.1. 
data sheet of a perforation gun is shown in Fig 3.26 Approximare Jet Velocities and Pressures. 
3-1 
Table 3.1 - Schlumberger Perforation Data 
DESIGNATION 
GUN 
1.318" D.S. 
1.318" D.S. 
1-11/16" OS 
1-11/18" DS 
1-11116"DS 
1.11/16"FJ 
2-118' W 
1.11116"EJ 
2.118" OS 
2-1/8" DS 
2.118' DS 
2.11B'DS 
2-118" FJ 
2-718' DS 
3918' PPG 
3-3/8" PPG 
3-318" PPG 
3.318' PPG 
4" PPG 
4" PPG 
4'PPG 
4" PPG 
4' PPG 
4" PPG 
5" PPG 
5" PPG 
2.718' HSD 
2-718" HSD 
2-718" HSD 
3-38" HSD 
3.318' HSD 
3916 HSD 
3-38' HSD 
3-3/8" HSD 
34/8' HSD 
3.38" HSD 
4" LSD 
4.112'HSD 
4.112" HSD 
5'5 SPF HSD 
5" HSD 
5" HSD 
5-112" HSD 
5-1/2" HSD 
6' HSD 
6' HSD 
6" HSD 
5" HSD 
r HSD 
7-114" HSD 
7-114' HSD 
- 
SPF 
4 
4 
4 
4 
4 
416 
416 
416 
4 
4 
4 
4 
416 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
4 
12 
12 
12 
5 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
CHARGE NAME 
16A HD, RDX 
20A HD, RDX 
16A HD, PSF 
20A HD, HMX 
2OA HD, PSF 
1-11/16"EJ.RDX 
1-11116"FJ,RDX 
2-1/6" EJ, RDX 
25A HD. RDX25A HD, PSF 
25A UJ. HMX 
25A UJ. RDX 
2-116" FJ, RDX 
3681 HD. RDX 
36A UJ. RDX 
368 HJ, PSF 
388 BJ, RDX 
36A UJ, HMX 
418 HJ 11, RDX 
41A HJ 111, ROX 
418 HJ 11, HMX 
418 HJ 11, PSF 
418 UP. RDX 
418 UP. RDX 
518 UP, RDX 
518 UP, RDX 
38A UJ. RDX 
38A UJ. HMX 
368 W, RDX 
368 BJ, RDX 
36A UJ, RDX 
38A UJ, HMX 
418 11, HMX 
418 HJ II, RDX 
41A UJ, HNS 
418 UP, RDX 
418 HJ, SX1 
348 HJ 11, RDX 
43C UP, RDX 
518 HJ 11, ROX 
418 HJ 11, RDX 
418 UP 11, RDX 
418 HJ 11, RDX 
418 UP 11, RDX 
418 HJ 11, ROX 
418 HJ 11, HMX 
51C UP, RDX' 
518 UP 11, RDX' 
64C UP, ROX 
518 HJ 11, RDX' 
51C UP 11, RDX' 
PART 
NUMBEF 
HZ24509 
HZ47295 
HZ24234 
H334539 
Mi00171 
HZ24372 
P274035 
P273920 
HZ24470 
- 
H224577 
H304920 
H334542 
Mi00173 
HZ24353 
- 
HZ47776 
H334970 
H304637 
H421504 
1334534 
HZ47262 
HZ47776 
HZ47776 
-1304043 
-1304750 
i304765 
.I334974 
i304750 
i304765 
i304953 
i304952 
1304881 
u296ffi 
*a442 
i42839 
1304953 
1334852 
1304953 
43W52 
- 
1334497 
i304952 
i304930 
i304865 
1428136 
1 3 ~ 7 7 
1304956 - 
- 
1 HR 
lEMP 
RATE 
435 
330 
330 
400 
435 
300 
330 
330 
435 
330 
400 
330 
300 
330 
435 
330 
330 
4 w 
330 
- 
- 
- 
- 
330 
400 
435 
330 
330 
330 
330 
330 
400 
330 
330 
330 
ow 
330 
400 
5M) 
330 
210 
330 
330 
330 
330 
3 3 0 
330 
IW 
130 
330 
330 
330 
130 
)30 
130 
- 
T 
- 
EXPL 
LOAD 
GMS 
1.8 
1.6 
3.2 
3.2 
3.2 
9.5 
13.5 
11.0 
6.5 
6.5 
6.5 
15.5 
6.5 
14.0 
13.5 
15.0 
15.0 
22.0 
15.0 
22.0 
22.0 
22.0 
22.0 
22.0 
32.0 
32.0 
15.0 
15.0 
15.0 
15.0 
15.0 
22.0 
15.0 
22.0 
22.0 
22.0 
22.7 
24.0 
21.0 
37.0 
22.0 
22.0 
22.0 
22.0 
22.0 
22.0 
32.0 
32.0 
66.0 
37.0 
35.1 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
- 
T - - - CASING 
4-112 
4-1/2 
4.1/2 
4-11,? 
5.112 
5-112 
512 
4.12 
4-112 
4-112 
4-1/2 
5.112 
4-112 
5-112 
4-112 
4.112 
4-1 I2 
5-1/2 
51.2 
5112 
5 1 2 
7 
7 
4-112 
4-112 
4.112 
4-12 
4-12 
4-112 
4112 
4-1.2 
4-12 
51fZ 
5.1.2 
7 
7 
7 
7 
7 
7-518 
7518 
4-(/2 
- 
_. 
x 
- 
9.516 
- 
- 
- 
9518 
%5/8 
45/8 
$518 
$516 
9.516 
$518 __ 
- 
iCTll 
0.21 
E.H 
0.19 
0.21 
0.21 
0.23 
0.33 
0.30 
0.34 
0.32 
0.30 
0.32 
0.31 
0.29 
0.37 
0.34 
0.35 
0.37 
0.30 
0.44 
0.36 
0.40 
0.38 
0.66 
0.79 
0.67 
0.77 
0.24 
0.32 
0.39 
0.37 
0.31 
0.30 
0.40 
0.42 
0.31 
0.63 
0.46 
0.71 
0.42 
- - 
- 
- 
- 
- 
- 
- 
- 
0.46 
0.41 
0.68 
0.40 
0.38 
0.66 
0.40 
0.63 
0.76 
1 23 
0.47 
0.60 - 
- - PENET 
5.75 
6.69 
5.10 
6.69 
6.02 
6.03 
13.26 
16.41 
10.64 
10.36 
12.64 
12.78 
12.97 
16.12 
20.18 
10.65 
16.64 
20.44 
23.99 
- 
- 
- 
20.97 
20.28 
16.22 
16.65 
14.17 
16.57 
16.66 
19.19 
14.40 
16.76 
19.04 
20.05 
19.73 
16.43 
19.30 
21.43 
7.75 
17.10 
6.35 
25.80 
16.07 
8.80 
1 1.60 
19.06 
21.07 
20.40 
8.40 
6.65 
7.07 
27.86 
12.25 
19.81 
- 
_. 
- 
- 
- 
- 
)-FOURTH Et 
SEI 
0.23 
CFE E.H. 
0.81 
0.19 0.64 
0.27 0.67 
0.27 0.67 
0.25 0.76 - - 
0.34 
0.76 0.34 
0.79 0.36 
0.90 
0.33 0.80 
0.33 0.70 
0.36 0.69 - - 
0.40 
0.67 0.43 
1.03 0.34 
0.78 
- - 
0.36 
0.92 0.45 
1.03 0.41 
1.11 0.36 
0.73 
0.86 0.91 
0.66 1.02 
0.88 0.69 
0.73 0.76 
0.84 0.40 
- - - - 
- - 
- - 
0.34 0.75 - - 
0.46 0.80 - - - 
- - - - - - - 
- - - - 
0.50 0.62 
- - 
0.49 0.73 
- - 
0.47 
1.23 0.48 
0.75 
- - - - - 
0.54 0.W 
- 
- - 
ION 
ION II 
i 7 P OAP DATE 
TESl 
3.30 4.26 11.74 
" 
- 
" 
" 
- 
- 
- 
" 
- 
3.79 07.76 
5.21 12-84 
5.21 12-64 
4.03 03.74 - 03-87 
6.11 09.78 
11 .OO 06-74 
6.63 09-74 
9.50 05-83 
7.40 03-75 
6.33 10-65 
- 0347 
10.98 03-74 
7.73 07.72 
13.44 06-65 
- 01-67 
12.61 03-64 
15.42 07-65 
13.50 08.85 
11.96 06.64 
11.88 0574 
11.17 03-60 
9.73 09-61 
9.04 06.61 
9.55 09-81 
- 02-67 - 01-68 
- 01-67 
- 02-67 
12.36 02-65 
12.57 11.84 
- 01-66 
- 01-88 
- 09-67 
- 10-65 
- 04-66 
- 0647 - 08-87 
11.46 12-63 
- 04-65 
11.77 12-83 
- 11.64 
12.00 12-63 
- 11.64 
12.17 09-64 - 01-65 
- 01-85 
- 04-68 
16.67 06-64 - 11.64 
3-2 
PRESSURE LOSS IN PERFORATIONS 
The effect of perforations on the productivity of wells can be quite substantial. Therefore, 
much work is needed to calculate the pressure loss through perforation tunnels. A very 
brief review of the background work in the area was presented by Karakas and Tariq 
(1988). Most of the calculations on perforation pressure losses are based on single phase 
gas or liquid flow. It is generally believed that if the reservoir pressure is below the 
bubblepoint causing two phase flow through the perforations, the pressure loss may be an 
order of magnitude higher than that for single phase flow. Perez and Kelkar (1988) 
presented a new method for calculating two phase pressure loss across perforations. For 
this manual, two methods of calculating the pressure loss in perforations are presented 
with appropriate examples. These methods were proposed by McLeod (1983) and 
Karakas and Tariq (1988). 
McLeod Method 
Pressure loss in a perforation is calculated using 
the modified Jones, Blount, and Glaze equations 
proposed by McLeod. McLeod (1988) treated an 
individual perforation tunnel as a miniature well 
with a compacted zone of reduced permeability 
around the tunnel. It is believed that the com- 
pacted zone is created due to the impact of the 
shaped charge jet on the rock. However, there is 
no physical means to actually calculate the perme- 
ability of the compacted zone. McLeod suggested 
compacted zone is - 
from his experience that the permeability of the 
10% of the formation permeability, if 
perforated overbalanced, and 
40% of the formation permeability, if 
perforated underbalanced. I " d - I 
These numbers may be different in different areas. I I I 
The thickness of the crushed zone is assumed to 
Fig 3.3 Flow Inro a Perforation. 
be 0.5 in. The massive reservoir rock surrounding a well perforation tunnel renders it fea- 
sible to assume a model of an infinite reservoir 'surrounding the well of the perforation 
tunnel. Thus, in the application of Darcy's law, -0.75 in the denominator can be ne- 
The pressure loss equations through perforations are given below. 
glected. A cross section of McLeod's perforation flow model is presented in Fig. 3.3. 
Oil Well 
pwfs - pwf = as, + 2 
where the constants a and b are defined in Appendix A. Note that the flow rate qo in this 
equation is not the well production rate but the flow rate through an individual perfora- 
tion. 
3-3 
Gas Well 
The constants a and b are adequately defined in Appendix A. Again, the gas flow rate qg 
is the flow rate through an individual perforation. 
Example Problem No. 3-1 
Oil Well 
Make a completion sensitivity study for the following well: 
= 20md 
= 3,000 psia 
= 2000 ft 
= 25 ft 
= 20ft 
= 0.021 ft 
= 0.883 ft 
= 0.4 x k md 
= 0.063 ft 
= 0.365 ft 
= 35O 
= 0.65 
= 1.2 (rb/stb) 
= 1cp 
Calculate the pressure loss through perforations for shot densities 2, 4, 8, 12, 20, 
and 24 in the flow rate range from 100 to 1.200 stb/D. 
Plot completion sensitivities q vs Ap. 
3-4 
Solution 
Pressure loss through individual perforation, 
where, 
a = \ . 
p0 Bo (In $) 
b = 
7.08 X 10-3 Lp kp 
Calculations - 
Bo = 2.33 x 1010 2.33 x lOl0 = ,.9175 E9 - - 
k'd20' 
(0.4 x 20)1.201 
a = 
2.30 x 10-14 x 1.22 x Po x 53.03 x 31.75 
0.8832 
b = = 26.3598 
7.08 X 10-3 X 0.883 X 8 
Therefore, 
4 = 0.1371 + 26.3598 qo 
where, 
qo = oil flow rate per perforation, BPD 
- - well flow rate 
(shot per foot) x (perforated interval) 
3-5 
Table 3.2 -Well Flow Rate (BPD) 
Gas Well 
For gas wells, 
The constants a and b are calculated ex- 
actly the same way as shown in the oil 
well example using the gas well equations 
given in Appendix A, Section VB. To cal- 
culate the pressure loss through perfora- 
for the given well flow rate and then uwf is 
tions, pwfs is calculated from the IPR curve 
x 800 - 
U 
0 
0 600 
w 
0: 
d 
2 4w 
3t: 
2w 
'0 200 400 6W 800 1000 1200 
FLOW RATE, (BPD) 
calculakd as 
. ... 
~ 
Fig. 3.4 Plot offlow rate vs pressure drop for 
varying shot densities. 
Pwf = 4- 
AP = Pwfs - Pwf 
Karakas and Tariq Method 
For practical purposes, McLeod's method gives fair estimates of pressure loss through 
perforations. However, this model is not very sophisticated to consider the effects of the 
phasing and spiral distribution of the perforations around the wellbore. Karakas and Tariq 
(1988) presented a semianalytical solution to the complex problem of three-dimensional 
(3D) flow into a spiral system of perforations around the wellbore. These solutions are 
presented for two cases. 
A two-dimensional (2D) flow problem valid for small dimensionless perforation 
component of flow into perforations is neglected. 
spacings (large perforation penetration or high-shot density). The vertical 
A 3D flow problem around the perforation tunnel, valid in low-shot density 
perforations. 
3-6 
0 Karakas and Tariq (1988) presented the perforation pressure losses in terms of pseu- doskins, enabling the modification of the IPR curves to include the effect of perforations 
on the well performance as follows. 
For steady-state flow into a perforated well, 
where 
St = total skin factor including pseudoskins due to perforation (obtained from 
well test), 
k = formation permeability, and 
h = net thickness of the formation. 
For the total skin, 
St = sp + sdp. 
SP = perforation skin factor, and 
sdp = damage skin factor. 
The damaged skin factor, Sdp, is the treat- 
able skin component in perforated com- 
pletion. The perforation skin, sp, is a func- 
tion of the perforation phase angle 8, the 
perforation tunnel length lPI the perforation 
n,, and the wellbore radius rw. The follow- 
hole radius rp, the perforatlon shot density 
correlate the different components of the 
ing dimensionless parameters are used to 
perforation skin, sp. 
Dimensionless Perforation Height 
hD - - 
1P 
Dimensionless Perforation Radius 
Dimensionless Well Radius, 
F 
-t 
-J 'I 
(CRUSHEOZONE 
I D'A?ETER 
DlWETER 
LENGTH 
I 
I 
I 
I 
I I 
+ - PHASE ANGLE 
rw 
(Ip + rw) 
. ig. 3.5 Perforation geometry. 
rwD = 
3-7 
The calculation of perforation skin, sp, is essential to estimate the damage skin, sdp. from a 
prior knowledge of total skin, s,, determined from well tests. Karakas and Tarlq (1988) 
characterized the perforation skin, sp as, 
SP = sh + Swb f sv 
where, 
sh = pseudoskin due to phase (horizontal flow) effects, 
Swb = pseudoskin due to wellbore effects (dominant in zero degree phasing), 
sv = pseudoskin due to vertical converging flow effects (negligible in the case 
of high-shot density; 3D effect), 
sh 
r 
= In ( r w e 7 e ) ) 
where rwe(0) is the effective wellbore radius as a function of the phasing angle 0 and 
perforation tunnel length. 
rWeW = 
0.25 1, ifO=O" 
ae (r, + lp) otherwise 
a Table 3.2. ag is a correlating parameter used by Karakas and Tariq (1988) and presented in 
Table 3.2 - Dependence of 0, on phasing. 
Perforation 
Phasing 
(360") 0" 0.250 
180" 0.500 
1 20' 
0.726 90' 
0.648 
0.860 45' 
0.813 60" 
0 0 
The pseudoskin effect due to the wellbore, Swb can be calculated by using the following 
empirical relationship - 
This component of perforation pseudoskin is rather significant in case of 0' phasing. 
However, for rwd < 0.5, the wellbore effect can be considered negligible for phasing 
smaller than 120'. Table 3.3 presents the coefficients C1 and C2 as functions of phasing 
angle 8. 
3-8 
Table 3.3 - Variables C1 and C2 
Perforating Phasing I c1 c2 
(360") On 
180" 
120" 
90" 
60" 
450 
3.0E4 1 1.6E1 2.675 6.63-3 5.320 7.509 2.6E-2 4.532 1.9E-3 6.155 
4.6E-5 8.791 
It is interesting to note, that for high-shot density and unidirectional perforations or where 
sv is negligible, the perforation skin, sp, is independent of the perforation hole diameter. 
Karakas and Tariq (1988) suggested that for a low-shot density or high dimensionless 
perforation height, hD, the pseudovertical skin sv can be estimated with the following 
equation 
sv = 10ahD pD b-1 r b 
where the coefficients a and b are given by 
a = a1 log rpD + a2 
b = bl rpD + b2 
The constants al, a,, bl, and bz are presented in Table 3.4,a s functions of the phasing 
angle 8. 
Table 3.4 -Vertical skin correlation coefficients 
Phasing I a1 bl 1 bz a, 
0" (360') 
1.6392 1.1915 0.2398 -1.788 45O 
1.6490 1.3654 0.1023 -1.898 60" 
1.6935 1 S674 0.1038 -1.905 90" 
1.7770 1.6136 0.0634 -2.018 1 20° 
1.8115 3.0373 0.0943 -2.025 180" 
1.8672 5.313 0.0453 -2.091 
3-9 
e Example Problem No. 3-2 
Given- 
r, = 0.5 ft 
Ip = 1.25 ft 
n, = 16 
(a) Calculate the perforation pseudoskin s , for 0' phasing. 
Solution: 
sp = sh + Swb (sv is negligible for 16 shots per foot) 
Sh = 0.25 X 1.25 = 0.31 
r, 0.5 
rwD = r, + 1, + 0.5+ 1.25 - 1.75 
0.5 
" = 0.29 
From Table 3.3, C, = 0.16 and C, = 2.675. 
(b) If this well is tested and the total skin calculated from buildup is 4, what is the 
treatable skin? 
St = 4 = s P -t sdp 
sdp = 4 - Sp = 4 - 0.65 = 3.35. 
This is just an estimate and is more accurately characterized later. 
Crushed Zone Effect 
For conditions of linear flow into perforations, the effect of a crushed or compacted zone 
may be neglected. In the case of 3D flow, an additional skin due to the crushed zone may 
be calculated as follows - 
where the crushed zone permeability and radius (kc and rc) can be calculated using the 
McLeod (1983) method. 
3-10 
a Anixotropy Effects 
The formation anisotropy affects the pseudovertical skin s,. The flow into perforations in 
the vertical plane is elliptical (otherwise radial) in anisotropic formations. The effective 
equivalent perforation radius in this case is given by 
Damaged Zone Effects 
In a perforated completion, the contribution of a damaged zone to the total skin largely 
depends on the relative position of the perforations with respect to the damaged zone 
radius. Karakas and Tariq (1988) showed that the skin damage for perforations 
terminating inside the damaged zone can be approximated by 
where s, is a pseudoskin to take into account the boundary effects for perforations 
terminating close to the damaged zone boundary. s, is negligible for r, > 1.5 (r, + I,). 
= Permeability of damaged zone 
Id = Length of damaged zone 
rd = Radius of damaged zone 
Table 3.5 presents values of s, for 180' phasing. 
Table 3.5 - Skin due to boundary effect, 180" phasing. 
re 
(rw + sx 
18.0 
-0.002 2.0 
-0.001 10.0 
O.Oo0 
I 1.5 -0.085 1.2 -0.024 
For perforations extending beyond the damaged zone (b = O) , Karakas and Tariq (1988) 
contended, the total skin st equals the pseudoskin due to perforations sp. That is, 
3-1 1 
The pseudoskin due to perforation sp, is calculated using modified 1, and modified rw as 
rp = lp- ( 1 +)Lj 
Example Problem No. 3-3 
Given - 
For Example 3-2, calculate the perforation skin spr if the perforation tunnel extends 
beyond the damaged zone where, 
Id = 2 ft, 
k = 2md, 
kd = 1.0md 
Solution: 
2=1.25-- X 2f t 1 2 
= 0.25 
rtW = r w - ( l - T ) x 2=0 .5+1=1 .5 f t 1 
sh = 0.25 X 1 = 0.25 X 0.25 = 0.0625 P 
rwD = 1.5 +0.25 =m- 1 .5 l S -0.86 
Swb = 0.16 eXp (2.675 X 0.86) = 1.60 
Sp = Sh+Swb=o.o6+ 1.60s 1.66. 
Remember, this perforation skin sp is also the total skin st, and it includes the effect of 
damage. 
3-12 
Pressure Loss in Gravel Pack r 
Gravel packing is done to control sand 
production from oil and gas wells. Sand 
production often becomes a production 
problem causing a reduction in hydrocar- 
bon production, eroding surface and down- 
hole equipment leading to casing collapse. 
Although the dynamics of gravel packing 
is not in the scope of this manual, a typical 
cross section of a gravel-packed well is 
presented in Fig.3.4a and Fig. 3.4b. Thefigures show the flow path the reservoir 
fluid has to follow to produce into the 
wellbore. Evaluation of well performance 
in a gravel-packed well thus requires an 
accounting of the pressure losses caused 
Jones, Blount, and Glaze equations are 
by the flow through the gravel packs. 
adapted with minor modifications to ac- 
count for the turbulence effects for the cal- 
culation of the pressure loss through the 
gravel packs. These modified equations for 
oil and gas cases are presented. 
Oil Wells 
pWr,-pWr = Ap=aq2+bq 
where, 
L; USED BY SOME COMPANYS 
Fig.3.6~ Gravel Pack Schematic. 
CYLINDAICAL- 
TUNNEL FLOW 
Fig. 3.66 Cross Section of Gravel Pack Across a 
Perforation Tunnel. 
q = flow rate, BPD, 
pwr = pressure, well flowing (wellbore), psi, 
pwrs = flowing bottomhole pressure at the sandface, 
/3 = turbulence coefficient, ft-1 
3-13 
For gravel, the equation for p is: 
where 
Bo = formation volume factor, rb/stb, 
P = fluid density, lb/ft), 
L = length of linear flow path, ft, 
A = total area open to flow, f t2 
k, = permeability of gravel, md. 
(A = area of one perforation X shot density x perforated interval), 
Gas Wells 
2 
Pwfs - Pwf = aq2+bq 
where, 
a = 
1.247 x 10-10 p yg TZL 
A2 
8.93 X 103 p TZL 
kg A 
b = 
q = flow rate, Mcf/D, 
pwfs = flowing bottomhole pressure at the sandface, psia, 
pwf = flowing bottomhole pressure in the wellbore, psia, 
P = turbulence factor, ft-1. 
1.47 X 107 
yg = gas specific gravity, dimensionless, 
T = temperature, OR (OF + 460), 
Z = supercompressibility, dimensionless, 
L = linear flow path, 
A = total area open to flow 
(A = area of one perforation X shot density x perforated interval), 
p = viscosity, cp. 
3-14 
References 
1. Karakas, M. and Tariq, S.: “Semi-Analytical Productivity Models for Perforated 
Completions,” paper SPE 18271, 1988. 
2. Perez, G. and Kelkar, B.G.: “A New Method to Predict Two-Phase Pressure Drop 
Across Perforations,” paper SPE 18248, 1988. 
3. Jones, L.G., Blount, E.M., and Glaze, O.H.: “Use of Short-Term MultipleRate Flow 
Tests to Predict Performance of Wells Having Turbulence,” paper SPE 6133, 1976. 
4. McLeod Jr., H.O.: “The Effect of Perforating Conditions on Well Performance,” 
JPT (Jan. 1983) 31-39. 
3-15 
CHAPTER 4 - FLOW THROUGH TUBING AND 
FLOWLINES 
Fluid flow through the reservoir and completion are discussed in the previous chapters. 
However, the performance evaluation of a well is not complete until the effects of the 
tubing and flowline are considered with the other effects discussed in the previous 
plumbing system includes the tubing and the flowline. The objective here is to calculate 
chapters. This chapter discusses the flow through the plumbing system in the well. The 
flowing phases. In most oil and gas wells, two- or three-phase (oil, gas, and water) flow 
the pressure loss in the tubing or in the flowline as a function of flow rates of different 
occurs in the plumbing system. Consequently, a brief discussion on the theory of 
phase flow. 
multiphase flow in pipes is presented. This theory is an extension of the theory of single- 
The pressure gradient equation under a steady-state flow condition for any single-phase 
incompressible fluid can be written as 
where, 
= pressure drop per unit length of pipe, (psi/ft) , 
p = density of fluid, (lbm/ft3), 
e = angle of inclination of pipe, 
v = fluid velocity (ft/sec), 
f = friction factor, 
d = internal diameter of the pipe (ft), 
a = correction factor to compensate for the velocity variation over the 
pipe cross section. It varies from 0.5 for laminar flow to 1.0 for 
fully developed turbulent flow. 
This equation applies to any fluid in a steady state flow condition. An important thing to 
components. 
note in this equation is that the total pressure gradient is the sum of three principal 
Hydrostatic gradient (p sin 0) 
Friction gradient - (f2ygzcdp ) 
4- 1 
Acceleration gradient - (;:2 ) 
The friction factor f for laminar single-phase flow, is calculated using an analytical 
expression such as, 
where NRc is the Reynolds Number and is defined as 
where, 
p = Viscosity of the flowing fluid. 
For turbulent flow when the Reynolds Number exceeds 2000, the relationship between 
the friction factor and Reynolds Number is empirical in nature. This relationship is very 
sensitive to the characteristics of the pipe wall and is a function of relative roughness e/d, 
where E is defined as the absolute roughness of the pipe. The most widely used method to 
calculate the friction factor in turbulent flow is the equation of Colebrook (1938) - 
" 
1 
c - 1.74-210g($+ N R ~ lge7 f0.5 ) 
Note that the friction factor f occurs on both sides of this equation requiring a trial and 
Moody (1944) in graphical form (Moody diagram) is widely used for the calculation of 
error solution procedure. For this reason, the solution of these equations presented by 
the friction factor. A Moody diagram is presented in Fig. 4.1. A very simple equation 
proposed by Jain (1976) reproduces the Colebrook equations over essentially the entire 
range of Reynolds Number and relative roughness of interest, and is presented as 
Selecting the absolute pipe roughness is often a difficult task because roughness can de- 
pend upon the pipe material, manufacturing process, age and type of fluids flowing 
through the pipe. Glass pipe and many types of plastic pipe can often be considered as 
smooth pipe. It is common to use a roughness of 0.00005 ft for well tubing. Commonly 
used values for line pipe range from 0.00015 ft for clean, new pipe to 0.00075 ft for very 
dirty pipe. An acceptable procedure used by many investigators is to adjust the absolute 
roughness to permit matching measured pressure gradients. 
4-2 
Fig. 4-1 Pipe Friction Factors for Turbulent Flow. (Modified after Moody, L.F., Trans. 
ASME, 66,671,1944.) 
Single-phase Gas Flow in Pipes 
For gas flow or compressible flow, the density of fluid is a function of pressure and 
temperature. The energy balance Equation 4-1 can be modified to account for pressure 
and temperature-dependent density. The energy balance equation for steady-state flow 
can be written as, 
-dp+g_sinedL+- 144 fv2 vdv 2gcd dL + " -0 gc P gc 
The energy loss term due to friction uses the Moody friction factor 'f'. The kinetic energy 
term (vdv)/gc, is negligible for all cases of gas flow as shown by Aziz (1963). Applying 
real gas law, the density of gas p becomes 
p (+ ) = 2.7047 3 
4-3 
Equation 4-2 can be rewritten as 
53.24 TZ +sinedL+-- - 0 fv*dL 
Yg P 2gcd 
The velocity of gas at in situ pressure and temperature conditions is 
v = 0.4152 
Pd 
where, 
q = gas production rate, MMscfD (14.65 psia, 6OoF), 
v = gas velocity in pipe, ft/sec, 
d = pipe diameter, ft, 
Ys - - gas gravity, (air = 1). 
Substituting the velocity term in Equation 4-3 gives 
53*24 TZ + sin 0 dL + 0.002679 - i ( % y q z d L = O 
Yg P 
(4-3) 
(4-4) 
This is the most practical form of an energy balance equation used for gas flow 
calculations. The friction factor is calculated using the Moody diagram (Fig, 4.1) or using 
any of the friction factor equations presented in the previous section of this chapter as 
functions of the Reynolds Number and relative roughness factor. For steady-state gas 
flow Reynolds Number is defined as 
where, the viscosity of gas ‘p’ is in centipoise. For diameter ‘d’ in inches, 
Estimation of Static Bottomhole Pressure 
Ignoring the friction loss term and appropriately integrating over the pressure and length, 
4-4 
Therefore, 
where 
pbh = static bottomhole pressure, psia, 
pwh = static wellhead pressure, psia, 
T = average temperature between bottomhole and surface, 
z = compressibility factor at average pressure and average 
- 
- 
temperature.Equation 4-5 is used extensively to calculate the weight of the gas column. The solution 
bottomhole pressure from known surface pressure and temperature, a value of bottomhole 
of this equation is iterative in pressure asT is a function of pressure. To calculate the 
pressure has to be assumed. Then the average pressure and average temperature knowing 
the geothermal gradient should be calculated and the average z-factor determined. Now, 
using Equation 4-5, a new bottomhole flowing pressure is calculated. If this calculated pbh 
continue until convergence of the assumed and calculated bottomhole pressures. 
does not compare with the assumed bottomhole pressure, the iterative procedure should 
Estimation of Flowing Bottomhole Pressure 
Cullender and Smith (1956) proposed a simple method of calculating the flowing 
bottomhole pressure based on Equation 4-4. Cullender and Smith rearranged Equation 
4-4 and integrated the pressures over the whole length of the pipe. Thus, 
Equation 4-6 can be solved by any standard numerical integration schemes such as Simp- 
son's rule. Equation 4-6 uses the Moody friction factor and the diameter 'd' is in feet. 
4-5 
Multiphase Flow 
The energy balance equation for multiphase flow is very similar to that of the single- 
phase flow. In this case, the velocities and fluid properties of the total fluid mixture are 
used instead of the single-phase fluid properties. However, the definition of a fluid mix- 
ture becomes complicated in this case. The quality of a fluid mixture changes with the 
pipe diameter, pipe inclination, temperature, pressure, etc., mainly due to slippage be- 
tween the phases. In the absence of slippage, the mixture properties should be the input 
volumetric fraction weighted average of all the phases constituting the mixture. For ex- 
ample, if the mixture contains 50% oil and 50% gas at the pipe entry, then the average 
mixture density should be 
Pm = Po X 0.5 + pg X 0.5 
When gas and liquid phases flow in pipes, due to buoyancy or density contrast between 
However, such averaging is not practically valid in the case of multiphase flow in pipes. 
phase. Thus, in the case of upward two-phase flow (production), the gas gains velocity in 
the phases, the gas phase tends to gain an upward velocity with respect to the liquid 
mass, the cross section of pipe occupied by a liquid or gas phase thus change continu- 
the direction of flow as liquid slips down or loses velocity. To satisfy the conservation of 
flow string is called the liquid holdup (HL). The complementary fraction of pipe cross 
ously. The fraction of pipe cross section occupied by liquid at any point in the multiphase 
multiphase flow should be the holdup weighted sum of the single phase fluid property. 
section occupied by gas is called the gas void fraction. The actual mixture property in a 
Since the liquid holdup continuously changes in the pipe, the phase velocities also 
change. This section presents definitions of some of the important flow properties (such 
as holdup) and different velocities used in multiphase flow calculations. 
Liquid Holdup 
In gasfiiquid two-phase flow, due to the contrast in phase densities the gas phase tends to 
move up while the liquid phase tends to move down with respect to the gas phase, creat- 
ing a slippage between the phases. As a result in upflow, a liquid loses velocity requiring 
increased pipe cross section to flow with the same volumetric flow rate. This phe- 
nomenon of slippage causes the flowing liquid content in a pipe to be different from the 
input liquid content. The flowing liquid content is called the liquid holdup. Liquid holdup 
is also defined as the ratio of the volume of a pipe segment occupied to the total volume 
of that pipe segment. That is, 
volume of liquid in pipe segment 
H~ = volume of pipe segment 
Liquid holdup is a fraction which varies from zero for single-phase gas flow to one for 
single-phase liquid flow. The most common method of measuring liquid holdup is to 
measure the liquid trapped. There are different mechanistic and empirical models for the 
isolate a segment of the flow stream between quick-closing valves and to physically 
prediction of liquid holdup. The remainder of the pipe segment is occupied by gas, which 
is referred to as gas holdup or void fraction. That is, 
4-6 
No-Slip Liquid Holdup 
No-slip holdup, sometimes called input liquid content, is defined as the ratio of the 
volume of liquid in a pipe segment divided by the volume of the pipe segment which 
would exist if the gas and liquid traveled at the input or entrance velocity (no slippage). It 
can be calculated directly from the known gas and liquid flow rates from 
where, qL and q, are the in-situ liquid and gas flow rates, respectively. The no-slip gas 
holdup or void fraction is defined as 
It is obvious that the difference between the liquid holdup and the no-slip holdup is a 
measure of the degree of slippage between the gas and liquid phases. Since no-slip holdup 
is an analytically determined parameter, it is often used as an independent variable to 
determine important two-phase flow parameters such as the liquid holdup. 
Superficial Velocity 
Many two-phase flow correlations are based on a variable called superficial velocity. The 
superficial velocity of a fluid phase is defined as the velocity which that phase would 
exhibit if it flowed through the complete cross section of the pipe. 
Thus, the superficial liquid velocity is 
and, the superficial gas velocity is 
v,, = A 
where, qr and 98 are liquid and gas flow rates, respectively, and A is the cross-sectional 
area of the pipe. 
The actual phase velocities are defined as 
and 
where, vL and vg are liquid and gas velocities, respectively, as they flow in the pipe. 
4-1 
Mixture Velocity 
The mixture velocity v, used in two-phase flow calculations is 
v, = VSL + Vsg 
It is an important correlating parameter in two-phase flow calculations. 
Slip Velocity 
The slip velocity is defined as the difference in the actual gas and liquid velocities - 
vs = vg - VL 
Liquid Density 
The total liquid density may be calculated from the oil and water densities and flow rates 
if no slippage between the oil and water phases is assumed - 
P L = Pofo + rcwfw 
where, 
fo = 
40 + qw io B~ + iw B, 
f, = 1 - f, 
d, WOR = water/oil ratio = 7 
90 
q(0.W) = oil or water flow rate (stb/D) 
Two-Phase Density 
The calculation of two-phase density requires knowledge of the liquid holdup. Three 
equations for two-phase density are used by various investigators in two-phase flow. 
Ps = P L H L + P g H g 
Pn = P L h + P g A g 
4-8 
ps is used by most investigators to determine the pressure gradient due to the elevation 
change. Some correlations are based on the assumption of no-slippage and, therefore, use 
pn for two-phase density. pk is used by some investigators such as Dukler (1969) to define 
the mixture density used in the friction loss term and Reynolds Number. 
Viscosity 
The viscosity of an oiVwater mixture is usually calculated using the water/oil ratio as a 
weighting factor. The equation is 
PL = Pofo + Pwfw 
Two-Phase Viscosity 
The following equations have been used to calculate a two-phase viscosity. 
p,, = J L L ~ L + no slip mixture viscosity 
Ps = P L ~ L + pghg , slip mixture viscosity 
Surface Tension 
Correlations for the interfacial tension between water and natural gas at various pressures 
tension between natural gas and crude oil depends on oil gravity, temperature, and 
and temperature are obtained from measured data or PVT correlation. The interfacial 
dissolved gas, among other variables. 
When the liquid phase contains both water and oil, the same weighting factors as used for 
calculating density and viscosity are used. That is, 
OL =oofo +awfw 
where, 
a, = surface tension of oil, 
a, = surface tension of water, 
and fo, fw are oil and water fractions, respectively. 
Multiphase-Flow Pressure Gradient Equations 
The pressure gradient equation for single-phase flow can now be extended for multiphase 
flow by replacing the flow and fluid properties by the mixture properties. Thus, 
4-9 
where, 
p = density, 
v = velocity, 
d = pipe diameter, (ID), 
g = acceleration due to gravity, 
g, = gravity conversion factor, 
f = friction factor, 
= pressure gradient, 
m = mixture properties, 
0 = angle of inclination from horizontal. 
The equation is usually adapted for two-phase flow by.assuming that the gasfliquid 
mixture can be considered to be homogeneous over a finite volume of the pipe. For two- 
phase flow, the hydrostatic gradient is 
where, p. is the density of the gasfliquid mixture in the pipe element. 
Considering a pipe element which contains liquid and gas, the density of the mixture can 
be calculated from 
ps = p ~ H ~ + p g H g 
The friction loss component becomes 
where, ftp and pf are defined differently by several investigators, Le., Duns and Ros 
(1963), Hagedom and Brown (1965), etc. 
Two-Phase Friction 
Previously, it was shown that the term (dp/dL)f represents the pressure losses due to 
4-10 
friction when gas and liquid flow simultaneously in pipes. This term is not analytically 
predictable except for the case of laminar single-phase flow. Therefore, it must be 
determined by experimental means or by analogies to single-phase flow. The method 
which has received by far the most attention is the oa ne resulting in two-phase friction 
factors. The different expressions for the calculation of two-phase friction gradient are the 
following - 
@ FRICTION gc - - fL PL dl. (used in bubble flow) 
In general, the two-phase friction factor methods differ only in the way the friction factor 
is determined and to a large extent on the flow pattern. For example, in the mist-flow 
pattern, the equation based on gas is normally used; whereas, in the bubble-flow regime, 
the equation based on liquid is frequently used. The definition of pr can differ widely 
depending on the investigator as discussed previously. (Please refer to the appropriate 
papers for any detailed discussion of this topic.) 
Number. Recall that the single-phase Reynolds Number is defined as: 
Most correlations attempt to correlate friction factors with some form of a Reynolds 
One consistent set of units frequently used for calculating NRc is 
p = density, lbm/ft3, 
v = velocity, ft/sec, 
d = pipe diameter, ft, 
p = viscosity, lbm/ft-sec. 
Since viscosity is more commonly used in centipoise, the Reynolds Number with p in cp 
is 
4-1 1 
Hydrostatic Component 
From the pressure gradient equation in single and multiphase flow, it becomes evident 
that the elevation component drops out in horizontal flow. However, the elevation or the 
hydrostatic gradient component is by far the most important of all the three components 
in vertical and inclined flow. It is the principal component that causes wells to load up 
and die. Gas well loading is a typical example where the hydrostatic component builds up 
in the well due to liquid slippage and overcomes reservoir pressure, reducing the gas in- 
take. 
Friction Component 
This component is always more dominant in horizontal flow. Also in vertical or inclined 
gas, gas condensate or high GLR multiphase flow, the friction loss can be very dominant. 
due to very high friction losses compared to hydrostatic losses. In fact, by injecting more 
In gas lift wells, injection above an optimum GLR causes a reversal of the tubing gradient 
gas, one may lose oil production in a gas-lift well. 
Acceleration Component 
The acceleration component, which sometimes is referred to as the kinetic energy term, 
constitutes a velocity-squared term (Eq. 4-7) and is based on a changing velocity that 
must occur between various positions in the pipe. In about 98% of the actual field cases, 
this term approaches zero but can be significant in some instances, showing up to 10% of 
the total pressure loss. In those cases of low pressure and hence low densities and high 
gas volumes or high gas-oil ratios, a rapid change in velocity occurs and the acceleration 
component may become significant. It should always be included in any computer 
calculations. 
The acceleration component is completely ignored by some investigators and ignored in 
some flow regimes by others. When it is considered, various assumptions are made 
procedure to determine the pressure drop due to the kinetic energy change. This pressure 
regarding the relative magnitudes of parameters involved to arrive at some simplified 
gradient component is important near the surface in high GLR wells. 
From the discussion of the various components contributing to the total pressure gradient, 
it is essential that methods to predict the liquid holdup and two-phase friction factor be 
developed. This is the approach followed by most researchers in the study of two-phase 
flowing pressure gradients. 
4- 12 
Superficial Oil Velocity, V., , ft.lsec. 
1 
1 
c 
II 
Fig. 4.26 Figure showing the Liquid 
Velocity Profile in Stratified 
Flow. 
Flow Patterns 
Whenever two fluids with different physi- 
:a1 properties flow simultaneously in a 
ipe, there is a wide range of possible flow 
latterns. The flow pattern indicates the ge- 
)metric distribution of each Dhase in the L J ( 
Fig. 4 . 2 ~ Flow patterns for 20.09-centipoise pipe relative to the other phaae. Many in- 
viscosity, 0.861 -specific gravity oil, vestigators such as Mukherjee and Brill 
and water mktures in a 1.04-inch (1985) have attempted to predict the flow 
pipe based on observations of pattern that will exist for various flow 
a u k r , S d l i v a n , and wood, 1961. conditions. This is particularly important 
dent on the flow pattern. In recent studies, it was confirmed that the flow pattern is also 
as the liquid holdup is found to be depen- 
dependent on the angle of inclination of the pipe and direction of flow (e.g., production or 
injection). Consequently, some of the more reliable pressure loss correlations are depen- 
dent on the accurate prediction of a flow pattern. 
There are basically four important flow patterns. 
Bubble Flow (can be present in both upflow or downflow) 
Slug Flow (can be present in both upflow or downflow) 
Annular/Mist Flow (can be present in both upflow or downflow) 
Stratified Flow (only possible in downflow) 
4-13 
102 
t I . O @ O 0 
GAS VELOCIM NUMBER INQV) - 
Fig. 4.3 Predicted Flow Pattern Tram 
sition Lines Superimposed on the 
Kerosene in Vertical Uphill Flow. 
Observed Flow PatternMap for 
10' 
Fig. 4.4 
- - 
- 
10. - 
Predicted Flow Pattern Transition 
Lines Superimposed on the Observed 
Flow pattern Map for Kerosene in 
Uuhill SO'Flow 
Bubble flow in gashiquid two-phase flow 
is defined as the flow regime where both 
the phases are almost homogeneously 
,ol 
mixed or the gas phase travels as small 
uids followed by slugs of gas. Due to con- 
- 1 through the pipe as discrete slugs of liq- 
are longer than one pipe diameter and flow 
as the flow condition where gas bubbles 
- 5 10 The slug flow on the other hand is defined 
t bubbles in a continuous liquid medium. 
' 
tinuous segregation of phases in the direc- 
tion of flow, slug flow results in substan- 
tial pressure fluctuations in the pipe. This d- 
creates production problems, e.g., separa- 
lift valves, etc. Annular flow is defined as 
tor flooding, improper functioning of gas 
the flow pattern where the gas phase flows IO" I 10 I O 1 
as a core with the liquid flowing as an an- 
IPS 
nular film adjacent to the pipe wall. This Fig. 4.5 predicted ~l~~ patternTransition 
happens at a very high gas velocity. The 
stratified flow only occurs in two-phase 
Lines Superimposed on the Observed 
downflow. This flow pattern is character- 
Flow Pattern Map for Kerosene in 
ized by fluid stratification along the cross 
Horizontal Flow. 
section of the flow conduit or pipe. The heavier fluid flows through the bottom of the 
pipe, whereas the lighter fluid/gas occupies the upper cross section of the pipe. Figure 4.2 
presents a geometric configuration of gashiquid control volume in different flow patterns. 
In two-phase gasfliquid flow, the momentum balance equation (Eq. 4-7) depends on the 
flow pattern. The prediction of flow patterns is possible using the Mukherjee and Brill 
(1979) or Bamea et al. (1982) and Taitel et al. (1980) methods. Figures 4.3 through 4.5 
0 0 
nusslE '8 0 O 8 Ova* 
3 0 0 0 0 8L"O 
$ 8TR*TIFIED 
I d- 
GA6VELOClWNUMBERINGV)- 
4-14 
present some of the flow pattern maps for vertical upflow to horizontal flow. The flow 
pattern maps are presented with liquid and gas velocity numbers as the independent 
variables. These are defined as follows. 
Liquid Velocity Number = NLV = 1.938 V s ~ 
Gas Velocity Number = Ngv = 1.938 V, 
where, 
V s ~ = superficial liquid velocity, ft/sec, 
V,, = superficial gas velocity, ft/sec, 
pL = lbm/ft-', 
oL = surface tension of liquid, dynes/cm. 
Example Problem No. 4-1 -Application: 
Given: 
Tubing ID = 2.441 in, (2 7/8-in. tubing), APIgravity = 30" 
Surface tension of oil = 26 dynes/cm 
Oil rate = 250 stb/D GOR = 56-5 scf 
Determine the flow pattern for this vertical producing well. 
Solution 
'pecific gravity Of Oil = 131.5 + API - 131.5 +3O 141.5 - 141'5' = 0.88 
Tubing cross-section = - -- - = 0.0325 ft2 n d2 n 2.441 4 - 4 ( 12 y 
Superfkial oil velocity, VSL = 250 x 5.615 
86,400 X 0.0325 
ft/sec 
4-15 
= 0.5 ft/sec 
Liquid velocity number, NL" = 1.938 x 0.5 3- 
= 1.938 X 0.5 X 1.2055 
= 1.2 
Superficial gas velocity, V,, = 250 x 56 ft/sec 
86,400 x 0.0325 
= 5 ft/sec 
Gas velocity number = 1.938 x 5 x 1.2055 
= 11.7 
From Fig. 4.3 for NLV = 1.2 and Ngv = 11.7, the predicted flow pattern is slug flow. 
Calculation of Pressure Traverses 
A number of methods have been orooosed to calculate the oressure loss when pas and 
liquid simultaneously flow through a'pipeline. These methois provide means topredict 
flow patterns for given flow and fluid parameters such as the individual phase flow rates, 
fluid properties, pumping system dimensions, and one of the terminal pressures 
factors are calculated to determine the hydrostatic gradient and friction gradient. A 
(wellhead/separator pressure). For the predicted flow patterns, liquid holdup and friction 
detailed discussion on some of these methods is presented by Brown and Beggs (1977). 
The NODAL/GLADt and STARt/WPTt programs on the Schlumberger Wireline 
Services computer system offer the following pressure loss correlations: 
Duns and Ros (1 963) 
Orkiszewski (1967) 
Hagedorn and Brown (1965) 
Beggs and Brill (1973) 
Mukherjee and Brill (1985) 
Dukler (1964) 
0 t Mark of Schlumberger 
4-16 
The first three correlations are developed for vertical upflow or for production wells. 
Only the Beggs and Brill and Mukherjee and Brill correlations are developed for inclined 
multiphase flow, and are valid for both production and injection wells as well as for hilly 
terrain pipelines. These are also valid for horizontal single or multiphase flow. The 
Dukler correlation is only valid for horizontal flow. All these correlation programs can 
Brill program in the above list predicts the flow pattern transitions in inclined two-phase 
also be used for single-phase gas or single-phase liquid flow. Only the Mukherjee and 
flow. 
Gradient Curves 
Gradient curves are graphical presentations of pressure vs length or depth of flowline or 
tubing, respectively, for a set of fixed flow and fluid parameters. Figure 4.6 is a typical 
gradient curve for 2 7/8-in. tubing with 1000 BPD liquid production at 50% oil. The fixed 
fluid properties, such as specific gravity of gas, etc. are presented on the top right comer 
of the plot. On each gradient curve, a family of curves is presented for a number of 
gasiliquid ratios. These curves are computer generated and are used for design calcula- 
tions in the absence of a computer program. Gradient curves are used to calculate one of 
the terminal pressures when the other terminal pressure and the appropriate flow and fluid 
properties are known. 
Brown et al. (1980) presented a number of gradient curves for a wide range of tubing size 
and flow rates using the Hagedorn and Brown (1965) correlation. A few of these are 
appended for the solution of some of the problems. Figures 4.6 through 4.9 are a set of 
sample gradient curves presented by Brown. The gradient curves for the horizontal flow 
zero length or depth. To use these gradient curves for a nonatmospheric separator or 
(Fig. 4.8) and vertical flow in tubing (Figs. 4.6 and 4.7) start at atmospheric pressure at 
curves is presented in Example Problem 4-2. 
wellhead pressures, a concept of equivalent length is used. The use of these gradient 
Example Problem No. 4-2 -Application 
Given, 
Pwh = 100 psig Wellhead temperature = 70°F 
GLR = 400 scfbbl T,, = 140°F 
yg = 0.65 Depth = 5,000 ft (mid-perf.) 
Tubing ID = 2 in. API Gravity = 3 5 O A P I 
Calculate and plot the tubing intake curve. 
Solution 
A plot of the bottomhole flowing pressure vs flow rate is obtained based on pressure 
gradients in the piping. 
4-17 
1 
Fig. 4.6 Vertical Multiphase Flaw: Haw to Find the Flawing Bottomhole Pressure 
4-18 
160 PSI PRESSURE, 100 PSI ,3,360 PSI 
0 
50 
100 
200 
300 
400 
Fig. 4.7 Vertical Multiphase Flow: How to Find the Flowing Wellhead Pressure 
320, PSI PRESSURE, 100 PSI ?,80° 
6 
0 
50 
100 
200 
300 
400 
4-19 
100. PSI I PRESSURE, 100 PSI 
490 PSI 
I 
10.000 
18 
20 'I 
Fig. 4.8 Horizontal Multiphase Flow: How to Find the Flowng Wellhead 
Pressuure 
4-20 
TUBlNQ SIZE - 2.431 IN. ID 
AVERAGE FLOWING TEMPERATURE. 1ZO'F 
WATER SPECIFIC GRAVITY - 1.07 
00 
2,000 
0 
3,000 
4,000 
5,000 
6,000 
7,000 
8.000 
Fig. 4.9 Vertical Water Injection: How to Find Discharge Pressure 
Using the vertical multiphase flow correlations in Figs. 4.10 through 4.13, assume various 
flow rates and determine the tubing intake pressure P,f. Construct a table as follows. 
4-21 
200 730 
400 800 
600 910 
800 1,080 
Sample Calculation 
Using Fig. 4.10, start at the top of the gradient curve at a pressure of 100 psig. Proceed 
vertically downward to a gas liquid ratio of 400 scfbbl. Proceed horizontally from this 
point and read an equivalent depth of 1,600 ft. Add the equivalent depth to the depth of 
the well at mid-perforation. Calculate a depth of 6,600 ft. on the vertical axis, and proceed 
horizontally to the 400 scfbbl GLR curve. From this point, proceed vertically upward and 
read a tubing intake pressure for 200 BPD of 730 psig. 
Repeat this procedure for flow rates of 400, 600, and 800 BPD using Figs. 4.11 through 
4.13, respectively. 
Plot the PwF vs q values tabulated above as shown in Fig. 4.14 to complete the desired 
tubing intake curve. 
4-22 
PRESSURE in 100 PSlG 
0 4 8 12 18 20 24 28 
1 
Gas Specific Gravity 
2 Average Flowing Temp. 
3 
4 t 
E! 
E 5 .- 
7 
F 
3 
$? 
6 
7 
0 
8 38 
80 
9 
10 
Fig. 4.10 Vertical Flowing Pressure Gradients 
4-23 
PRESSURE in 100 PSIG 
0 4 0 12 16 20 24 28 
1 
Gas Specific Gravity 
2 
3 
ti 
E 
: 
.- E 5 
F 
5 
4 
0 
0 
6 
7 
0 
8 $8 
s, 
9 
10 
Fig. 4.11 This figure was used to determine P w f = 800 PSIG for a rate of 400 BPD 
through 2 inch ID tubing 
4-24 
7 
8 
9 
10 
Fig. 4.12 This figure was used to determineP w f = 910 PSIG for a flow rate of 600 BPD 
through 2 inch tubing. 
4-25 
Li 
F! 
w 
Y 
0 
0 
Fig. 4.13 This figure was used to determine Purfs - 1080 PSIG for a flow rate of 800 BPD 
through 2 inch ID tubing. 
4-26 
Fig. 4.14 
RATE (b/d) 
wellhead pressure of100 PSIG. 
This figure shows a tubing intake or outflow performance curve for a 
4-27 
REFERENCES 
1. Colebrook, C.F.: “Turbulent Flow in Pipes with Particular Reference to the 
Transition Region Between the Smooth and Rough Pipe Laws,’’ J. Inst. Civil Engrs. 
(London), 11, (1938-1939). 133-156. 
2. Moody, L.F.: “Friction Factors for Pipe Flow,”Trans. ASME 66, (Nov. 1944), 671- 
684. 
3. Jain, A.K.:“Accurate Explicit Equation for Friction Factor,” J.Hydraulics Div., 
ASC, 102 (HY5), (1976), 674. 
4. Aziz, K.: “Ways to Calculate Gas Flow and Static Head,” Handbook Reprint from 
Pet. Eng. (1963), Dallas. TX. 
5. Cullender, M.H. and Smith, R.V.: “Practical Solution for Gas-Flow Equations for 
Wells and Pipelines with Large Temperature Gradients,” Trans., AIME, 207,281- 
287. 
6. Dukler, A.E., et al.: “Gas-Liquid Flow in Pipelines,” I. Research Results, AGA-API 
Project NX-28, (May 1969). 
7. Duns, H., Jr. and Ros, N.C.J.: “Vertical Flow of’Gas and Liquid Mixtures in Wells,” 
Proc. Sixth World Pet. Congress, (1963), 451. 
8. Mukherjee, H. and Brill, J.P.: “Empirical Models to Predict Flow-Patterns in Two- 
Phase Inclined Flow,” Int. J. Multiphase Flow, in press. 
9. Bamea, D., Shoham, 0. and Taitel, Y. “Flow Pattern Transition for Downward 
Inclined Two Phase Flow; Horizontal to Vertical,” Chem. Engr. Sci. (1982) 37, 
735-740. 
10. Taitel, Y., Bamea, D. and Dukler, A.E.: “Modeling Flow Pattern Transitions for 
Steady Upward Gas-Liquid Flow in Vertical Tubes,” AIChE J. (1980) 2,3,345-354. 
11. Brown, K.E. and Beggs, H.D.: “The Technology of Artificial Lift Methods,” 
PennWell Publishing, (1977). 1. 
12. Orkiszewski, J.: “Predicting Two-Phase Pressure Drops in Vertical Pipes.” JFT, 
(June 1967), 829-838. 
13. Hagedorn, A.R. and Brown, K.E.: “Experimental Study of Pressure Gradients 
Occurring During Continuous Two-Phase Flow in Small Diameter Vertical 
Conduits,” JPT, (April 1965), 475-484. 
14. Beggs, H.D. and Brill, J.P.: “A Study of Two-Phase Flow in Inclined Pipes,” JPT, 
(May 1973), 607-617. 
15. Mukherjee, H. and Brill, J.P.: “Liquid Holdup Correlations for Inclined Two-Phase 
Flow,” JPT, (May 1983). 
4-28 
16. Dukler, A.E., Wicks, M., In and Cleveland, R.G.: “Frictional Pressure Drop in 
Two-Phase mow: B. An Approach Through Similarity Analysis,” AIChE J. (1964) 
10, 1. 
17. Brown, K.E.: “The Technology of Artificial Lift Methods,” PennWell Publishing, 
(1980), 3. 
4-29 
CHAPTER 5 - WELL PERFORMANCE 
EVALUATION OF STIMULATED 
WELLS 
An effective way to evaluate 
stimulation or to compare dif- 
- 20.- ferent stimulation designs is by 
30- 
stimulation over time. If a par- 
comparing net payout due to 
in five months (whereas an al- 
yields a net revenue of x dollars 
out the cost of stimulation and 
ticular stimulation design pays 
months), the first design un- 
temative design does it in 10 
able or sellable design. Fig. 5.1 
doubtedly is the most accept- 
vs time. 
is an example plot of net payout 
COMPETITORS 
DESIQN 
-50 
TIME" 
Fig. 5.1 Net Payout at any time = Extra revenuejiom oil or gas 
production due to stimulation at any rime, t - cost of 
stimulation. 
0 ARTIFICIAL LIFT pwh PRESSURE 
I R 
Artificial lift methods are used in oil wells 
that have adequate productivity but inade- 
quate pressure to lift the oil to the surface. 
There are basically two methods of artifi- 
cial lift. E 
Pumping 
Gas Lift PRESSURE 
w 
CI 
DISCHARGE 
Pumping Wells PRESSURE SUCTION GRADIENT 
DEAD OIL 
Downhole pumps add pressure to the 
flowing system. As shown in Fig. 5.1, the PNI pr 
dead oil column is stagnant and the 
comes the reservoir pressure stopping the 
fixed pressure gain between the suction and discharge sides of the pump. When properly 
inflow into the wellbore. Installation of a pump modifies the pressure profile by adding a 
designed, this pressure gain allows the fluid to flow to the surface at a fixed wellhead 
pressure. Pumps always operate with a positive suction pressure provided by a fluid 
column in the annulus above the pump level. This fluid level in the annulus can be 
monitored by an echometer. Before stimulating a pumping well, the fluid level in the 
annulus should be monitored to make the post-stimulation troubleshooting possible. 
- 
hydrostatic pressure of the column over- Fig. 5.2 Effect Of SUbsUrfaCe Pumps Of well 
pressure profile. 
5-1 
Diagnostic of potential stimulation 
P, needs in pumped oil wells - - 
In general, if the fluid level rises 
and the pump discharge rate falls, 
the problem is in the pump, (Case 1 
of Fig. 5.3). It is not uncommon to 
encounter such problems after t 
stimulation of a pumping well. In 3 
% most of these cases, the old pump 3 needs to be replaced or repaired. 
The other common problem is 
when the flow rate falls and the 
fluid level stays the same or re- 
cedes. This is commonly due to a , 
reservoir problem, such as deple- 
tion or skin buildup (Case 2 of q(STPD) -c 
Fig. 5.3). Fig. 5.3 Showing potenrialproblems in a pumping well 
Note also that in a uumuinrr well 
IPR WITH HIGHER SKIN 
ORIGINAL IPR CURVE 
q, q, 
through IPR curves. 
after a successful stimulaiion: the pumps may need to be redesigned for optimum flow. It 
is very possible that after a successful stimulation in a pumping well, the post stimulation 
production did not increase substantially due to existing pump limitations. 
Gas Lift Wells 
Gas lift is an artificial lift method where gas is injected to the liquid production string, 
normally through the tubing-casing annulus to aerate the liquid column, reducing the 
hydrostatic head of the liquid column. This reduces the bottomhole flowing pressure 
increasing production. The deeper the injection point, the longer the column of tubing 
fluid is aerated and the lower the bottomhole pressure. Thus, the objective of gas lift is to 
inject the optimum gas at the deepest possible point in the tubing. An optimum gas 
volume injection is very important because any higher volume leads to an excessive 
friction pressure loss in the tubing, thus overcoming the hydrostatic pressure gain. This 
situation results in an increase in the bottomhole flowing pressure, reducing production. 
Figure 5.4 shows a typical gas injection sequence used to unload or kick off a gas lift 
well. Gas lift valves are used to close and open at fixed casing or tubing pressures. The 
objective of unloading is to start aerating a fluid column in smaller lengths beginning at 
the top and then close the top valve to aerate through the second valve, and so on until the 
injection valve is reached. This valve is set in such a way that it remains open all the time. 
This stepwise unloading is done to kick off a well with limited surface injection pressure. 
5-2 
L 
PRESSURE (100 PSI) -c 
0 4 8 12 16 20 24 28 
1 
2 
3 
4 
5 
6 
7 
0 
9 . . - D. 
Fig. 5.4 Unloading wells with Gas L$t 
Effect of Stimulation of Gas Lift Wells 
After stimulation with the improved IPR curve, a redesign of the gas lift system is 
possible that after stimulation a gas lift well loses production due to gas lift design 
normally required for optimized flow. This requires new setting of gas lift valves. It is 
problems. This section is to caution DS operations people against such gas lift system 
failures in a successfully stimulated well. 
5-3 
Example Problem 5.1 - Clay Consolidation (Clay Acid) 
(Effect of moving damage away fromthe wellbore) 
I 
I 
\ 
\ 
Fig. 5.5 
log - re rw 
1 1 re Average Permeability, F = %+- 1ogk-E- log- -G logrw kd rx ko rc 
Percentage of original Permeability = E X 100 
Given: 
r, = 0.365 ft 
Formation Permeability, k = 100 md 
Spacing = 160 acres 
(a) Calculate the percentage of original productivity due to 80% damage I ft 
deep around the wellbore. 
(b) Calculate the percentage of original productivity due to an 80% damage 
collar, 1 ft wide and 4 ft from the wellbore. 
5-4 
Solution: 
- 3.6106 - 0.0286 + 0.0304 
= 61.2md 
:. Percentage of original productivity = 61% 
1,489 
log (m) 
(b) k80% = 1 4.365 1 5.365 1 
100 lOg0.365 + B log4.365+100 log5.365 
1,489 
- 3.6106 - 0.01078 + 0.00448 + 0.02443 
= 91 md 
;. Percentage of original productivity = 91 % 
Example Problem 5.2 - Pre- and Post-acid Evaluation 
Summary 
An offshore Louisiana well was tested following its completion in the Pliocene formation. 
It produced 1,200 BPD at a wellhead pressure of 1,632 psig from a 71 ft. gravel-packed 
unconsolidated sandstone reservoir. 
Analysis of the test data identified severe wellbore damage which was restricting produc- 
BOPD at the same wellhead pressure should that damage be removed. 
tion (Skin = 210). It also showed that the production rate could be increased to 6,850 
To treat the damage effectively, a clear understanding of its origin is needed. The analysis 
of the test data indicated inadequate perforations and a high probability of formation 
damage. This was confirmed by core analysis and production logs run after the test. An 
acid treatment was formulated and the post-acid test indicated a significant improvement 
in skin (Skin = 15). The production rate increased to 4,400 BOPD at a wellhead pressure 
of 2,060 psig.' 
Pre-Acid Test Results 
The main results are summarized on page 1 of the referred paper.1 The test procedure and 
analysis plots are given in pages 2 through 5. The Model Verified Interpretation (page 3) 
1 For more details refer to SPE 14820 presented at the 1986 SPE Symposium on Formation Damage 
Control, Lafayette, LA, February 26-27. 1986 
indicates a high permeability homogeneous reservoir with wellbore storage and severe 
skin effect. The Nodal analysis (page 4) shows that the production rate is significantly 
restricted by the skin effect, and projects a rate increase of 5,650 BOPD if the wellbore 
damage is removed. Finally, the shot density sensitivity plot (page 5) suggests adequate 
perforations and the likelihood of formation damage. The interpretation charts and 
computation sheets are presented. 
Production Logs Results 
The production logging data indicate that all of the 40 ft. perforated zone is contributing 
to the flow rate except the bottom 5 to 6 ft. Since the permeability variation in the 
perforated interval is minimal and the flow profile appears nonuniform, it is assumed that 
formation damage has affected the producing zone unevenly. 
Post-Acid Test Results 
production rate matches the prediction of the Nodal analysis. The charts and computation 
Significant improvement in the wellbore condition is noticed. The resulting increase in 
sheets are presented in this section. 
PRE-ACII 
NODAL 
Test Identification 
Test Type ...................................... SPRO 
Test No. ........................................ 1 
Formation..... ................................. 5 3 SAND 
Test Interval (ft) ............................ 11942-11982 
Completion Configuration 
Total Depth (MD/l'VD) (ft) .......... 11,920/10,800 
Casingniner ID (in.) ..................... 6.094 
Hole Size (in.) ................................ 8.5 
Perforated Interval (ft) .................. 40 
Shot Density (shots/ft) .................. 12 
Perforation Diameter (in.) ............. 0.610 
Net pay (ft) .................................... 71 
Interpretation Results 
Model of Behavior ........................ Homogeneous 
Fluid Type used for Analysis ........ Liquid 
Reservoir Pressure (psi) ................ 5585 
Transmissibility (md-ft/cp) ........... 53390 
Effective Permeability (md) .......... 526.0 
Skin Factor. .................................... 210.0 
MAXIMUM PRODUCTION R 
INALYSIS 
YALYSIS 
Test String Configuration 
Tubing Vertical Multiphase 
Flow ......................................... Hagedom-Brown 
Tubing Length (ft)/ID (in.) ........... 11.830/2.992 
Packer Depth (ft) ........................... 11,826 
Gauge Depth (ft)flype .................. 11,92O/DPTT 
Tubing Absolute Roughness (ft) .. 5.0E-05 
RockiFluld/Wellbore Properties 
Oil Density (" AFT) ........................ 29.5 
Gas gravity .................................... 0.600 
GOR (scf/STB) ............................. 628 
Water Cut (a) ............................... 0 
Viscosity (cp) ................................. 0.70 
Total Compressibility (Vpsi) ........ 9.00E-06 
Reservoir Temperature (OF) 218 
Porosity (%) 28 
Form. Vol. Factor @bl/STB) ......... 1.37 
Bubble Point Pressure, psi ............ 5120 
Wellhead Pressure (psig) .............. 1632 
Wellhead Temp. (OF) ..................... 100.0 
Production Time (days) ................. 3.0 
TE DURING TEST: 1,200 BPD 
.................................. 
.......... 
5-6 
Test Objectives 
The objectives of this test were to evaluate the completion efficiency and estimate the 
production potential of the well. 
Comments 
The test procedure and measurements are summarized on the following pages. The 
well and reservoir parameters listed above reveal a high permeability formation and a 
system behaved as a well in a homogeneous reservoir with wellbore storage and skin. The 
severely damaged wellbore. Removing this damage would result in increasing the 
jeopardizing the integrity of the gravel pack. The shot density sensitivity plot suggests 
production rate to 6,850 BOPD at the same wellhead pressure of 1,632 psig, without 
adequate perforations and high formation damage. This could be confirmed by production 
logs and core analysis. Acid treatment is recommended for removing the wellbore 
damage and increasing the production. Note that the skin due to partial penetration cannot 
be eliminated by acidizing - consequently the ideal production rate may not be achieved. 
PRE-ACID TEST COMPUTATION SHEET 
1. LOG-LOG ANALYSIS 
1.1 Match Parameters 
Model: Homogeneous, WBS & S CDe*S = 1.0E185 
Pressure Match P d M = 0.23 
Time Match: (Ta/CD)/At = 1,700 
1.2 Reservoir Parameter Calculations 
I 
= 0.0093 bbllpsi 
match 
= 370.1 
= 210 
5-7 
2. GENERALIZED HORNER ANALYSIS 
2.1 Straight Line Parameters 
Superposition slope: m' = 4.1112E-03 
P (intercept): P* = 5,585 psia 
Pressure at one hour: P (1 hr) = 5,575 psia 
Pressure at time zero: P (0) = 4,622 psia 
2.2 Reservoir Parameter Calculations 
kh = 162'6 m' Bo = 37,929 md-ft 
Nomenclature 
permeability, md 
formation height, ft 
wellbore storage constant, bbl/psi 
scientific notation 
oil flow rate BPD 
dimensionless pressure 
pressure change, psi 
dimensionless time 
dimensionless wellbore storage constant 
time change, hr 
oil formation volume factor, bbl/STB 
oil viscosity, cp 
formation porosity 
5-8 
4000 
0.00 2.00 4.00 6.00 8.00 10.00 12.00 
ELAPSED TIME lHOURSI 
L 
Fig. 5.6 Pressure IFIawrare His&ory 
SEQUENCE OF EVENTS 
EVENT TIME DATE WHP BHP ELAPSED DESCRIPTION 
NO. (HR:W TIME (PSIA) (PSIA) 
(HRMIN) 
I 1 I 23-APRI 1228 I Run in Hole Flowing I 048 I 1613.0 1 1636.0 I 
2 
2434.0 5579.0 945 End Shut-In, POOH 21:25 23-APR 4 
1648.0 4623.0 428 End Flow & Start Shut-In 1608 23-APR 3 
1649.0 4621.0 400 Start Monitoring Flow 1540 23-APR 
SUMMARY OF now PERIODS 
PERIOD 
(WCHES) 
CHOKE SIZE FLOWRATE PRESSURE (PSIA) DURATION 
(HR:MIN) 
START STOP (MMSCFD) (BD) 
GAS on 
#1. DD 
- 0 0 5579.0 4623.0 5:17 #2, BU 
0164 0.754 1200.0 4623.0 1613.0 3:40 
5-9I O 0 10' 102 1 o3 1 o4 
DIMENSIONLESS TIME, TD/CD 
Fig. 5.7 Diagnostic Plot 
.. - -7 . . . .. 5400 -- a 
0 UI 
0 
3 8 51 00 -- 
w a n 
" 
a 
" 
4800 -- 0 Data for #2. BU " 
Slopern'"4.11121E-03 
P (intercept) = 5584.5 
4500 
0 
I P 
1500 3000 4500 6000 7500 
SUPERPOSITION TIME FUNCTION 
Fig. 5.8 Dimensionless Superposition 
5-10 
PRE-ACID TEST 
Buildup Data 
Delta time (hours) Bonornhole 
Pressure (psial 1 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
0.00000E+00 
1.50000E-03 
2.83333E-03 
4.16667E-03 
5.66667E-03 
7.00000E-03 
6.33333E-03 
9.83333E-03 
1.1 1667E-02 
1.25000E-02 
1.40000E-02 
1.53333E-02 
1.66667E-02 
1.81667E-02 
1.95000E-02 
2.08333E-02 
2.23333E-02 
2.36667E-02 
2.5OOOOE-02 
2.65000E-02 
2.78333E-02 
2.91667E-02 
3.06867E-02 
3.20WOE-02 
3.33333E-02 
3.48333E-02 
3.61 667E-02 
3.75000E-02 
3.90000E-02 
4.03333E-02 
4.16667E-02 
4622.6 
4624.3 
4635.5 
4647.4 
4656.5 
4664.6 
4672.7 
4681 .O 
4689.4 
4697.6 
4705.9 
4714.0 
4722.1 
4730.3 
4738.4 
4746.4 
4754.5 
4762.6 
4770.8 
4778.7 
4786.8 
4794.4 
4802.4 
4810.1 
4817.9 
4825.7 
4833.4 
4841.2 
4648.9 
4856.5 
4864.1 
32 4.31667E-02 4871.6 
Delta time (hours) Bonornhole 
Pressure (psis) 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
40 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 
61 
62 
63 
4.45000E-02 
4.58333E-02 
4.73333E-02 
4.86667E-02 
5.00000E-02 
5.56667E-02 
6.1 1667E-02 
6.66667E-02 
7.23334E-02 
7.78333E-02 
8.33334E-02 
8.9OOOOE-02 
9.45000E-02 
0.10000 
0.10567 
0.11117 
0.11667 
0.12233 
0.13333 
0.1 5000 
0.16667 
0.18333 
0.20000 
0.21687 
0.23333 
0.25000 
0.26667 
0.28333 
0.30000 
0.31667 
0.32783 
4879.3 
4866.8 
4694.3 
4901.6 
4909.1 
4938.5 
4967.3 
4995.6 
5023.2 
5050.2 
5076.6 
5102.4 
5127.6 
5152.0 
5175.7 
5196.6 
5220.8 
5242.4 
5283.0 
5338.3 
5386.8 
5427.9 
5462.8 
5491.8 
5515.2 
5534.0 
5548.4 
5559.5 
5567.5 
5573.1 
5576.0 
64 0.37783 5581.7 
5-1 1 
Delta time (hours) Bonomhole 
Pressure (psia) 
65 
66 
67 
68 
69 
70 
71 
72 
73 
74 
75 
76 
77 
78 
79 
80 
81 
62 
83 
84 
85 
66 
87 
88 
89 
90 
91 
92 
93 
94 
95 
96 
97 
0.42763 
0.47783 
0.52783 
0.57783 
0.62763 
0.69450 
0.74450 
0.79450 
0.84450 
0.89450 
0.94450 
0.99450 
1.0445 
1.0945 
1.1445 
1.1945 
1.2445 
1.2945 
1.3445 
1.3945 
1 A 4 5 
1.4945 
1.5445 
1.5945 
1.6445 
1 .e945 
1.7445 
1.7945 
1 a445 
1.6945 
1.9445 
1.9945 
2.0445 
5582.3 
5580.6 
5578.2 
5576.1 
5574.0 
5573.8 
5574.1 
5574.4 
5574.5 
5574.6 
5574.9 
5574.9 
5575.1 
5575.2 
5575.3 
5575.5 
5575.5 
5575.7 
5575.7 
5575.9 
5575.9 
5576.0 
5576.1 
5576.1 
5576.2 
5576.2 
5576.2 
5576.4 
5576.4 
5576.5 
5576.4 
5576.5 
5576.6 
98 2.0945 5576.7 
Delta time (hours) Bonomhole 
Pressure (psia) 
99 
100 
101 
102 
103 
104 
105 
106 
107 
106 
109 
110 
111 
112 
113 
114 
115 
116 
117 
118 
119 
120 
121 
122 
123 
1 24 
125 
126 
127 
128 
129 
130 
2.2612 
2.3445 
2.51 12 
2.6778 
2.8445 
3.01 12 
3.1778 
3.3445 
3.4278 
3.8612 
3.6945 
3.9278 
4.0945 
4.261 2 
4.4278 
4.5945 
4.7612 
4.9278 
5.0945 
5.1333 
5.1362 
5.1390 
5.1417 
5.1473 
5.1500 
5.1526 
5.1557 
5.1583 
5.1612 
5.1945 
5.2278 
5.2812 
5576.6 
5576.9 
5577.0 
5577.2 
5577.4 
5577.5 
5577.7 
5577.8 
5577.9 
5577.9 
5578.0 
5576.2 
5578.3 
5576.5 
5578.5 
5578.6 
5578.7 
5576.7 
5578.9 
5576.9 
5576.9 
5579.0 
5579.0 
5578.9 
5579.0 
5579.0 
5579.0 
5579.0 
5579.0 
5576.9 
5578.9 
5579.0 
131 5.2778 5579.0 
5-12 
NODAL PLOT 
" 
" 
" 
" 
3800 t t 
Fig. 5.9 Production Potential Evaluation, Nodal Plot 
RATE vs. WELLHEAD PRESSURE 
450 826 1200 1575 IS50 
WELLHEAD PRESSURE (psi) 
Fig. 5.10 Production Potenrial Evaluation, Rate vs. Wellhead Pressure, 
5-13 
WELL PERFORMANCE RATE vs. SHOT DENSITY 
7000 I I 
I 0 
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 
PERFORATION DENSITY (shotslft) 
Fig. 5.11 Production Potential Evaluation, Well Performance Rate vs. Shot Density 
POST-ACID ANALYSIS 
NODAL ANALYSIS 
Test Identification Test String Configuration 
Test Type ...................................... SPRO 
Test No. ........................................ 2 
Tubing Length (fi)/I.D. (in) .......... 11,830/2.992 
Formation E 3 S A N D 
.......................... Packer Depth (ft) 11,826 
Test Interval (ft) ............................ 11942-11982 Downhole Valve (Y/N)nype ....... N 
...................................... Gauge Depth (ft)nype .................. 11,920iDPTT 
Completion Configuration Test Condition 
Total Depth (MDITVD) (ft) .......... 11,920/10.800 
Casingniner LD. (in) .................... 6.094 Tubinwellhead Pressure (psi) .... 2.060 
Wellhead Temperature (OF) ........... 100.0 Hole Size (in) ................................. 8.5 
Separator Pressure (psi) ................ 150 
Perforated Interval (ft) .................. 40 
Shot Density (shots/ft) .................. 12 
Perforation Diameter (in) ............... 0.610 
Net pay (ft) .................................... 71 
Interpretation Results 
RocWFluidWellbore Properties 
Oil Density (" API) ........................ 29.5 
Gas gravity .................................... 0.600 
GOR (scf/STB) .............................. 1,013 
Water Cut (%) ............................... 0 
Model of Behavior ........................ Homo geneous 
Total Compressibility (l/psi) ........ 9.00E-06 Fluid Type used for Analysis ........ Liquid 
viscosity (CP) ................................. 0.70 . . 
Reservoir Pressure (psi) ................ 543 1 
Transmissibility (md ft/cp) ........... 53751 Porosity (%) .................................. 28 
Form. Vol. Factor (bbl/STB) ......... 1.37 Effective Permeability (md) ......... 530 
Reservoir Temperature (OF) .......... 218 
................................ Skin Factor.... 15 I Production Time (days) ................. 2.5 
MAXIMUM PRODUCTION RATE DURING TEST: 4,398 BPD 
5-14 
Test Objectives 
The objective of the test was to evaluate the effectiveness of the acid stimulation 
treatment. 
Comment: 
The test procedure and measurements are summarized on the next page. The acid 
treatment was effective in removing the formation damage. Analysis of the data revealed 
a significant improvement in the wellbore condition resulting in over a 3,000 BOPD 
increase in production at 428 psi higher wellhead pressure. 
POST-ACID TEST COMPUTATION SHEET 
1. LOG-LOG ANALYSIS 
1.1 Match Parameters 
Model: Homogeneous, W B S & S CDe2S = 1.OE16 
Pressure Match: PdAF' = 0.06318 
Time Match (TdCD)/At = 1.300 
1.2 Reservoir Parameters Calculations 
kh = 141.2Q p 3 2 
' O (.P )match = 37,626 md-ft 
= 0.122 bbl/psi 
match 
= 486 
2. GENERALIZED HORNER ANALYSIS 
2.1 Straight Line Parameters 
Superposition slope: m' = 4.14328 E-03 
P (intercept): P* = 5,430 psia 
Pressure at one hour: P (1 hr) = 5,401 psia 
5-15 
Pressure at time zero: P (0) = 5,041 psia 
2.2 Reservoir Parameter Calculations 
kh = 162'6 m' Bo = 37,635 md-ft 
(' hr)-P 
('))-log ( k )+ 3.2,) = 15 4) Po Ct 'w 
Nomenclature 
k = permeability, md 
h = formation height, ft 
C = wellbore storage constant, bbl/psi 
E = scientific notation 
Q, = oil flow rate, BPD 
PI, = dimensionless pressure 
AP = pressure change, psi 
To = dimensionless time 
CD = dimensionless wellbore storage constant 
At = time change, hr 
Bo = oil formation volume factor, bbl/STB 
po = oil viscosity, cp 
9 = formation porosity 
5-16 
PRESSURVFLOWRATE HISTORY 
"" 
5300 
4850 
0.00 1.50 3.00 4.50 6.00 7.50 8.00 
ELAPSED TIME (HOURS) 
Fig. 5.12PressurelFlowrate History 
SEQUENCE OF EVENTS 
SUMMARY OF FL.0W PERIODS 
5-17 
DIAGNOSTIC PLOT 
X Derivalive Data. t 4 . Eu 
Pressure Match - 8.3181E-02 
Time Match - 1300 
10’’ IO’ 1 o2 1 o3 1 o4 
DIMENSIONLESS TIME, TD/CD 
Fig. 5.13 Posr-Acid Test Validation, Diagnosric Plot 
e DIMENSIONLESS SUPERPOSITION 
5470 j 
I I I 
I 
I 
SUPERPOSITION TIME FUNCTION 
Fig. 5.14 Post-Acid Tesr Validation, Dimensionless Superposition 
5-18 
POST-ACID TEST 
Buildup Data 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
0.00000E+00 
1.33338E-03 
2.83330E-03 
4.16667E-03 
5.50003E-03 
6.99997E-03 
8.33333E-03 
9.66670E-03 
1.1 1666E-02 
1.25000E-02 
1.38334E-02 
1.53333E-02 
1.66667E-02 
1.60000E-02 
1.95000E-02 
2.08333E-02 
2.21667E-02 
2.36666E-02 
2.50000E-02 
2.63334E-02 
2.78333E-02 
2.91667E-02 
3.05000E-02 
3.20000E-02 
3.33333E-02 
3.46667E-02 
3.61666E-02 
5040.6 
5040.7 
5040.7 
5040.8 
5040.8 
5040.8 
5041.9 
5049.3 
5058.2 
5067.5 
5076.5 
5065.5 
5099.5 
5122.5 
5144.3 
5066.5 
5184.7 
5203.2 
5220.2 
5236.1 
5250.6 
5264.0 
5276.3 
5267.4 
5297.4 
5306.4 
5314.4 
Delta time (hours) Bonomhole 
Pressure (psia) 
28 3.75000E-02 5321.5 
Deita time (hours) Bonomhole 
Pressure (psiar 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
46 
49 
50 
51 
52 
53 
54 
55 
3.88334E-02 
4.03333E-02 
4.1 6667E-02 
4.56333E-02 
5.00000E-02 
5.41667E-02 
5.83333E-02 
6.25000E-02 
6.66667E-02 
7.08333E-02 
7.50000E-02 
7.91667E-02 
8.33333E-02 
8.75000E-02 
9.16667E-02 
9.58333E-02 
0.10000 
0.10417 
0.1 0833 
0.11250 
0.11667 
0.12083 
0.1 2500 
0.12917 
0.1 3333 
0.13750 
0.14167 
5327.7 
5333.3 
5338.1 
5348.6 
5356.2 
5361.1 
5364.7 
5367.5 
5369.7 
5371.4 
5372.9 
5374.1 
5375.0 
5376.0 
5376.8 
5165.2 
5378.2 
5378.8 
5379.5 
5380.1 
5380.6 
5381.1 
5381.5 
5382.0 
5382.5 
5382.9 
5383.3 
56 0.14583 5363.8 
5-19 
Della time (hours) Bonomhole 
Pressure (psia] 
57 
58 
59 
60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
70 
71 
72 
73 
74 
75 
76 
77 
78 
79 
80 
81 
82 
83 
84 
65 
86 
0.15000 
0.15417 
0.15967 
0.16800 
0.17633 
0.18487 
0.19300 
0.20133 
0.20967 
0.21800 
0.22633 
0.23467 
0.24300 
0.25133 
0.25967 
0.26800 
0.27633 
0.28467 
0.29300 
0.30133 
0.30967 
0.31800 
0.32633 
0.33467 
0.34300 
0.35133 
0.35967 
0.36800 
0.37633 
0.38467 
5383.9 
5384.2 
5384.6 
5385.2 
5386.9 
5386.3 
5386.9 
5387.3 
5387.6 
5388.0 
5386.4 
5368.8 
5389.0 
5389.4 
5389.8 
5390.0 
5390.4 
5390.6 
5390.8 
5391.1 
5391.4 
5391.8 
5391.9 
5392.2 
5392.4 
5392.5 
5392.8 
5392.9 
5393.2 
5393.2 
87 0.39300 5393.5 
Delta time (hours) Bonomhole 
Pressure (psia 
66 
89 
90 
91 
92 
93 
94 
95 
96 
97 
98 
99 
100 
101 
102 
103 
104 
105 
106 
107 
108 
109 
110 
111 
112 
113 
114 
115 
116 
117 
0.401 33 
0.40967 
0.41 800 
0.42633 
0.43467 
0.44300 
0.45133 
0.45967 
0.48467 
0.50967 
0.53467 
0.55967 
0.58467 
0.60967 
0.63467 
0.65967 
0.70967 
0.75967 
0.80967 
0.85967 
0.90967 
0.95667 
1.0097 
1.0597 
1.1097 
1.1597 
2.1 638 
2.1 763 
2.1 668 
2.2013 
5393.6 
5393.8 
5393.9 
5394.2 
5394.3 
5394.5 
5394.8 
5394.8 
5394.9 
5395.4 
5395.5 
5395.9 
5396.4 
5396.5 
5397.2 
5397.4 
5398.0 
5398.7 
5399.3 
5399.5 
5400.0 
5400.5 
5401.0 
5401.2 
5401.6 
5402.0 
5406.1 
5406.3 
5406.2 
5406.3 
118 2.2138 5406.3 
5-20 
Delta time (hours) Bonomhoie 
Pressure (psia) 
119 2.2263 5406.4 
120 2.2368 5406.4 
121 2.3430 5406.6 
122 2.5055 5409.0 
123 2.5097 5409.0 
124 2.5138 5409.0 
125 2.5638 5409.1 
126 2.6138 5409.3 
127 2.6638 5409.4 
128 2.7138 5409.3 
129 2.7638 5409.5 
130 2.8136 5409.9 
131 2.8638 5409.8 
Delta time (hours) Bonomhoie 
Pressure (psial 
132 2.9138 5409.8 
133 2.8638 5410.2 
134 30.138 5410.0 
135 3.0638 5410.3 
136 3.1138 5410.2 
137 3.1638 5410.4 
138 3.2138 5410.8 
139 3.2638 5410.8 
140 3.3138 5410.9 
141 3.3638 5410.9 
142. 3.4138 5411.1 
143 3.4638 541 1 .O 
NODAL PLOT 
5500 i I I I I I I 1 
4000 I I 1 I I I 1 
0 907 1815 2722 3630 4537 5444 €352 
PRODUCTION RATE (STBID) 
Fig. S.1S Post-Acid Production Evaluation, Nodal Plot 
5-21 
RATE vs. WELLHEAD PRESSURE 
80OOA t 
7000- 
$ 6000" 
2 5000" 
In .- 
2 
8 F 4000- 
0 
2 
@ 3000" 
$ 9 2000" 
1000" 
0, I- 
900 1350 1800 2250 2700 
WELLHEAD PRESSURE (PSI) 
Fig. 5.16 Post-AcidProduction Evaluation. Rate vs. Wellhead Pressure 
Example Problem 5-3: Producing Well 
Using tubing data from Example Problem 4-2 and reservoir parameters from Example 
Problem 2-3(b) (s = - 3 , calculate the natural production of the well. 
k = 5md 
h = 20ft 
p.0 = 1.1 cp 
- pr =2,500 psig 
s = -5 
Bo = 1.2 RB/STB 
Spacing = 80 acres r,., = 0.365 ft 
Solution: 
Drainage radius, re = = 1,053 ft 
AOFP = q = 7.08 X 10-3 kh p; 
F~ Bo[ In (: )- 0.75 + s ] 
5-22 
- - 7.08 x 10-3 x 5 x 20 x 2,500 
1.1 x 1 . 2 [ h ( g ) - - 0 . 7 5 - 5 ] 
= 604 STBD 
From example 4-2, the following tubing intake pressures are calculated for different flow 
rates: 
400 
600 
800 
1 .OS0 800 
910 
"0 200 400 600 800 1000 
J 
FLOW RATE (STBID) 
Fig. 5.17 Example Problem 5.3. IPR and Tubing Intake Curve 
curve and the IPR curve gives the natural production of the well, i.e., 410 STB/D. 
These values are plotted on the figure shown above. The intersection of the tubing intake 
5-23 
Example Problem 5-4 
Solve Example Problem 5-3 for varying rw, Le., rw = 100 ft, 200 ft, 400 ft, and 800 ft. 
Make a plot of q vs r,. (Use a skin factor of +2.) 
Solution: 
The tubing intake curve is plotted as shown in Example Problem 5-3 with the following 
points: 
Using data from Example Problem 5-3, the value of production rate q is calculated for 
different values of rw and plotted 
(i) rw = 100 ft 
AOFP = 9 = 7.08 X 10-3 kh pr 
1 
- - 7.08 x 10-3 x 5 x 20 x 2,500 
l . l x l . 2 [ h ( ~ ) - o . 7 5 + 2 ] 
= 372STBD 
Similarly, the flow rates at other values of rw are calculated and plotted 
200 
879 800 
605 400 
46 1 
5-24 
00 
FLOW RATE, (STD/D) 
Fig. 5.18 PIOI of tubing intake vsproduction ratesfor dilferent rw. 
From the above plot, production rate is read-off at the intersection Of the tubing intake 
curves and the IPR curves for the different values of effective wellbore radius. These are 
tabulated and plotted - 
"0 200 400 600 800 
EFFECTIVE WELLBORE RADIUS, IW H 
Fig. 5.19 Plot offow rate vs effective wellbore 
radius. 
Note: Hydraulically induced fractures increase the effective wellbore radius (Rats, 1961 
- Appendix F). 
5-25 
Example Problem 5-5 
Using the data from Example Problem 5-3 (tubing intake and IPR) and Example Problem 
3-1 (Table 3-3, do a shot density sensitivity analysis. 
Solution 
Calculate and plot the response curve from Fig. 5.17 (Example Problem 5-3) as follows: 
Response Curve Calculation 
4 (STB/D) 
938 200 
A P 
0 410 
40 400 
244 350 
488 300 
713 250 
Using data from Table 3.2, plot the pressure drop vs flow rate for different shot densities 
on the same plot as the response curve. 
The intersection of the response curve with the shot density curves gives the production 
rate for different shot densities. 
Fig. 5.20 PIor offrow rate vspressure drop for varying 
shot densities. 
5-26 
These values are read off and tabulated as 
Shot Density 
(BPW W F ) 
Flow Rate 
2 
408 24 
405 20 
400 12 
390 8 
378 4 
350 
These values are then plotted as shown here. 
500 
400 
E 
300 
2 
3 200 
2 
100 
0 
0 5 10 15 20 
SHOT DENSITY (SPF) 
Fig. 5.21 Plor of shot density vsflow rate. 
5 
Exercises 
1. For the following welldata, calculate the absolute open flow potential of the well. 
k, = 30md PC = 3,000 psia 
h = 40ft GOR = 300scf/STB 
API = 30 hp - - 10 ft 
Reservoir Temperature = 200°F 
"Is = 0.7 160 acre spacing 
(Produces all oil) Drilled hole size = 12-1/4 in. 
Casing size = 7 in. 
5-27 
2. Calculate In (re&) for r, = 7 in. and for drainage areas of 20,40, 80, 160 and 320 
acres. 
Hint: make a table. 
Drainage Acres re rw In (re/ r,) 
20 
40 
80 
160 
3. Draw the IPR for a well with the following data - 
k = 50md Depth = 5,000 ft 
h = 1 0 0 f t 
p, = 2,000 psia (Producing all oil) 
Determine the absolute open flow potential and an estimated production if you 
designed the tubular. 
4. Draw the IPR in Problem #I for skin of -50, +5. 
5. Using Vogel’s IPR relationship, construct an IPR for the following cases - 
(a) P, = P b . - - 3,000 psia 
AOFP = 10,WBOPD 
(b) Pr = 2,500 psia P b > P, 
q o = 100BPD 
pwf = 1,800 psia 
6. Given: P, = 2,000 psia 
pb = 1,500 psia 
PI = 4.7 BPD/psi 
construct the IPR curve. 
5-28 
7. The following data are obtained from a four-point test - 
Pr = 2,500 psi Pb = 3,000 psia 
Test # 
500 1,752 4 
1,000 1,595 3 
1,500 1,320 2 
2,000 880 1 
PH (psia) 9. (BPW 
Calculate - 
1. Value of C and n. 
2. Absolute open flow potential where: 
qo = c (P:-Pwf) 2 2 
8, The well in Problem #1 is fractured with the best proppant available, and the 
fracture half-length is 500 ft. Draw the post-frac IPR. 
9. Construct JPRs for the following well as a function of penneabilities - 
P, = 2,000 psi re = 2,000ft 
s = o r, = 0.5 ft 
h = 50ft Bo = 1.2 RB/STB 
= 2cp 
k = 1,lO. 100,1,ooO, 5,000 md 
10. For Problem No. 1, assume k = 100 rnd and construct IPR curves for skin. 
Skin = -5, -1,O, 1,5, 10,50,70 
1 1. Draw a sensitivity of 90 vs S from Problem No. 2. 
12. Given- 
p=* = 200 psia GLR = 800 scf/bbl 
Flowline Length = 400 ft Fw = 0.5 
Flowline ID = 2.5 in. Tubing ID = 2.5 in. 
Depth = 5,000 ft Oil Gravity = 35 A P I 
Water Sp. Gr. = 1.074 Gas Sp. Gr. = 0.65 
Bottom Hole Temp. = 180" F Surface Temp. = 60' F 
5-29 
Reservoir data for the construction of an IPR - 
pr = 4,000 psia qo = 3,000BPD 
- 3,000 psia PWf = 2,000 psia Pb - 
Draw the IPR and intake curves and predict the flow rate in this well. 
13. Make a tubing ID sensitivity and recommend the best tubing size for the following 
data - 
GLR - 800 scf/stb API - - 
Yg - 0.65 pwh - - 
FW - 0 T - - 
Depth - - 5,000 ft Tubing ID = 
IPR from Problem No. 1 
- 
- 
- 
14. Make a completion sensitivity study for the following well - 
Pwh - - 200 psia GLR - 
API - 35 F W - - 
Yg 
rP - 0.021 ft rC - - 
1, - 0.883 ft kP 
re - 2,000 ft h - - 
P, - 3,000 psia hP 
r, - 0.365 ft k - - 
P o - 1.2 cp Tubing ID = 
Depth - - 5,000 ft 
- 
- 
- - 0.65 BO - - 
- 
- - - 
- 
- - - 
- 
- 
Use the McLeod equations. 
35 
200 psia 
140' F 
2, 2.5, 3,4 in. 
800 scf/STB 
0 (all oil) 
1 
0.063 ft 
0.4K 
25 ft 
20 ft 
20 md 
2.0 in. 
5-30 
APPENDIX A 
Equation Summary 
I. Oil IPR Equations 
A. Darcy’s Law 
B. Vogel; Test Data; pr S pb 
C. Combination Vogel - Darcy; Test Data 
1. For test when pwf test > pb 
2. For test when pwftest < pb 
D. Jones IPR 
II. Gas IPR Equations 
A. Darcy’s Law (Gas) 
B. Jones’ Gas P R 
m. Backpressure Equation 
IV. Transient Period Equations 
A. Time to Pseudosteady State 
B. Oil IPR (Transient) 
C. Gas IPR (Transient) 
V. Completion Pressure Drop Equations 
A. Gravel-Packed Wells 
1. Oil Wells 
2. Gas Wells 
B. Open Perforation Pressure Drop 
1. Oil Wells 
2. Gas Wells 
A- 1 
I. Oil IPR Equations 
A. Darcy’s Law 
AOF = (PI) (F1-0) 
where, 
9 = oil flow rate, BPD, 
AOF = absolute open flow potential, BPD, 
k = permeability, md, 
h = net vertical formation thickness, ft, 
PI = average formation pressure (shut-in BHP), psi, 
- pwfs = average flowing bottomhole pressure at the sandface, psi, 
P o = average viscosity, cp, 
Bo = formation volume factor, RB/STB, 
re = drainage radius, ft, 
rw = wellbore radius, ft, 
S = skin factor, dimensionless, 
PI = Productivity Index, BPD/psi. 
B. Vogel; Test Data; pr S pb 
= 1-0.2%- 0.8 
qornax PI 
A-2 
C. Combination Vogel = Darcy; > Pb; Test Data 
1. For test when pwftest > Pb. 
PI = 4 - 
Pr -Pwfs 
qb = PI(Fr-Pb) 
2. For test when pwftest < Pb 
PI = q 
( f i - P b ) + f i [ 1 - 0 . 2 g ) - 0 . 8 ( E r ] 
A-3 
where, 
qo = flow rate, BPD, 
q, = flow rate at bubblepoint, 
pb = bubblepoint pressure, 
qomm = maximum flow rate (Vogel or combination), 
PI = Productivity Index BPD/psi. 
D. Jones IPR 
pr-pWfs = aq2+ b 
Pr -Pwfs = 10-14 /3 Bi p ) q2 + [ 
(2.30 X h rw 
2 7.08 X 103 MI 4 
AOF = -bfdb2+4a(p,-O) 2a 
where, 
b= ~ B o [ l n [ 0 . 4 7 2 ( k ) ] + s ] ] rw 
7.08 X 10-3 MI 4 
q = flow rate, BPD, 
pr = average reservoir pressure (shutin BHP). psi, 
- 
pwfs = flowing BHP at sandface, psi, 
/3 = turbulence coefficient, ft.1, 
p k1.201 
2.33 x 1010 (after Katz), 
Bo = formation volume factor, RB/STB, 
A-4 
p = fluid density, lb/ft3, 
hp = perforated interval, ft, 
p., = viscosity, cp, 
re = drainage radius, ft, 
r, = wellbore radius, ft, 
S = skin factor, dimensionless, 
k = permeability, md, 
a = turbulence term, 
b = darcyflow term. 
II. Gas IPR Equations 
A. Darcy's Law (Gas) 
where, 
q = flow rate, McfD, 
k = permeability, md, 
h = net vertical thickness, ft, 
ijr = average formation pressure (shut-in BHP), psia, 
pwfs = sandface flowing BHP, psia, 
p = viscosity, cp, 
T = temperature, O R (OF + 460), 
Z = supercompressibility, dimensionless, 
re = drainage radius, ft, 
r, = wellbore radius, ft, 
S = skin factor, dimensionless. 
A-5 
B. Jones' Gas IPR (General Form) 
-2 "2 
Pr - P wfs = a$+ bq 
3.16 x 10-12 p yg TZ 1.424 x 103 p TZ In (0.472 ") "2 - 2 [ TW + S I 
Pr -Pwfs = $ rw 9 2 4 kh 9 
-b 4 4- 
2a AOF" = 
where, 
3 . 1 6 ~ 10-12pygTZ 
a = 
rw 
1 .424~ lO3pTZ [ ln(0.472l")+s] 
kh b = 
q = flow rate, Mcf/D, 
a = turbulence term, 
b = darcy Flow term, 
pr = reservoir pressure (shutin BHP), psia, 
r w 
pwtl = sandface flowing BHP, psia. 
fi = turbulence coefficient, fV, 
2.33 x 1010 P = k1.201 
yg = gas specific gravity, dimensionless, 
T = reservoir temperature, OR (OF + 460), 
Z = supercompressibility, dimensionless, 
h, = perforated interval, ft, 
p = viscosity, cp, 
re = drainage radius, ft, 
rw = wellbore radius, ft. 
A-6 
III. Backpressure Equation 
where, 
c = 703 x 10-6 kh 
P = [ q 3 - 4 + s ] 
n = 0 . 5 < n < 1.0, 
q, = flow rate, Mcf/D, 
k = permeability, md, 
h = net vertical thickness, ft, 
pr = average formation pressure (shutin BHP), psia, 
- 
. . 
pds = sandface flowing BHP, psia, 
p = viscosity, cp, 
T = temperature, O R (OF + 460), 
Z = supercompressibility, dimensionless, 
re = drainage radius, fi, 
r, = wellbore radius, ft, 
S = skin factor, dimensionless. 
A-7 
IV. Transient Period Equations 
A. Time to Pseudosteady State 
where, 
4 = porosity, fraction, 
p = viscosity, cp, 
c, = total system compressibility, psi", 
re = drainage radius, ft, 
k = permeability, md, 
talab = time for pressure transient to reach re, hr. 
B. Oil IPR (Transient) 
90 = kh @I - PWfS ) 
162.6 po B,r log f kt I \ -x23 + 0.87 S 1 
where, 
k = permeability, md, 
h = net vertical thickness, ft, 
p = viscosity, cp, 
Bo = formation volume factor, RB/STB, 
t = time of interest; t < tslab (hr), 
0 = porosity, fraction, 
c, = total system compressibility, 
r, = wellbore radius, ft 
S = skin factor, dimensionless. 
A-8 
C. Gas IPR (Transient) 
kh (8 - Pw’f, ) 
9g = 
1,638 1.1 TZ[ log ( )- 3.23 + 0.87 S kt 
($1 Ct rw 1 
where, 
qs = flow rate, Mcf/D, 
k = permeability, md, 
pr = reservoir pressure (shutin BHP), psia, 
- 
pwr = flowing bottomhole pressure at sandface, psia, 
1.1 = viscosity, cp, 
T = temperature, O R (“F + 460), 
Z = supercompressibility, dimensionless,t = time of interest; t < tsmb (hr), 
f = porosity, fraction, 
ct = total system compressibility, 
r, = wellbore radius, ft, 
S = skin factor, dimensionless. 
A-9 
V. Completion Pressure Drop Equations 
A. Gravel-packed wells 
1. Oil Wells (General) 
pwfs - pwf = AP = aq2 + bq 
Ap = 9.08 x 1 0 1 3 p B: p L 1.1 B,L q2 + A2 1.127 x 10-3 kg A 9 
where, 
a = 
9.08 X 10-13 p B: p L 
A2 
pB,L 
1.127 x I03 kg A ’ 
b = 
q = flow rate, BPD, 
pwf = pressure well flowing (wellbore), psi, 
pwfs = flowing BHP at sandface, psi, 
b = turbulence coefficient, ft.1 . 
For GP wells: 
Bo = formation volume factor, RB/STB, 
p = fluid density, lb/ftj, 
L = length of linear flow path, ft, 
A = total area open to flow, ft2, 
(A = area of one perforation x shot density X perforated interval), 
kg = permeability of gravel, md. 
A- 10 
2. Gas Wells (General) 
PvA-df = aq2+bq 
Pwfs-pwf - 2 2 - 1.247 x 10-10 p yg TZL 8.93 X 103 1.1 TZL A2 92 + kg A q 
where, 
a = 1.247 x 1010 p yg TZL A2 
b = 8.93 X 103 p TZL 43 '4 
q = flow rate, McfD, 
pwrs = flowing pressure at the sandface, psia, 
pwr = flowing bottomhole pressure in wellbore, psia, 
p = turbulence factor, ft", 
1.47 X 107 
p = ,O.SS 
B 
= gas specific gravity, dimensionless, 
T = temperature, OR ("F + 460). 
Z = supercompressibility, dimensionless, 
L = linear flow path, 
A = total area open to flow, 
(A = area of one perforation X shot density X perforated interval), 
p = viscosity, cp. 
A-1 1 
B. Open Perforation Pressure Drop 
1. Oil Wells (General) 
where, 
2 .30xl014PBip 
a = 
b = 
7.08 x 10-3 kp ’ 
qo = flow rate/perforation (q/perforation), BPD, 
p = turbulence factor, ft-1, 
P = . 1.201 ’ 2.33 X 1010 
KP 
Bo = formation volume factor, RB/STB, 
p = fluid density, lb/ft3, 
Lp = perforation tunnel length, ft, 
p = viscosity, cp, 
kp = permeability of compacted zone, md, 
kp = 0.1 k formation if shot overbalanced, 
kp = 0.4 k formation if shot underbalanced, 
rp = radius of perforation tunnel, ft , 
r, = radius of compact zone, ft , 
(re = rp + 0.5 in.). 
A-12 
2. Gas Wells (General) 
Pwfs-Pwf = aq2+bq 
2 2 
- - 
where, 
a = 
b = 
qo = 
P = 
Yg = 
T = 
z = 
r, = 
- 
‘P - 
L p = 
r . 1 6 ~ 1012;TZ(--- r: :) + [1.4*4~ 103pTZ(h:)] 
kP LP 9 
flow rate/perforation (q/perforation), BPD, 
turbulence factor, ft-1, 
P = 2.33 x 1010 
G 2 0 1 
gas specific gravity, dimensionless, 
temperature, O R (“F + 460), 
supercompressibility factor, dimensionless, 
radius of compact zone, ft, 
(rc = rp + 0.5 in.), 
radius of perforation, ft , 
perforation tunnel‘length, ft, 
m = viscosity, cp, 
kp = permeability of compacted zone, md, 
kp = 0.1 k formation if shot overbalanced, 
kp = 0.4 k formation if shot underbalanced, 
A-13 
APPENDIX B 
Fluid Physical Properties Correlations 
Oil Properties 
Oil in the absence of gas in solution is called dead oil. The physical properties of dead oil 
oil is defined as 
are a function of the API gravity of oil and pressure and temperature. The A P I gravity of 
= SP.GR. @ 60°F 
141.5 - 131.5 
The A P I gravity of water is 10. With gas in solution, oil properties also depend on gas 
normally represented by R,. 
solubility in addition to the pressure, temperature, and API gravity of oil. Gas solubility is 
Gas Solubility (R,) 
Gas solubility is defined as the volume of I 
gas dissolved in one stock tank barrel of 
oil at a fixed pressure and temperature. 
Gas solubility in oil increases as the \&) 
pressure increases up to the bubble point 
pressure of the oil. Above the bubble point 
pressure, gas solubility stays constant 
(Fig. B-1). 
There are different correlations to calculate pig. ~ - 1 variation ofgas $olubi/iry with pressure 
correlation, Lassater correlation, etc. The Standing correlation states 
the gas solubility such as the Standing and temperature. 
Gas solubility 
$/- Temp. - T 
Prass"r0 (psla) 
Pb - Bubble poinl p re~ l l~ re h 
where, 
Ye = specific gravity of gas (air = 1 .O), 
P = pressure of oil, psia, 
T = temperature of oil, "F, 
API = A P I gravity of oil, "MI. 
B-1 
Formation Volume Factor (Bo) of Oil 
The volume in barrels occupied by one 
stock tank barrel of oil with the dissolved 
gas perature at any is defined elevated as pressure the formation and tem- vol- w &) +r ume factor of oil. The Bo unit is reservoir 
barrels .per stock tank barrel which is 
dimensionless. It measures the volumetric 
shrinkage of oil from the reservoir to the Preeaure (prla) 
surface conditions. The formation volume 
Pb 
Pb I Bubble point pressure 
gas ‘above the bubble point pressuk, the formation volume factor decreases due to the 
compressibility of the liquid. 
There are different correlations for calculating the formation volume factor of oil. These 
correlations are empirical and based on data from different oil provinces in the United 
States. The Standing correlation developed from California crude is one of the oldest and 
quite commonly used. The Standing correlation can be written as 
Bo = 0.972 + 0.000147 X F1.175 
where, 
F = Rs t) 0’5 + 1.25 (T) 
RS = gas solubility (m ) , scf 
T = temperature of oil, OF 
calculation of the formation volume factor this graphical method is very convenient. 
Standing also presented his correlation in graphical form (Fig..B-3). For noncomputer 
For the calculation of gas solubility R, and formation volume factor Bo. a knowledge of 
the bubble point pressure pb is necessary. Standing presented a nomograph (Fig. B-4) to 
determine the bubble point pressure. Gas solubility calculated at the bubble point pressure 
remains constant above the bubble point pressure. However, Standing’s or any other 
correlation for formation volume factor cannot be used above the bubble point pressure p. 
To calculate the formation volume factor of oil above bubble point pressure, the 
following equation is used - 
Bo = Bob exP [-co (P - Pb)] 
where Pb and Bob are calculated from the Standing or Lassater correlation using R, = R,. 
R being the produced gas oil ratio. The parameter Co is not a constant and can be 
calculated from the correlation presented by Trube as foliows- 
B-2 
Oil Viscosity 
The oil viscosity of reservoir oil containing 
solution gas decreases with pressure up to 
the bubble point pressure. Above the 
bubble point pressure, the viscosity in- 
creases (Fig. B-5). In the absence of labo- 
ratory determined data for the oil viscosity 
at any specified pressure and temperature, 
the Beal correlation is used. Beal corre- 
lated the absolute viscosity of gas free oil 
with the API gravity of crude at atmo- 
tures (Fig. B-6). The viscosity of gas satu- 
spheric conditions for different tempera- 
rated crude oil was correlated by Chew 
viscosity and gas solubility (Fig. B-7). 
and Connally with the gas free crude oil 
Beal also presented a correlation to esti- 
mate the viscosity increase from the bub- 
late the viscosity of oil above the bubble 
ble point pressure (cp/l,OOO psi) to calcu- 
point pressure is known (Fig. B-8). 
point pressure if the viscosity at the bubble 
The laboratory determined data on Bo, R,, 
and ko are recommended for use in any 
calculation wherever available. 
Gas Physical Properties 
The specific gravity of gas is a very 
important correlating parameter for the gas 
property evaluation. Normally, it can be 
very easily determined in the laboratory. In 
value, the specific gravity of gas can be 
the absence of a laboratory determined 
knowing the molecular weight (M) of gas. 
calculated from the following relationship 
where the molecular weight of air is 29. Thus, the specific gravity of air is 1. 
Gas density can be easily determined from the real gas law, 
pg =0.0433 yg zT 
where, 
Ps = density of gas (g/cc),B-5 
I 0' 
10 20 30 40 50 60 70 
OIL GRAVITY, API AT 60" F AND ATMOSPHERIC PRESSURE 
Fig. 8-6 Dead oil viscosify at reservoir femperfure and atmospheric pressure. Afer Beal. 
B-6 
0.3 1 .o 10 100 
(AT RESERVOIR TEMPERATURE AND ATMOSPHERIC PRESSURE) 
VlSCOSrpl OF DEAD OIL, CP 
Fig. 8-7 Viscosity of gas-saturated crude oil at reservoir temperature and pressure. Dead oil viscosity 
from laboratory data, or from Fig. 8-6. Afer Chew and Connaily. 
"Is = specific gravity of free gas (air = l), 
P = pressure of gas, psia, 
T = absolute temperature of gas, OR, 
Z = real gas deviation factor. 
= 460 + temperature, OF, 
B-7 
Real Gas Deviation Factor (z) 
The real gas deviation factor is a very im- 
portant variable used in calculating the gas 
To determine this parameter, Standing 
density and gas formation volume factor. 
law states that at the same reduced pres- 
used the law of corresponding states. This 
carbon gases have the same gas deviation 
sure and reduced temperature, all hydro- 
factor. The reduced pressure and reduced 
temperature are defined as follows - 
ppr = reduced pressure = " 
PPC 
Tpr = reduced temperature = - T 
TPC 
where p and T are the absolute pressure 
and absolute temperature of gas. 
ppc = pseudo-critical pressure, Fig. B-9 Correlation of pseudocritical properties of 
TF = pseudo-critical temperature. condensate wellfluids and miscellaneous natural gas with fluid gravity. After Brown 
Pseudo-critical pressure and pseudo-criti- et al. 
cal temperature are correlated by Brown et al. with the specific gravity Of gas (Fig. B-9). 
After determining the pseudo-critical pressure and pseudo-critical temperature from the 
correlation in Fig. B-9, the reduced pressure and reduced temperature are calculated using 
the definition presented earlier. For these calculated reduced pressures and reduced 
temperatures, the gas deviation factor can be calculated using the appropriate correlations 
after Standing and Katz (Fig. B-10). 
Gas formation volume factor B, can be calculated from 
a 
Bg@) = 0.0283- ZT 
P 
where, 
P = pressure, psia, 
T = absolute temperature, OR. 
B-8 
1.1 
1 .o 
0.9 
0.8 
N 
PSEUDO-REDUCED PRESSURE, D.. 
1.1 
I .o 
1.95 
- 
1.7 
1.6 
1.5 
I .4 
I .3 
I .2 
1.1 
I .o 
1.9 
Fig. B-10 Real gas deviation factor for natural gases as a function ofpseudoreduced pressure and 
temperature. After Standing and Katz. 
B-9 
Gas Viscosity (pg) 
Can; Kobayashi, and Burrows presented a correlation for estimating natural gas viscosity 
as a function of gas gravity, pressure, and temperature. This correlation also includes 
correlations for the presence of nonhydrocarbon gases in the natural hydrocarbon gas. 
Carr et al. correlated the viscosity of natural gases at one atmospheric pressure with the 
specific gravity of gas and the temperature of gas (Fig. B-1 1). The viscosity of natural gas 
at atmospheric pressure is then corrected for pressure through the second correlation (Fig. 
B-12). To use Fig, B-12, the pseudo-reduced pressure and pseudo-reduced temperature 
need to be calculated. This correlation presents the viscosity ratio of the viscosity of gas 
at the appropriate pressure and temperature to the viscosity of gas at atmospheric pressure 
and given temperature. 
B-10 
10s 
9,0 
8.0 
1.0 
pb - 
A 6.0 
m 
$- 
c 8 5.0 
:: 
5 
4,O 
3.0 
1.0 
1.0, 
PSEUDOREDUCED TEMPERATURE, Tar 
Fig. B-12 Effect of temperature and pressure on gas viscosity: After Carr, Kobayashi, and Burrows 
B-I1 
Rock and Fluid Compressibility 
Compressibility is defined as the change in 
volume per unit volume for unit change in 
pressure at constant temperature. 
c = compressibility = 
The unit of compressibility is reciprocal of 
pressure (l/psi). 
Oil Compressibility (co) 
The isothermal compressibility of an un- 
dersaturated oil (above the bubblepoint 
pressure) can be defined as 
PSEUDOREDUCED PRESSURE, ppr 
L 
Oil compressibility is always positive as 
the volume of an undersaturated liquid de- ibility for an undersaturated oil. Afer 
creases as the pressure increases. Oil com- 
pressibility can be determined from labora- 
- 
tory data, oil compressibility can also be 
tory experiments. In the absence of labora- 
determined from the Trube correlation (Fig. 
B-13). Trube correlated pseudo-reduced 
compressibility, cpr with the pseudo- 
reduced pressure ppr and pseudo-reduced 
temperature Tpr The oil compressibility can 
then be estimated from 
Fig. B-I3 Correlation ofpseudoreduced compress- 
- 
Trube. 
1300 
SPEClFlCQRAVITYOFUNDERSANRATED RESERVOIR 
LIWID AT RESERVOIR PRESSURE CORRECTED TO W F 
1 
I 
Fig. 8-14 Approximate correlation of liquidpseudo- where, critical pressure and temperature with 
specific gravity. After Trube. 
PPE - - pseudocritical pressure estimated from Fig. B-14, 
TPE = pseudo-critical temperature estimated from Fig. B-14. 
The apparent compressibility of oil c,, below the bubblepoint pressure can be calculated 
taking into account the gas in solution by 
R 
= (0.83p:21.75) x % Bo 
B-12 
e For an isothermal condition, the compressibility of oilfield water can be defined as: 
cw = " - iW (2) T 
where, 
B, = formation volume factor of water. 
Dodson and Standing presented a correlation for estimating the compressibility of water 
(Fig. B-15). Since the gas solubility in water is very low, the effect of gas solubility is 
ignored in this manual. 
Gas Compressibility (cg) 
conditions can be defined as 
Gas compressibility under isothermal 
where. 
Z = gas deviation factor at absolute 
pressure p in psia and absolute 
temperature T in "R. 
mation of gas compressibility. Trube de- 
Trube presented a correlation for the esti- 
fined gas compressibility as the ratio of 
pseudo-critical compressibility to the 
pseudo-critical pressure as 
To estimate the gas compressibility, Trube 
4.0 
3.6 
3.2 
2.8 
2.4 
60 100 140 180 220 260 
TEMPERATURE. 'F 
1.0 
0 5 10 15 20 25 
GASWATER RATIO. CU FllBBL 
presented correlations to estimate the cpr Fig. B-,s Efecr ofdissolved gas on 
function of pseudo-reduced pressure and 
pseudo-reduced temperature (Figs. B-16 
and B-17). Note that these two correlations 
are similar. They present pseudo-reduced 
compressibility at two different ranges of compressibility values. 
L 
pressibiliry. After Dodson 
water com- 
and Standing, 
B-13 
Rock Pore Volume Compressibility (c,) 
Rock compressibility under isothermal 
conditions can be defined as 
Cf = "4 
;Pr; I T 
There are different correlations for rock 
compressibility, each for a fixed type of 
pressibility correlation after Newman. It is 
rock. Figure B-18 presents the rock com- 
very strongly recommended to use labora- 
tory data for this parameter wherever pos- 
sible. From Fig. B-18, it is clear that these 
correlations are questionable at best. 
However, for any well performance calcu- 
component of the total compressibility c, 
lations' rock compressibility a minor Fig, 8-16 Correlarjon ofpseudoreducedcompress- 
defined as 
c, = c, so + c, s, + c, s, +Cf 
where, 
S = saturation of fluid where 
subscript o is used for oil, g 
for gas, and w for water. 
S , + S , + S , = l 
Gas compressibility is an order of magni- 
pressibility. In gas reservoirs, it is often as- 
tude higher than the rock or liquid com- 
sumed that 
't '8 
It may also be noted that whereas the gas 
compressibility is in the order of 10-6, the 
liquid or rock compressibilities are typi- 
cally in the order of 1 0 5 or 10-4. 
Fig. 8-17 Correlation of pseudoreduced compress- 
ibiliry for natural gases. After Trube. 
B-14 
e 
INITIALPOROSITYATZERO NETPRESSURE. (I 
Fig, ~ - 1 8 Pore-voIume compressibitity at 75-per- Fig. B-19 Pore-volume compressiblity at 75-percent 
cent lithostoticpressure vs. initial lithostatic pressure vs. initial sample 
sample porosity for limestones. After porosity for friable sandstones. After 
Newman. Newman. 
7 
$ 8 
: e 
i : 
-% 
$ 2 
i? -; 10 
" 8 
c 7 
8 
5 . : : 1 : 
Z Z 
9 Fig. B-20 Pore-volume compressibility at 75-per- 
cent lithostatic pressure vs. initial 
sample porosity for consolidated 
sandstones. Ajier Newman. 
5 
INITIAL wwsm *T ZERO NET PRESSURE. o, 
B-15 
APPENDIX C 
c- 1 
0 4 
PRESSURE in 100 PSlG 
0 12 16 20 24 20 
7 
8 
9 
10 
c-2 
c-3 
1 
2 
3 
7 
9 
10 
c-4 
t 
W 
LL 
0 
0 
0 
.- c 
F 
w 
1 
2 
3 
4 
5 
6 
7 
6 
9 
10 
c-5 
PRESSURE in 100 PSlG 
n A R 12 16 20 24 28 
C-6 
1 
2 
3 
4 
t; w 
U 
7 
8 
9 
10 
c-7 
C-8 
e 
e 
0 4 
PRESSURE in 100 PSlG 
a 12 16 20 24 26 
t 
ttl 
c-9 
c-10 
c-11 
c-12 
C-13 
4 
t- 
w W 
U 
0 
r 
8 
.- c 5 
L c 
6 
7 
a 
9 
10 
PRESSURE in 100 PSlG 
0 4 6 12 16 20 24 26 
0 
$8 
80 
GO 
%%% % % P c& 9 00 
C-14 
PRESSURE in 100 PSlG 
0 4 8 12 16 20 24 28 
Gas Specific Gravity 
Average Flowing Temp. 
0 
.'s 
80 
's, 
C-15 
PRESSURE in 100 PSiG 
C-16 
a APPENDIX D 
Calculation of Gas Velocity 
4E 
ZT TS 
where, 
q, = gas flow rate, MMscfd, 
Therefore, 
q = gas flow rate, MMcfd, 
p = pressure, psia, 
T = temperature, OR, 
d = diameter, ft, 
f = Moody friction factor. 
= 14.7365 $ ft/sec 
Therefore, 
V (ft/sec) = 14.7365 x 0.02873 
Pdz 
= 0.415273 ST 
Calculation of Friction Loss Term 
= 0.002679 (& ) $7 q: dL 
D- 1 
e Reynolds' Number 
d = diameter, ft, 
v = velocity, ft/sec, 
p = density, lbm/ft', 
p = viscosity, cp, 
'/g = specific gravity of gas (Air = l), 
i.e.,NRe = 1,488d (2.7047 g} 
Pd2P 
Also reported in literature as 20,050 ssrg for diameter d of pipe in inches. 
Pd 
Cullender and Smith Modification 
Divide Equation 4-4 by (Tz/p)2 and obtain, 
i.e., 
1.e.. 
D-2 
e APPENDIX E - PARTIAL PENETRATION 
Partial penetration occurs when the well has been drilled partially through the producing 
interval or when only part of the cased interval has open perforations (Fig. E.1). 
PARTIAL PENETRATION PARTIAL COMPLETION 
Fig. E.1 
The pseudo-skin factor (sR) due to partial penetration can be computed using the 
nomograph (Fig. E.2). 
PENETRATION RATIO (b) 
Fig. E.2 
E- 1 
APPENDIX F - PRAT’S CORRELATION 
Prat (1961) defined a correlation between the dimensionless wellbore radius (rwe/xf) and 
the dimensionless fracture conductivity. This is shown in Fig. F.l. 
1 
0.1 
0.01 
0.001 
0.1 1 10 100 1000 10,000 
FCD 
Fig. F.1 Dimensionless wellbore radius vs FCD 
Here, FCD = !F 
kxf 
F- 1 
* APPENDIX G - SLIDES 
NODAL ANALYSIS/PRODUCTION SYSTEMS ANALYSIS 
OVERVIEW: 
THREE LEVELS OF OIUGAS FIELD MANAGEMENT 
FIELD LEVEL 
SPECIFICATIONS: RATES (GOR, FW , 
PRESSURES, ET J . 
CONSTRAINTS: RATES, GOR, FW, ETC. 
GROUP LEVEL 
SPECIFICATIONS: RATES (GOR, FW), 
PRESSURES, SOLIDS 
CONSTRAINTS: RATES, GOR, FW 
WELL LEVEL 
SPECIFICATIONS: RATES (GOR, FW), 
CONSTRAINTS: ALL SPECIFIED PARAMETERS 
PRESSURES, SOLIDS 
G- I 
WHY MANAGE? 
e 
EARNINGS & INVESTMENT APPRAISAL 
REVENUE-PROJECTIONS DEPENDING ON OIL & GAS 
PRICE 
INVESTMENT DECISIONS BASED ON REVENUE 
OVERALL ENGlNEERlNGflECHNlCAL STRATEGY FOR THE 
DEVELOPMENT OF RESERVOIR 
PRIMARY RECOVERY/ARTIFICIAL LIFT 
SECONDARYRECOVERY 
TERTIARY RECOVERY 
POSSIBLE ENVIRONMENTAL IMPACT 
e 
G-2 
e ENGINEERING TOOLS USED IN RESERVOIR MANAGEMENT 
FORMATION EVALUATION 
(BASED ON GEOLOGICAL INFORMATIONS) 
RESERVOIR SIMULATOR 
PREDICTION 
HISTORY MATCHING 
WELUCOMPLETION SIMULATOR (NODAU 
PRODUCTION SYSTEMS ANALYSIS) 
MANAGEMENT OF A RESERVOIR AT THE WELL LEVEL 
G-3 
HISTORICAL PREVIEW 
NODAL ANALYSIS 
W. E. GILBERT (1 944,1954) OF SHELL 
FLOWING AND GAS LIFT WELL PERFORMANCE 
(CONCEPT) 
PROBLEMS 
LACK OF UNDERSTANDING OF MULTIPHASE FLOW IN PIPES 
AND PIPING COMPONENTS (E.G. CHOKES, ETC.) 
- 
COMPUTERS (??) 
POETTMAN AND CARPENTER 1952) 
WATER, OIL AND GAS FLOW I r!i VERTICAL PIPES 
G-4 
HISTORICAL PREVIEW (CONT) 
REVOLUTION IN COMPUTER TECHNOLOGY 
A. E. DUKLER (1969) 
H. D. BEGGS AND J. P. BRILL (1973) 
GAS-LIQUID FLOW IN INCLINED PIPES 
JOE MACH, E. A. PROANO AND KERMIT E. BROWN (1977), 
NODAL APPROACH AND APPLICATION OF SYSTEMS 
ANALYSIS TO THE FLOWING AND ARTIFICIAL LIFT WELLS. 
0-5 
a OBJECTIVE AND IMPORTANCE 
OPTIMIZATION OF OIUGAS WELL PRODUCTION 
MINIMIZATION OF UNNECESSARY WELL COST 
OPTIMUM TUBING SIZE 
OPTIMUM SEPARATOR PRESSURE 
OPTIMUM COMPLETION 
SHOT DENSITY 
STIMULATION (ACID, FRACTURE HALF LENGTH) 
.e OPTIMIZE ARTIFICIAL LIFT 
PREDICTION AND CONTROL OF UNACCEPTABLE 
PRODUCTION CONDITIONS (SURGING) 
G-6 
ECON.OMIC . . IMPACT 
0 MAXIMIZATION OF ROI (RETURN ON INVEST.MENT) 
G-7 
	Table of Contents
	chapter_1_introduction
	chapter_2_reservoir_system
	chapter_3_completion_system
	chapter_4_flow_through_tubing_and_flowlines
	chapter_5_well_performance_evaluation_of_stimulated_wells
	appendix_a
	appendix_b
	appendix_c
	appendix_d
	appendix_e
	appendix_f
	appendix_g