vaneverdingen1949

Prévia do material em texto

T.P. 2732 
THE APPLICATION OF THE LAPLACE TRANSFORMATION 
TO FLOW PROBLEMS IN RESERVOIRS 
A. F. VAN EVERDINGEN, SHELL OIL CO., HOUSTON, AND W. HURST, PETROLEUM 
CONSULTANT, HOUSTON, MEMBERS AIME 
ABSTRACT 
For several years the authors have felt the need for a source 
from which reservoir engineers could obtain fundamental 
theory and data on the flow of fluids through permeable media 
in the unsteady state. The data on the unsteady state flow are 
composed of solutions of the equation 
O'P + ~ oP = oP 
or' r Or at 
Two sets of solutions of this equation are developed, namely, 
for "the constant terminal pressure ca;;e" and "the constant 
terminal rate case." In the constant terminal pressure case the 
pressure at the terminal boundary is lowered by unity at zero 
time, kept constant thereafter, and the cumulative amount of 
fluid flowing across the boundary is computed, as a function 
of the time. In the constant terminal rate case a unit rate 
of production is made to flow across the terminal boundary 
(from time zero onward) and the ensuing pressure drop is 
computed as a function of the time. Considerable effort has 
been made to compile complete tables from which curves can 
be constructed for the constant terminal pressure and constant 
terminal rate cases, both for finite and infinite reservoirs. 
These curves can be employed to reproduce the effect of any 
pressure or rate history encountered in practice. 
Most of the information is obtained by the help of the 
Laplace transformations, which proved to be extremely helpful 
for analyzing the problems encountered in fluid flow. Tht' 
application of this method simplifies the mOTe tedious mathe-
matical analyses employed in the past. With the help of La-
place transformations some original developments were ob-
tained (and presented) which could not have been easily 
foreseen by the earlier methods. 
INTRODUCTION 
This paper represents a compilation of the work done over 
the past few years on the flow of fluid in porous media. It 
concerns itself primarily with the transient conditions prevail-
ing in oil reservoirs during the time they are produced. The 
study is limited to conditions where the flow of fluid obeys the 
Manuscript received at office of Petroleum Branch January 12, 1949. 
Paper presented at the AIME Annual Meeting in San Francisco, Febru-
ary 13-17. 1949. 
1 References are given at end of paper. 
diffusivity equation. Multiple-phase fluid flow has not been 
considered. 
A previous publication by Hurst' shows that when the pres-
sure history of a reservoir is known, this information can be 
used to calculate the water influx, an essential term in the 
material balance equation. An example is offered in the lit-
erature by Old' in the study of the Jones Sand, Schuler Field, 
Arkansas. The present paper contains extensive tabulated 
data (from which work curves can be constructed), which data 
are derived by a more rigorous treatment of the subject mat-
ter than available in an earlier publication. ' The applicatIon of 
this information will enable those concerned with the analysis 
of the behavior of a reservoir to obtain quantitatively correct 
expressions for the amount of water that has flowed into the 
reservoirs, thereby satisfying all the terms that appear in the 
material balance equation. This work is likewise applicable to 
the flow of fluid to a well whenever the flow conditions are 
such that the diffusivity equation is obeyed. 
DIFFUSITY EQUATION 
The most commonly encountered flow system is radial flow 
toward the well bore or field. The volume of fluid which flows 
per unit of time through each unit area of sand is expressed 
by Darcy's equation as 
K oP 
v = 
fJ. Or 
where K is the permeability, fJ. the viscosity and oP lor the 
pressure gradient at the radial distance r. A material balance 
on a concentric element AB, expresses the net fluid traversing 
the surfaces A and B, which must equal the fluid lost from 
within the element. Thus, if the density of the fluid is ex-
pressed by p, then the weight of fluid per unit time and per 
unit sand thickness, flowing past Surface A, the surface near-
est the well bore, is given as 
2~rp ~ ~~ = 2~fJ.K ( pr ~~) 
The weight of fluid flowing past Surface B, an infinitesimal 
distance or, removed from Surface A, is expressed as 
oP o( pr g; ) 
[pr - + 
or or 
2~K 
or] 
December, 1949 PETROLEUM TRANSACTIONS, AIME 305 
T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS 
IN RESERVOIRS 
The difference between these two terms, namely, 
o( pr 'O_~) 
27rK or 
- -- -------- or, 
p or 
is equal to the weight of fluid lo:t by the element AB, ()j' 
OP 
- 27rfr -- or 
aT 
where f is the porosity of the formation. 
This relation gives tf:e equat:on of continuity for the radial 
system, namely, 
a (pr .Q~-) 
K Or OP 
- ---- fr --- (II-I) 
p or aT 
From the physical characteristics of fluids. it is known 
that density is a function of pressure and that the density 01 
a fluid decreases with decreasing pressure due to the fact that 
the fluid expands. This trend expres~ed in exponential form 
is 
p = p"e-"(I',,-I') (II-2) 
where P is less than P,,, and c the compressibility of the fluid. 
If we substitute Eq. II-2 in Eq_ II-I, the diffusivity equation 
can be expressed using density as a function of radius and 
time. or 
( 02p + 2:.. 2!_) ~_ = ~_ (I1-3) 
or' r Or fllc aT 
For liquids which are only slightly compressible, Eq. II-2 
simplifies to p ~ Po [1- c (Po - P)] which further modifies 
Eq. 1I-3 to give 
( o~_ -+ _1 __ OP ) ~ = 1l.!'... Furthermore, if the or- r or fpc aT 
radius of the well or field. R h, is referred to as a unit 
radius, then the relation simplifies to 
o'P 1 oP oP - - + -- -- == ------
or' r Or at 
(II-4) 
where t = KT /fJlcR,,' and r now expresses the distance as a 
multiple of R h , the unit radius. The units appearing in this 
paper are always med in connection with Darcy's equation, so 
that the permeability K must be expressed in darcys; the 
time T in seconds. the porosity f as a fraction, the viscosity f' 
in centipoises. the compressibility c as volume per volume 
per atmosphere, and the radius Rb in centimeters. 
LAPLACE TRANSFORMATION 
In all publications, the treatment of the diffusivity equation 
has been essentially the orthodox application of the Fourier-
Bessel series. This paper presents a new approach to the 
solution of problems encountered in the study of flowing fluids, 
namely, the Laplace transformation, since it was recognized 
that Laplace transformations offer a useful tool for solving 
difficult problems in less time than by the use of Fourier-
Bessel series. Also, original developments have been obtained 
which are not easily foreseen by the orthodox methods. 
If p(t) is a pressure at a point in the sand and a function 
of time, then its Laplace transformation is expressed by the 
infinite integral 
(III-l) 
where the constant p in this relationship is referred to as the 
operator. If we treat the diffusivity equation by the process 
implied by Eq. Ill-I, the partial differential can be trans-
formed to a total differential equation. This is performed by 
multiplying each term in Eq. II-4 by e-'" and integrating with 
respect to time between zero and infinity, as follows; 
'L _ . ., (o'P 1 oP ) 
,ie' -,-+---
o Or- r or 
x oP 
dt = f e-;'t --dt 
o· at 
(III-2) 
Since P is a function of radius and time, the integration with 
respect to time will automatically remove the time function 
and leave P a function of radius only. This reduces the left 
side to a total differential with respect to r, namely, 
x O'l' 
J e-:" 
oar' 
Jo 
a')' 1 e-JO ' P dt f 
d'P,JO) 
dt = -----._- = _.-
or' dr' 
and Eq. HI-2 hecomes 
dr' 
P, PRESSURE 
q(t), RATE 
I dP"" 
r dr 
dP 
dt 
t, t2 t3 
t, TIME 
dt 
etc. 
FIG. lA - SEQUENCE CONSTANT TERMINAL PRESSURES. 
1 B - SEQUENCE CONSTANT TERMINAL RATES. 
306 PETROlEUM TRANSACTIONS, AIME December, 1949 
A. F. VAN EVERDINGENAND W. HURST T.P. 2732 
Furthermore, if we consider that P (l) is a cumulative pressure 
drop, and that initially the pressure in the reservoir is every· 
where constant so that the cumulative pressure drop p(t~O)=O, 
the integration of the right hand side of the equation becomes 
dP 
00 
As this term is also a Laplace transform, Eq. III·2 can be writ· 
ten as a total differential equation, or 
d'P(p) + 1 dP,p) 
dr' r dr 
(III.3) 
y 
8 
i! PLANE 
c~ ________ ~----~ 
--------------________ hM __ ~~(T~O~)--x 
Dr-----~~------~ 
A 
FIG. 2 - CONTOUR INTEGRATION IN ESTABLISHING THE CONSTANT 
TERMINAL RATE CASE FOR INFINITE EXTENT. 
y 
i! 
PLANE 
-1~~rt-+-1~-+-+~~~4-~~--+---x 
(cr ,0) 
FIG. 3 - CONTOUR INTEGRATION IN ESTABLISHING THE CONSTANT 
TERMINAL RATE CASE FOR LIMITED RESERVOIR. 
The next step in the development i, to reproduce the boun· 
dary condition at the wdl bore or field radius, r = 1, as a 
Laplace transformation and introduce this in the general solu· 
tion for Eq. III·3 to give an explicit relation 
By inverting the term on the right by the Mellin's inversion 
formula, or other methods, we obtain the solution for the 
cumulative pressure drop as an explicit function of radius 
and time. 
ENGINEERING CONCEPTS 
Before applying the Laplace transformation to develop the 
necessary work·curves, there are some fundamental engineer· 
ing concepts to be considered that will allow the interpreta· 
tion of these curves. Two cases are of paramount importance 
in making reservoir studies, namely, the constant terminal 
pressure case and the constant terminal rate case. If we know 
the explicit solution for the first case, we can reproduce any 
variable pressure history at the terminal boundary to deter· 
mine the cumulative influx of fluid. Likewise, if the rate of 
fluid influx varies, the constant terminal rate case can be used 
to calculate the total pressure drop. The constant terminal 
pressure and the constant terminal rate calOe are not inde· 
pendent of one another, as knowing the operational form of 
one, the other can be determined, as will be shown later. 
Constant Terminal Pressure Case 
The constant terminal pre3sure case is defined as follows: 
At time zero the pressure at all points in the formation is con· 
stant and equal to unity, and when the well or reservoir is 
opened, the pressure at the well or reservoir boundary, r = 1, 
immediately drops to zero and remains zero for the duration 
of the production history. 
If we treat the constant terminal pressure case symbolically, 
the solution of the problem at any radius and time is given 
by P = p(,.,t). The rate of fluid influx per unit sand thickness 
under these conditions is given by Darcy's equation 
q(T) = 21TK (r OP) " (IV.I) 
/L or r = 1 
If we wish to determine the cumulative influx of fluid in 
absolute time T, and having expressed time in the diffusivity 
equation as t = KT/f/LcRb" then 
T 21TK f,acRo' t 
Q('I') = f q(T) dT = --x-~ J 
o· /L K 0 
= 21TfcR h 2 Q(t) 
where 
( OP) -- dt or r = 1 
(IV·2) 
Q«) = / (OP ) dt (IV.3) 
o or r = 1 
In brief, knowing the general solution implied by Eq. IV·3, 
which expresses the integration in dimensionless time, t, of the 
pressure gradient at radius unity for a pressure drop of one 
atmosphere, the cumulative influx into the well bore or into the 
oil.bearing portion of the field can be determined by Eq. IV·2. 
Furthermore, for any pressure drop, f,P, Eq. IV·2 expresses 
the cumulative influx as 
Q('I') = 21TfcR,,' f,P Q", (IV·4) 
per unit sand thickness.* 
* The set of symbols now introduced and the symbo~s reoorted in 
Hurst's1 earlier paper on water-drive are related as follows: 
t 
G(o;' O/R') = Q(l) and G(o;' B/R') r Q(t) dt where 
o· 
0;' e/R' = t 
December, 1949 PETROlEUM TRANSACTIONS, AIME 307 
T.P. 2732 THE APPLICATION OF THE lAPLACE TRANSFORMATION TO FLOW PROBLEMS 
IN RESERVOIRS 
When an oil reservoir and the adjoining water-bearing for-
mations are contained between two parallel and sealing fault-
ing planes, the flow of fluid is essentially parallel to these 
planes and is "linear." The constant terminal pressure case 
can also be applied to this case. The basic equation for linear 
flow is given by 
O'P 
Ox' 
oP 
at 
(IV-S) 
where now t = KT / fl'c and x is the absolute distance meas· 
ured from the plane of influx extending out into the water-
bearing sand. If we assume the same boundary conditions as 
in radial flow, with P = P(x, t) as the solution, then by 
Darcy's law, the rate of fluid influx across the original water-
oil contact per unit of cross-sectional area is expressed by 
qUi = ~ ( ~:-) x=o (IV-6) 
The total fluid influx is given by 
! K fl'c .t ( oP ) Q(T) = j q('l') dT = --. --- j -- dt 
o I' K 0 Ox x=o 
= f C Q(l) (IV-7) 
where Q(" lS the generalized ~olution for linear flow and is 
equal to 
~ ( OF ) Q(l) = J .-
o OX 
dt (IV-8) 
x==o 
Therefore, for any over-all pressure drop L.F, Eq. IV-7 gives 
Q{'j') = fcL.P Q,,) (IV-9) 
per unit of cross-sectional area. 
Constant Terminal Rate Case 
In the constant terminal rate ca:-;e it is likewise assumed that 
initially the pressure everywhere in the formation is constant 
but that from the time zero onward the fluid is withdrawn 
from the well bore or reservoir boundary at a unit rate. The 
pressure drop is given by P = p(,.,t), and at the boundary of 
the field, where r = 1, (OP/Or)..=l = -1. The minus sign 
is introduced because the gradient for the pressure drop rela-
tive to the radius of the well or reoervoir is negative. If the 
cumulative pressure drop is expressed as L.P, then 
.' (IV-IO) 
where q(t) is a constant relating the cumulative pressure drop 
with the pressure change for a unit rate of production. By 
applying Darcy's equation for the rate of fluid flowing into 
the well or reservoir per unit sand thickness 
where q(T) is the rate of water encroachment per unit area of 
cross-ECction, and P tt ) is the cumulative pressure drop at the 
sand face per unit rate of production. 
Superposition Theorem 
With these fundamental relationships available. it remams 
to be shown how the constant pressure case can be interpreted 
for variable terminal pressures, or in the constant rate case, 
for variable rates. The linearity of the diffusivity equation al-
lows the application of the superposition theorem as a se-
quence of constant terminal pre~sures or constant rates in 
such a fashion that it reproduces the pressure or production 
hiHory at the boundary, r = 1. This is essentially Duhamel's 
principle, for which reference can be made to transient electric 
circuit theory in texts by Karman and Biot,S and Bush." It has 
been applied t olhe flow of fluids by Muskat,' Schilthuis and 
Hurst,' in employing the variable rate case in calculating the 
pressure drop in the East Texas Field: 
The physical significance can best be realized by an appli-
cation. Fig. I-A shows the pressure decline in the well bore 
or a field that has been flowing and for which we wish to ob-
tain the amount of fluid produced. As shown, the pressure 
history is reproduced as a series of pressure plateaus which 
repre~ent a sequence of constant terminal pressures. Therefore, 
hy the application of Eq. IV-4, the cumulative fluid produced 
in time t by· the pressure drop L.P", operative since zero time, 
is expre,'ed hy Q(T) = 27rfcR b' ,0,1'" Q't). If we next consider 
r-Q(t) 
30~--------~------------~r--------' 
q(T! = -21rK ( QL.P) =-21rK q(,) (oP(r,t)) 
I' Or" = 1 I' or r = 1 101---/ 
h · h ' l·fi q('nl' Th f w IC sImp I es to q(t) = --. ere ore, for any constant 
21rK 
rate of production the cumulative pressure drop at the field 
radius is given by 
P _ qcnl' P 
,0, - 27rK (t) (IV-ll) 
Similarly, for the constant rate of production m linear flow, 
the cumulative pressure drop is expressed by 
L.P = qcnl' p 
K (ti (IV-I2) 
0~1----------~5-------------J10~------~ 
FIG, 4 - RADIAL FLOW, CONSTANT TERMINAL PRESSURE CASE, INFIN-
ITE RESERVOIR, CUMULATIVEPRODUCTION VS. TIME. 
308 PETROlEUM TRANSACTIONS, AIME December, 1949 
A. F. VAN EVERDINGEN AND W. HURST T.P. 2732 
the pressure drop ,6P" which occurs in time t" and treat this 
as a separate entity, but take cognizance of its time of incep-
tion t
" 
then the cumulative fluid produced by this increment 
of pressure drop is Q(t) = 2trfcRb ' ,6P, Q(t-tl)' By super-
imposing all the.'e effects of pressure changes, the total influx 
in time t is expressed as 
Q(T) = 27rfcR h' [,6Po Q(t) + ,6P,Q(t-t,) + 
,6P,Q(tt,) + ,6P,Q(t-t
3
) + ] (IV-I3) 
when t > I,. To reproduce the smooth curve relationship of 
Fig. I-A, these pressure plateaus can be taken as infinitesim-
ally small, which give the summation of Eq. IV-13 by the 
integral 
, ~ o,6P 
QfT) = 27rfcR,,- j ---- Q(t-t') dt' . 
o· at' (IV-I4) 
By considering variable rates of fluid production, such as 
shown in Fig. I-B, and reproducing these rates as a series of 
constant rate plateaus, then by Eq. IV -11 the pressure drop in 
the well bore in time t, for the initial rate q" is ,6Po = qoP(t). 
At time t" the comparable increment for constant rate is ex-
pressed as .q, - qo, and the effect of this increment rate on 
the corresponding increment of pressure drop is ,6P, = 
(q, - qJ p(t-tl)' Again by superimposing all of these effects, 
the determination for the cumulative pressure drop is ex-
pressed by 
,6P = q(o) P tt ) + [q, (t, ) - q(O)] p(t-t,) + [q(t,) - q(t ,)] 
p(t-t .. ) + [q(t3) -q(f,)] p(t-t,) + (IV-I5) 
rr===-Q-(t)-,------r------,----~~----,_--__. 
35~---+---~---~ 
3.01-----+-----+----,~1__7"-------+_--__+---_____l 
2.5f--------+---V;:L--+---------::~---====+===1 
2.01----+---I'---T"---t------ir-------f-----__+--------j 
1.5r----__ -----!lr----__ --=l=~--;I~t_---A~SYrM-T-.:0-T~IC~VA-.:L;..:U-=E-I:.:..5::00".::J.\~ 
"R =2.0 
I. OJ----f--+-----+-----f-------+-----__+--------l 
ASYMTOTIC VALUE 0.625 
R = 1.5 
o 0 0;;--------;I-';;.0;-------:2t.0;;------;f3.0;;-----~40;;------;05L,;0:------d6.0 
FIG. 5 - RADIAL FLOW, CONSTANT TERMINAL PRESSURE CASE, 
CUMULATIVE PRODUCTION VS. TIME FOR LIMITED RESERVOIRS. 
If the increments are infinitesimal, or the smooth curve rela-
tionship applies, Eq. IV-I5 becomes 
t dq(t') 
,6P = q(o) P(t) + J -- p(t-t') dt' 
o dt' 
If q(o) = 0, Eq. IV-I6 can also be expressed as 
t 
(IV-I6) 
,6P = J q(t') p'(t-t') dt' (IV-I7) 
o 
where p'(t) is the derivative of Pit) with respect to t. 
Since Eqs. IV-I3 and IV-I5 are of such simple algebraic 
forms, they are most practical to use with production history 
in making reservoir studies. In applying the pressure or rate 
plateaus as shown in Fig. 1, it must be realized that the time 
interval for each plateau should be taken as small as possible, 
so as to reproduce within engineering accuracy the trend of 
the curves. Naturally, if an exact interpretation is desired, Eqs. 
IV-I4 and IV-I6 apply. 
FUNDAMENTAL CONSIDERATIONS 
In applying the Laplace transformation, there are certain 
fundamental operations that must be clarified. It has been 
stated that if P (t) is a pressure drop, the transformation for 
Pit) is given by Eq. III-I, as 
To visualize more concretely the meaning of this equation, if 
the unit pressure drop at the boundary in the constant termi-
nal pressure case is employed in Eq. III-I, its transform is 
given by 
00 
-pt -e 1 
(V-I) PiP) = J e- pt 1 dt = --- 1 
o p p 
o 
The Laplace transformations of many transcendental functions 
have been developed and are available in tables, the most com-
plete of which is thc tract by Campbell and Foster.' It is there-
fore often possible after solving a total differential such as 
Eq. 1I1-3 to refer to a ~et of tables and transforms and deter-
mine the invcrse of PCP) or Pit). It is frequently necessary to 
simplify PiP) before an inversion can be made. However, Mel-
lin's inversion formula is always applicable, which requires 
analytical treatment whenever used. 
There are two possible simplifications for PCP) when time 
is small or time is large. This is evident from Eq. 111-3, where 
p can be interpreted by the operational calculus as the oper-
ator d/ dt. Therefore, if we consider this symbolic relation, 
then if t is lorge, p must be small, or inversely, if t is small, 
p will be large. To understand this, if PiP) is expressed by an 
involved Bessel relationship, the substitution for p as a small 
or large value will simplify Pcp) to give Pit) for the corre-
sponding times. 
Mellin's inversion formula is given on page 71 of Carslaw 
and Jaeger:' 
December, 1949 PETROlEUM TRANSACTIONS, AIME 309 
T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS 
IN RESERVOIRS 
1 
p(t·=--
, 271"i 
eAt P d A 
'Y-i~ 
(A) 
where P 
(A) 
is the transform P 
(p) 
Where this report is con-
corned with pressure drops, the above can be written as 
1 
P )-p -- r (t, (t2 ) - 2 .. At, Ato (e -e -) P dA. (V-2) (A) 71"1 
'Y--i r:JJ 
The integration is in the complex plane A = x + iy, along a 
line parallel to the y-axis, extending from minus to positive 
infinity, and a distance I' removed from the origin, so that all 
poles are to the left of this line, Fig. 2. The reader who has a 
comprehensive understanding of contour integrals will recog-
nize that this integral is equal to the integration a.round a 
semi-circle of infinite radius extending to the left of the line 
x = 1', and includes integration along the "cuts," which joins 
the poles to the semi-circle. Since the integration along the 
semi-circle in the second and third quadrant is zero for radius 
infinity and t>O, this leaves the integration along the "cuts" 
and the poles, where the latter, as expressed in Eq. V-2, are 
the residuals. 
Certain fundamental relationship3 III the Laplace trans-
formations are found useful: ll) 
Theorem A ~ If P,p, is the transform of p(», then 
or the transform 
= p fi,p, 
dP(t) 
of -- = p 
dt 
o 
p(t=O) 
approaches zero as time approaches infinity. 
00 
Theorem B ~ The transform of r p(t') dt' is expressed by 
o' 
00 t _e-Pt 
J e-pt J p(t') dt' dt = -- J p(t') dt' 
o 0 p 0 
p 
o 
1 IX) 
+ ~ J e-pt p(t) dt 
po 
or the transform of the integration p(t') with respect to t' 
_ t 
from zero to t is p'P)/p, if e-pt J p(t') dt' is zero for time 
o 
infinity. 
Theorem C ~ The transform for e±ct p,» is equal to 
CD IX) 
oJ e-pt e±ct P(l) dt = oJ e-(P:;:-O)t P,t} dt = P,p:cJ 
if p - c is positive. 
Theorem D ~ If P,(p) is the transform of P,(t), and P,(p) 
is the transform of P" t), then the product of these two trans-
forms is the transform of the integral 
t 
oJ p,(t') P"t-t') dt' 
-r-PRESSURE DROP IN ATMOSPHERES- P(t) 
1.80 
I. 9011----+--\-~11_\\--l__-+_-_+-_1--+_-__I 
2.0011---+-
2.101--+_-_+-\-_1--\ 
2.20~~~~---4~-\~-----+- -~---, 
FIG. 6 - RADIAL FLOW, CONSTANT TERMINAL RATE CASE, PRESSURE 
DROP VS. TIME, Pit) VS. t 
This integral is comparable to the integrals developed by the 
superimposition theorem, and of appreciable use in this 
paper. 
CONSTANT TERMINAL PRESSURE AND 
CONSTANT TERMINAL RATE CASES, 
INFINITE MEDIUM 
The analytics for the constant terminal pressure and ratc 
cases have been developed for limited reservoirs'" when the 
exterior boundary is considered closed or the production rate 
through this boundary is fixed. In determining the volume of 
water encroached into the oil-bearing portion of reservoirs, 
few cases have' been encountered which indicated that the 
sands in which the oil occurs are of limited extent. For the 
most part, the data show that the influx behaves as if the 
water-bearing parts of the formations are of infinite extent, 
because within the productive life of oil recervoirs, the rate of 
water encroachment does not reflect the influence of an ex-
terior boundary. In other words, whether or not the water sand 
is of limited extent, the rate of water encroachment is such as 
if supplied by an infinite medium. 
310 PETROLEUM TRANSACTIONS, AIME December, 1949 
A. F. VAN EVERDINGENAND W. HURST T.P. 2732 
Computing the water influx for an infinite reservoir with the 
help of Fourier-Bessel expansions, an exterior boundary can 
be assumed so far removed from the field radius that the pro-
duction for a considerable time will reflect the infinite caEe. 
Unfortunately, the poor convergence of these expansions inval-
idates this approach. An alternative method consists of using 
increasing values for exterior radius, evaluating the water in-
flux for each radius separately, and then drawing the envelope 
of these curves, which gives the infinite case, Fig. 5. In such 
a procedure, each of the branch curves reflects a water reser-
voir of limited extent. Inasmuch as the drawing of an envelope 
does not give a high degree of acuracy, the solutions for the 
constant terminal pressure and constant terminal rate cases 
for an infinite medium are presented here, with values for 
Q(t> and Pet) calculated directly. 
The constant terminal pressure case was first developed by 
Nicholson" by the application of Green's function to an instan-
taneous circular source in an infinite medium. Goldstein" pre-
sented this solution by the operational method, and Smith13 
employed Carslaw's contour method in its development. Cars-
law and Jaeger"'" later gave the explicit treatment of the 
constant terminal pressure case by the application of the La-
place transformation. The derivation of the constant terminal 
rate case is not given in the literature, and its development 
is presented here. 
The Constant Rate Case 
As already discussed, the boundary conditions for the con-
stant rate case in an infinite medium are that (1) the pres-
sure drop P «, t) is zero initially at every point in the forma-
tion, and (2) at the radius of the field (r = l) we have 
3.81--------1~--+--H~-+_r_=----___+ 
3.61------_t_---+ 
3.41------ _-T----4-+-il------4-------!-
3.2~----_#4_-.....,._t_-__=._._+_--
3.01~---// 
2.8 s 
IXIO 3 5 
( OP) . -- = -1 at all tImes. Or r=l 
A reference to a text on Bessel functions, such as Karman 
and Biot,' pp. 61-63, shows that the general solution for Eq. 
111-3 is given by 
(VI-I) 
where 10 (rYp) and Ko(rYp) are modified Bessel func-
tions of the first and second kind, respectively, and of zero 
order. A and B are two constants which satisfy a second order 
differential equation. Since P (r.p) is the transform of the 
pressure drop at a point in the formation, and because at a 
point not yet affected by production the absolute pressure 
equals the initial pressure, it is required that P (r,p) should 
approach zero as r becomes large. As shown in Karman and 
Biot,' 10 (r Y p ) becomes increasingly large and Ko (r V p) 
approaches zero as the argument (r V p ) increases. There-
fore, to obey the initial condition, the constant A must equal 
zero and (VI-l) becomes 
(VI-2) 
To fulfill the second boundary condition for unit rate of 
production, namely (oPlor).,", = -1, the transform for 
unity gives 
(-~~-)r=1= 
1 
p 
(VI-3) 
by Eq. V-I. The differentiation of the modified Bessel func-
tion of the second kind, Watson's Bessel Functions," W.B.F., 
p. 79, gives Ko'(z) = -K,(z). Therefore, differentiation Eq. 
6.8 
R-200 
6.6 
I 
R-S 6,4 
R-~OO 6.2 
6.0 
5.8 
R-300 
5.6 
3 5 8 
FIG. 7 - RADIAL FLOW, CONSTANT TERMINAL RATE CASE, CUMULATIVE PRESSURE DROP VS. TIME P(t) VS. t 
December, 1949 PETROLEUM TRANSACTIONS, AIME 311 
T,P, 2132 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS 
IN RESERVOIRS 
VI-2, with respect to r at r = 1, gives 
and since 
( OP) .-- = -By p or r=1 
1 
p 
K,\ 'v p ) 
the constant B = lip'!' K, (V p ). Therefore, the transform 
for the pressure drop for the constant rate case in an infinite 
medium is given by 
P,,·,p) = (VI-4) 
p'/'K, (V p ) 
To determine the inver3e of Eq. VI-4 in order to establish 
the pressure drop at radius unity, we can resort to the sim-
plification that for small times the operator p is large. Since 
Kn(z) = ,/~ e 
2z 
(VI-S) 
for z large, W.B.F., p. 202, thell 
1 
P(l,P) (VI-6l 
p"I' 
The inversion for thi" transform 
Foster, Eq. 516, as 
JS given in Campbell and 
2 
(VI-7) 
'/71" 
In brief, Eq. VI-7 states that when t = K T/f/LcRb' is small, 
which can he caw,ed by the boundary radius for the iield, R,,, 
being large, the pressure drop for the unit rate of production 
approximates the condition for linear flow. 
To justify this conclusion, the treatment of the linear flow 
equation, Eq. IV-S, by the Laplace transformation gives 
dx' 
pp'P) (VI-8) 
for which the general solution is the expression 
p'X,p) = Ae-xVP + Be+xV---;;- (VI-9) 
By repeating the reasoning already employed in this develop-
ment, the transform for the pressure drop at x = 0 gives 
P(OVp) = IIp'/' 
which is identical with (VI-6) with p the operator of t 
KT/f/Lc. 
The second simplification for the transform \ VI-4) is to 
consider p small, which is equivalent to considering time, t, 
large. The expansions for Ko (z) and K, (z) are given in Cars-
law and Jaeger," p. 248. 
Ko(~) = - Io(z) 1log~ + 'Y r + ( Z_)' 
2 L 
, (VI-IO) 
(1+~)(~-) (l+~+:)(;)' 
+ (2!)' +-- (3!)' + 
z 
Kn(z) =- (_1)"+1 In(z) 110g-+'Y l , 2 ( 
( _~)n+2' 1 00 'J 
+ - (_1)" ~ ------ [ :::; m-' + 
2 ,,0 r! (n+r)! m~1 
1 n-l ( Z )_n+2' (n-r-l)! + - :::; (-1)' - ----,-
2 ." 2 r! 
(VI-ll) 
where 'Y is Euler's constant 0.57722, and the logarithmic term 
consists of natural logarithms. When z is small 
z 
Ko (z) ~ - [log "2 -t- 'Y] 
K,(z) ~ liz 
Therefore, Eq. VI-4 becomes 
-log p + (Jog 2 - 'Y) 
P (',1') = --:)--
~p p 
(VI-12) 
(VI-l3) 
(VI-14) 
The inversion for the first term on the right is given by Camp-
bell and Foster, Eq. 892, and the inverse of the second term by 
FIG. 8 - CONSTANT RATE OF PRODUCTION IN THE STOCK TANK, 
ADJUSTING FOR THE UNLOADING OF FLUID IN THE ANNULUS, Pit) 
VERSUS t where Z = c/27rfcR,,', AND c is the VOLUME OF FLUID UN-
LOADED FROM THE ANNULUS, CORRECTED TO RESERVOIR CONDI-
TIONS, PER ATJvlOSPHERE BOTTOM-HOLE PRESSURE DROP, PER UNIT 
SAND THICKNESS. 
Eq. V-I. Therefore, the pressure drop at the boundary of the 
field when t i,; large is given by 
1 
p,,) = -2 [log 4t - 'Y ] 
1 
-- [log t + 0.80907 ] 
2 
(VI-IS) 
The solution given bv Eq. VI-IS is the solution of the con-
tinuous point source problem for large time 1. The relationship 
has been applied to the flow of fluids by Bruce," Elkins," and 
others, and is particularly applicable for study of interference 
between flowing welk 
The point source solution originally developed by Lord Kel-
vin and discm'~.ed in Carsl aw18 can be expressed as 
1 :r e-" 1 (1 ) 
P"',I) =- ,J -- dn =--) -Ei -- r 
2 It n 2 4t 
(VI-16l 
often referred to as the logarithmic integral or the Ei-func-
tion. Its values are given in Tahle" of Sine, Cosine, and Expo-
nential Integrah Volumes I and II, Federal Works Agency, 
W.P.A., City of New York. For large values of the time, t, 
I 
Eq. VI-16 reduces to P". 1) = -- [log 4t - 'Y] which is Eq. 
2 
VI-IS, and this relation is accurate for values of t> 100. 
312 PETROLEUM TRANSACTIONS, AIME December, 1949 
A. F. VAN EVERDINGEN AND W. HURST T.P. 2732 
By this development it is evident that the point source solu-
tion does not apply at a boundary for the determination of the 
pressure drop when t is small. However, when the radius, R b , 
is small, such as a well radius, even small values of the abso-
lute time, T will give large values of the dimensionless time t, 
and the point source solution is applicable. On the other 
hand, in considering the presmre drop at the periphery of a 
field (in which case Rb can have a large numerical value) the 
value of t can be easily less than 100 even for large values of 
absolute time, T. Therefore, for intermediate times, the rig-
orous solution of the constant rate case must be used, which 
we will now proceed to oLtain. 
To develop the explicit solution for the constant terminal 
rate case, it is necessary to invert the Laplace transform,Eq. 
V 1-4., by the Mellin's inversion formula. The path of integra-
tjon for this transform is described by the "cut" along the 
negative real axis, Fig. 2, which give6 a single valued function 
on each side of the "cut." That is to say that Path AB re-
quired by j<;q. V -2 is equal to the Pat11 AD and CB, both of 
which are descnbed by a semi-circle of radius infillity. Since 
lts integration is zero JIl the second and third quadrant, this 
leaves the mtegratlOn along l'atils Du and UC equal LO AB. 
The integration on tlie upper portion of the "cut' can be ob-
tained by making A = u' e +i~ which yields 
At, At, -
1 _JC(e -e )Ko(VAr) 
--:;-:-- J d A 
~'IT"1 0 
A"I' K,( V A 
-u't -u'L 
1 J:; (e '- e -) Ko (u e' r) du 
-~ J ~~-
trIO lrr 17r 
u' e K, ( u e' ) (VI-17) 
Carslaw and Jaeger" (page 249) shows that modified Bassel 
i7l" 
±-
2 
functions of the first and second kind of arguments z e 
can be expressed by the regular Bessel functions as complex 
values, as follows: 
and 
The 
171" 
±-
2 
10 (z e ) 
i7l" 
±-
2 
Ko (z e ) 
I, (z e 2) 
171" 
±-
2 
K,(z e ) 
L(z) 
± .L(z). 
71" 
-2 [J,(z) + i Y,(z) ] 
substitution of the corresponding 
(VI-I8) 
values for 
Likewise, the integration along the under portion of the 
negative real "Cilt" is expressed by A = 
At, At, -
} _~_-=-~l Ko (V A ~_ 
271" CXJ 
1 
A3;" K,(V A 
-171" 
u' e 
-1 CXJ (e-u't'_e-u2t2) Ko (u e-i7l"/2 r) du 
-J 
71" 0 -i7l"/2 -i7l"/2 
u e K,(u e ) 
Using Eq. VI-18, yields the relationship 
and 
-u2t, -u2t. 
1 CXJ(e -e -)[Y,(u)Jo(ur)-J,(u)YO(ur)] 
-;;:-) -----~,'(u) + Y,'(u)] 
du 
(VI-20) 
The integration along Paths DO and OC is the sum of the 
relations VI-I9 and VI-20, or 
Pcr. '1) - Per. t,) = 
2 ~(e-u'tl_e-u't2) [Y,(u) .To(ur) -J,(U) Yo(ur)] du 
-;;:-) u'[J,'(u) + Y,'(u)] 
Initially, that is at time zero, the cumulative pressure drop at 
any point in the formation is zero, Per. t~o) = O. Hence, the 
pressure drop since zero time equals: 
-u2t 
2 CXJ (1- e ) [J,(U) Yo(u r) - Y,(u) Jo(u r)] du 
Pe,·.t) = -;;:-). ---~-. u'[J,'(u) + Y,'(u)] 
(VI-2I) 
which is the explicit solution of the constant terminal rate case 
for an infinite medium. 
To determine the cumulative pressure drop for a unit rate 
of production at the well bore or field radius, (where r = 1) 
then Eq. VI-21 changes to 
-u't 
2 CXJ(I-e ) [J,(U) Yo(u)-Y,(u) Jo(u)] du 
P(l.t) =--:;;0.1 u' [J,'(u) +Y,'(u)] 
By the recurrence formula given in W.B.F., p. 77 
2 
J,(u) Yo(u) - L(u) Y,(u) = 
7I"U 
Equation VI-22 simplifies to 
4 CXJ (1- e-u't) du 
p(t)=,f 
71" 0 u" [J,2(U) + Y,'(u)] 
Constant Terminal Pressure Case 
(VI-22) 
(VI-23) 
(VI-24) 
As already shown, the transform of the pressure drop in 
an infinite medium is P (r.p) = B Ko ( v'p r). In the constant 
terminal pressure case it is assumed that at all times the pres-
sure drop at r = 1 will be unity, which is expressed as a 
transform by Eq. V-I 
P(1.P) = lip 
By solving for the constant B at r = 1 in the above formula, 
i7l"/2 i7l"/2 fidB I K(v') h Ko (u e r) and K, (u e ) from Eq. VI-18 in Eq. VI-17 we n = 1 PoP ,so t at the transform for the 
gives the integration along the upper portion of the negative pressure at any point in the reservoir is expressed by 
real "cut" as 
-u2t -u't, 
1 ct:J (e '_e 
r_~~~ 
) [Y,(u) -Jo(ur) -J,(U) Yo(ur)] du 
. u' [J,'(u) + Y,'(u) ] 
(VI-I9) 
where the imaginary term has been dropped. 
- Ko(Vp r) 
p(r.p) = ---- (VI-2S) 
p Ko( v'p) 
The comparable solution of VI-25 for a cumulative pressure 
drop can be developed as before by considering the paths of 
Fig. 2, with a pole at the origin, to give the solution 
December, 1949 PETROLEUM TRANSACTIONS, AIME 313 
T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS 
IN RESERVOIRS 
P(r, (1)-P(r, t,) = 
2 ~(e-u't'_e-u't')[lo(u) Yo(ur)-Yo(u) Jo(ur)]du 
71" ) u'[lo'(u) + Yo'(u)] (VI-26) 
If we are interesterl in the cumulative fluid influx at the field 
radius, r = 1, then the relationship Eq. IV-3' applies, or 
~ ( oP ) Q(t) = J -- dt 
o Of r= 1 
(IV-3) 
The determination of the transform of the gradient of the 
pressure drop at the field's edge follows from Eq. VI-25, 
(~~(:.~ )r=l= --~~/f~~;vp~) 
since K: (z) = - K, (z). Since the pressure drop P (r, t) corre-
sponds to the difference between the initial and actual pres· 
sure, the transform 0.£ the gradient of the actual pressure at 
r = 1 is given by 
(-~) or r=1 
or 
K,( V p ) 
which corresponds to the integrand of Eq. IV·3. Further, from 
the definition given by Theorem B, namely, that if P (p) is the 
t 
transform of P(th then the transform of oJ p(t') dt' is given by 
P (p) I p and the La place transform for Q,,) is expressed by 
(VI-27) 
The application of the Mellin's inversion formula to Eq. VI-27 
follows the paths shown in Fig. 2, giving 
, 
-u t 
(1- e .\ du 4 IX' 
Q(t) = - J -,---------
71"' 0 u [Jo'(u) + Yo2(U) ] 
(VI.28) 
With respect to the transform Q(P)' there is the simplification 
that for time small, p is large, or Eq. VI-27 reduces to 
Q(P) = lip'!' 
and the inversion is as before 
2 
Q(t) = --- 1'/' 
V--:;-
(VI.29) 
(VI-30) 
which is identical to the linear flow case. For all other values 
of the time, Eq. VI·28 must be solved numerically. 
Relation Between Q(p) and Pip) 
It is evident from the work that has already gone before, 
that the Laplace transformation and the superimposition the· 
orem offer a basis for interchanging the constant terminal 
pressure to the constant terminal rate case, and vice versa. In 
any reservoir study the essential interest is the analyses of 
the flow either at the well bore or the field boundary. The 
purpose of this work is to determine the relationship between 
Q (t), the constant terminal pressure case, and P (t), the con-
stant terminal rate case, which explicitly refer to the boundary 
r = 1. Therefore, if we conceive of the influx of fluid into a 
TABLE I - Radial Flow, Constant Terminal Pressure 
and Constant Terminal Rate Cases for Infinite 
Reservoirs 
1.0(10)-' 
5.0 " 
1.0(10)-1 
1.5 " 
2.0 " 
2.5 " 
3.0 
4.0 " 
5.0 " 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
1.0 
1.5 
2.0 
2.5 
3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 
1.0(10)1 
1.5 " 
2.0 H 
2.5 " 
3.0 " 
4.0 " 
5.0 " 
6.0 " 
7.0 " 
8.0 (( 
9.0 " 
1.0(10)' 
1.5 " 
2.0 " 
2.5 " 
3.0 " 
4.0 " 
5.0 " 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
1.0(10)' 
1. 5(10)1 
2.0 " 
2.5 " 
3.0 " 
4.0 " 
5.0 " 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
1.0(10)' 
1.5 " 
2.0 " 
2.5 " 
3.0 " 
4.0 " 
5.0 " 
6.0 " 
7.0 " 
8.0 H 
9.0 " 
1.0(10)' 
1.5 " 
2.0 " 
2.5 " 
3.0 H 
4.0 u 
5.0 " 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
1.0(10)11 
1.5 " 
2.0 " 
2.5 " 
3.0 " 
4.0 " 
5.0 " 
6.0 H 
7.0 " 
8.0 " 
9.0 " 
1.0(10)" 
0.112 
0.278 
0.404 
0.520 
0.606 
0.689 
0.758 
0.898 
1.020 
1.140 
1.251 
1.359 
1.469 
1.570 
2.032 
2.442 
2.838 
3.209 
3.897 
4.541 
5.148 
5.749 
6.314 
6.861 
7.417 
9.965 
1.229(10)1 
1. 455 " 
1.681 " 
2.088 !' 
2.482 " 
2.860 " 
3.228 " 
3.599 " 
3.942 " 
4.301 " 
5.980 " 
7.586 " 
9.120 " 
10.58 
13.48 " 
16.24 " 
18.97 " 
21. 60 " 
24.23 " 
26.77 " 
29.31 " 
P 
(t) 
0.112 
0.229 
0.315 
0.376 
0.424 
0.469 
0.503 
0.564 
0.616 
0.659 
0.702 
0.735 
0.772 
0.802 
0.927 
1.020 
1.101 
1.169 
1.275 
1.362 
1.436 
1.500 
1.556 
1.604 
1.651 
1.829 
1.960 
2.067 
2.147 
2.282 
2.388 
2.476 
2.550 
2.615 
2.672 
2.723 
2.921 
3.064 
3.173 
3.263 
3.406 
3.516 
3.608 
3.684 
3.750 
3.809 
3.860 
1.5(10)' 
2.0 " 
2.5 " 
3;0 " 
4.0 " 
5.0 " 
6.0 " 
7.0 H 
8.0 " 
9.0 " 
1.0(10)' 
1.5 " 
2.0 " 
2.5 " 
3.0 " 
4.0 " 
5.0 
6.0 " 
7.0 " 
8.0 (( 
9.0 " 
1.0(10)' 
1.5 " 
2.0 " 
2.5 " 
3.0 " 
4.0 " 
5.0 " 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
1.0(10)' 
1.5 " 
2.0 " 
2.5 " 
3.0 " 
4.0 '( 
5.0 " 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
1.0(10)1 
TABLE I - Continued 
1. 828(10)' 
2.398 " 
2.961 " 
3.517 " 
4.610 " 
5.689 " 
6.758 " 
7.816 " 
8.866 " 
9.911 " 
10.95 " 
1. 604(10)' 
2.108 " 
2.607 " 
3.100 " 
4.071 " 
5.032 " 
5.984 " 
6.928 " 
7.865 " 
8.797 " 
9.725 " 
1.429(10), 
1. 880 " 
2.328 " 
2.771 " 
3.645 " 
4.510 " 
5.368 " 
6.220 " 
7.066" 
7.909 " 
8.747 " 
1.288(10)' 
1. 697 " 
2.103 " 
2.505 " 
3.299 " 
4.087 " 
4.868 " 
5.643 " 
6.414 " 
7.183 " 
7.948 " 
1.5(10)" 
2.0 " 
2.5 " 
3.0 " 
4.0 " 
5.0 " 
6.0 " 
7.0 H 
8.0 " 
9.0 " 
1.0(10)1' 
1.5 " 
2.0 " 
4.136(10)' 
5.315 " 
6.466 " 
7.590 " 
9.757 
11.88 " 
13.95 " 
15.99 " 
18.00 " 
19.99 " 
21. 96 " 
3.146(10)3 
4.079 " 
4.994 " 
5.891 " 
7.634 " 
9.342 " 
11.03 " 
12.69 " 
14.33 " 
15.95 " 
17 .56 " 
2.538(10)' 
3.308 " 
4.066 
4.817 " 
6.267 " 
7.699 " 
9.113 " 
10.51 " 
11.89 " 
13.26 " 
14.62 " 
2126(10)5 
2.781 " 
3.427 " 
14.064 " 
5.313 " 
6.544 " 
7.761 " 
8.965 " 
10.16 " 
11.34 " 
12.52 " 
1.17(10)10 
1.55 " 
1.92 " 
2.29 " 
3.02 " 
3.75 " 
4.47 " 
5.19 " 
5.89 " 
6.58 " 
7.28 " 
1.08(10)" 
1.42 " 
314 PETROLEUM TRANSACTIONS, AIME December, 1949 
A. F. VAN EVERDINGEN AND W. HURST T.P. 2732 
T ABLE II - Constant Terminal Pressure Case 
Radial Flow, Limited Reservoirs 
R = 1.5 R -= 2.0 H == 2.5 
1 
R = 3.0 
", = 2.8899 
", = 9.3452 
", = 1.3606 
a~ == 4.6458 ,,"._', -_-- 0.8663 I 3.0875 
", = 0.6256 
", = 2.3041 
5.0(10)-2 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
1.0(10)-1 
1.1 " 
1.2 H 
1.3 " 
1.4 " 
1.5 " 
1.6 " 
1. 7 " 
I 8 " 
1: 9 " 
2.0 " 
2.1 
2.2 " 
2.3 " 
2.4 " 
2.5 " 
2.6 " 
2.8 " 
3.0 " 
3.2 " 
3.4 " 
3.6 I( 
3.8 " 
4.0 " 
4.5 " 
5.0 H 
6.0 " 
7.0 " 
8.0 " 
0.276 
0.304 
0.330 
0.354 
0.375 
0.395 
0.414 
0.431 
0.446 
0.461 
0.474 
0.486 
0.497 
0.507 
0.517 
0.525 
0.533 
0.541 
0.548 
0.554 
0.559 
0.565 
0.574 
0.582 
0.588 
0.594 
0.599 
0.603 
0.606 
0.613 
0.617 
0.621 
0.623 
0.624 
5.0(10)-' 
7.5 " 
10(10)-1 
1.25 " 
1.50 " 
1. 75 " 
2.00 " 
2.25 " 
2.50 " 
2.75 " 
3.00 " 
3.25 " 
3.50 " 
3.75 " 
4.00 " 
4.25 " 
4.50 " 
4.75 " 
5.00 " 
5.50 " 
6.00 " 
6.50 " 
7.00 " 
7.50 " 
8.00 " 
9.00 " 
1.00 
1.1 
1.2 
1.3 
1.4 
1.6 
1.7 
1.8 
2.0 
2.5 
3.0 
4.0 
5.0 
gm U(1,?)-l g:Eg~! U(1,?)-l 
0.404 2.0" 0.599 5.0" 
0.458 2.5" 0.681 6.0" 
0.507 3.0" I 0.758 7.0" 
0.553 3.5" 0.829 8.0" 
0.597 4.0" 0.897 9.0" 
0.638 4.5" 0.962 1.00 
0.678 5.0" 1.024 1.25 
0.715 5.5" 1.083 1.50 
0.751 6.0" 1.140 1.75 
0.785 6.5" 1.1951 2.00 
0.817 7.0" ;248 2.25 
0.848 7.5" 1.229 2.50 
0.877 8.0" 1.348 2.75 
0.905 8.5" 1.395 3.00 
0.932 9.0" 1.440 3.25 
0.958 9.5" 1.484 3.50 
0.983 1.0 1.526 3.75 
1. 028 1.1 I. 605 4.00 
1. 070 1.2 1. 679 4.25 
1.108 1.3 1. 747 4.50 
1.143 1.4 1.811 4.75 
1.174 1.5 1.870 5.00 
1.203 1.6 1.924 5.50 
1.253 1.7 1.975 6.00 
1.295 1.8 2.022 6.50 
1.330 2.0 2.106 7.00 
1.358 2.2 2.178 7.50 
1.382 2.4 2.241 8.00 
1.402 2.6 2.294 9.00 
1.432 2.8 2.340 10.00 
1.444 3.0 2.380 11.00 
1.453 3.4 2.444 12.00 
1.468 3.8 2.491 14.00 
;.487 4.2 2.525 16.00 
1.495 4.6 2.551 18.00 
1.499 5.0 2.570 20.00 
1.500 6.0 2.599 22.00 
8.0 2.619 
Q't) 
0.755 
0.895 
1.023 
1.143 
1.256 
1.363 
1.465 
1.563 
1.791 
!.D97 
2.184 
2.353 
2.507 
2.646 
2.772 
2.886 
2.990 
3.084 
3.170 
3.247 
3.317 
3.381 
3.439 
3.491 
3.581 
3.656 
3.717 
3.767 
3.809 
3.843 
3.894 
3.928 
3.951 
3.£67 
3.985 
3.993 
3.997 
3.999 
3.999 
4.000 
I 
7.0 2.613 24.00 
9.0 2.622 I 
_~~~~~~~~ ____ ..::10,-".0~_~2~.~62_4~. __ 
R = 3.5 
", = 0.4851 
", == 1.8374 
1.00 
1.20 
1. 40 
1.60 
1.80 
2.00 
2.20 
2.40 
2.60 
2.80 
3.00 
3.25 
3.50 
3.75 
4.00 
4.25 
4.50 
4.75 
5.00 
5.50 
6.00 
6.50 
7.00 
7.50 
8.00 
8.50 
9.00 
9.50 
10.00 
11 
12 
13 
14 
15 
16 
17 
18 
20 
25 
30 
35 
40 
1.571 
1. 761 
1. 940 
2.111 
2.273 
2.427 
2.574 
2.715 
2.849 
2.976 
3.098 
3.242 
3.379 
3.507 
3.628 
3.742 
3.850 
3.951 
4.047 
4.222 
4.378 
4.516 
4.639 
4.749 
4.846 
4.932 
5.009 
5.078 
5.138 
5.241 
5.321 
5.385 
5.435 
5.476 
5.506 
5.531 
5.551 
5.579 
5.611 
5.621 
5.624 
5.625 
TABLE II - Continued 
R = 4.0 
", = 0.3935 
", = 1.5267 
2.00 
2.20 
2.40 
2.60 
2.80 
3.00 
3.25 
3.50 
3.75 
4.00 
4.25 
4.50 
4.75 
5.00 
550\ 
6.00 
6.50 
7.00 
7.50 
8.00 
8.50 
9.00 
9.50 
10 
11 
12 
13 
14 
15 
16 
17 
18 
20 
22 
24 
26 
30 
34 
38 
42 
46 
50 
2.442 
2.598 
2.748 
2.893 
3.034 
3.170 
3.334 
3.493 
3.645 
3.792 
3.932 
4.068 
4.198 
4.323 
4.560 
4.779 
4.982 
5.169 
5.343 
5.504 
5.653 
5.790 
5.917 
6.035 
6.246 
6.425 
6.580 
6.7'2 
6.825 
6.922 
7.004 
7.076 
7.189 
7.272 
7.332 
7.377 
7.434 
7.464 
7.481 
7.490 
7.494 
7.497 
R = 4.5 
", = 0.3296 
", = 1.3051 
2.5 
3.0 
3.5 
4.0 
4.5 
5.0 
5.5 
6.0 
6.5 
7.0 
7.5 
8.0 
8.5 
9.0 
9.5 
10 
11 
12 
13 
14 
15 
16 
18 
20 
22 
24 
26 
28 
30 
34 
38 
42 
46 
50 
60 
70 
80 
90 
100 
2.835 
3.196 
3.537 
3.859 
4.165 
4.454 
4.727 
4.986 
5.231 
5.464 
5.684 
5.892 
6.089 
6.276 
6.453 
6.621 
6.930 
7.208 
7.457 
7.680 
7.880 
8.060 
8.365 
8.611 
8.809 
8.968 
9.097 
9.200 
9.283 
9.404 
9.481 
9.532 
9565 
9.586 
9.612 
9.621 
9.623 
9.624 
9.625 
R = 5.0 
", = 0.2823 
", = 1.1392 
3.0 
3.5 
4.0 
4.5 
5.0 
5.5 
6.0 
6.5 
7.0 
7.5 
8.0 
8.5 
9.0 
9.5 
10 
11 
12 
13 
14 
15 
16 
18 
20 
22 
24 
26 
28 
30 
34 
38 
42 
46 
50 
60 
70 
80 
90 
100 
120 
3.195 
3.542 
3.875 
4.193 
4.499 
4.792 
5.074 
5.345 
5.605 
5.854 
6.094 
6.325 
6.547 
6.760 
6.965 
7.350 
7.706 
8.035 
8.339 
8.620 
8.879 
9.338 
9.731 
10.07 
10.35 
10.59 
10.80 
10.98 
11.26 
11.46 
11.61 
11. 71 
11.79 
11.91 
11.96 
11.98 
11.99 
12.00 
12.0 
R = 6.0 
", = 0.2182 
", = 0.9025 
6.0 
6.5 
7.0 
7.5 
8.0 
8.5 
9.0 
9.5 
10.0 
10.5 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
22 
24 
25 
31 
35 
39 
51 
60 
70 
80 
PO 
100 
JlO 
120 
130 
140 
150 
160 
180 
200 
220 
5.148 
5.440 
5.724 
6.002 
6.273 
6.537 
6.795 
7.047 
7.293 
7.533 
7.767 
8.220 
8.651 
9.063 
9.456 
9.829 
10.19 
10.53 
10.85 
11.16 
11.74 
12.26 
12.50 
13.74 
14.40 
14.93 
16.05 
16.56 
16.91 
17.14 
17.27 
17.36 
17.41 
17.45 
17.46 
17.48 
17.49 
17.49 
17.50 
17.50 
17.50 
weB or field as a constant rate problem, then the actual cumu-
lative fluid produced as a function of the cumulative pressure 
drop is expressed by the superposition relationship in Eq. 
IV-14 as 
t d~P 
Q(T) = 27rfCRb ' J --- Q(t-t') dt' 
o dt' 
(IV-14) 
when ~P is the cumulative pressure drop at the well bore 
affected by producing the well at constant rate which is estab-
lished by 
q,'1") IL P(t) 
~P = ~~.~-
2rrK 
The substitution of Eg. 1'/-11 ill IV-14 give; 
q(T) flLCR b ' ~ d P(t') 
Q(T) = K ) ---;w-- Q(t'l') 
(IV-H) 
dt' 
Since the rate is constant, Q(T)=q(T) x T, and as t=KT/flLcR,; 
this relation becomes 
t dP(t') 
t = f -- Q(tl') dt' (VI-31) 
o dt' 
To express Eq. VI-31 in transformation form, the transform 
for t is lip', Campbell and Foster, Eq. 408.1. The transform 
for P (t) at r = I is P (p), and it follows from Theorem A that 
dP(t) 
the transform of ~~~ is pP(V) as the cumulative pressure 
dt 
drop P, t) for constant rate is zero at time zero. Finally from 
Theorem D, the transform for the integration of the form Eq. 
VI-31 is equal to the product of the transforms for each of the 
two terms in the integrand, or 
R _ 7.0 
", = 0.1767 
", = 0.7534 
9.00 
9.50 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
22 
24 
26 
28 
30 
35 
40 
45 
50 
60 
70 
80 
90 
100 
120 
140 
160 
180 
200 
500 
6.851 
7. ,27 
7.389 
7.902 
8.397 
8.876 
9.341 
9.791 
10.23 
10.65 
11.05 
11.46 
11.85 
12.58 
13.27 
13.92 
14.53 
15.11 
16.39 
17.49 
18.43 
19.24 
20.51 
21.45 
22.13 
22.63 
23.00 
23.47 
23.71 
23.85 
23.92 
23.96 
24.00 
TABLE II - Continued 
R _ 8.0 
", = 0.1476 
a, = 0.6438 
t 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
22 
24 
26 
28 
30 
34 
38 
40 
45 
50 
55 
60 
70 
80 
90 
100 
120 
140 
160 
180 
200 
240 
280 
320 
360 
400 
500 
6.861 
7.398 
7.920 
8.431 
8.930 
9.418 
9.895 
10.361 
10.82 
11.26 
11. 70 
12.13 
12.95 
13.74 
14.50 
15.23 
15.92 
17.22 
18.41 
18.97 
20.26 
21.42 
22.46 
23.40 
24.98 
26.26 
27.28 
28.11 
29.31 
30.08 
30.58 
30.91 
31.12 
31.34 
31.43 
31.47 
31.49 
31.50 
31.50 
R _ 9.0 
a, ~. 0.1264 
a, = 0.5740 
t 
10 
15 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
H 
46 
48 
50 
52 
54 
56 
58 
60 
65 
70 
75 
80 
85 
90 
95 
100 
120 
140 
160 
180 
200 
240 
280 
320 
360 
400 
440 
480 
7.41, 
9.945 
12.2613.13 
13.98 
14.79 
15.59 
16.35 
7.10 
17.82 
18.52 
19.19 
19.85 
20.48 
21.09 
21.69 
22.26 
22.82 
23.36 
23.89 
24.39 
24.88 
25.36 
26.48 
27.52 
28.48 
29.36 
30.18 
30.93 
31.63 
32.27 
34.39 
35.92 
37.04 
37.85 
38.44 
39.17 
39.56 
39.77 
39.88 
39.94 
39.97 
39.98 
R _10.0 
"1 = 0.1104 
", = 0.4979 
t 
15 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
44 
46 
48 
50 
52 
54 
56 
58 
60 
65 
70 
75 
80 
85 
90 
95 
100 
120 
140 
160 
180 
200 
240 
289 
320 
360 
400 
440 
480 
9.965 
12.32 
13.22 
14.09 
14.95 
15.78 
16.59 
17.38 
18.16 
18.91 
19.65 
20.37 
21.07 
21. 7(; 
22.42 
23.07 
23.71 
24.:13 
24.94 
25.53 
26.11 
26.67 
23.02 
29.29 
30.49 
31.61 
32.67 
33.66 
H.60 
35.48 
38.51 
40.89 
42.75 
44.21 
45.36 
46.95 
47.94 
48.54 
48.91 
49.14 
49.28 
49.36 
Oecember, 1949 PETROLEUM TRANSACTIONS, AIME 31! 
T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS 
IN RESERVOIRS 
1 
p' 
(VI-32) 
Evidence of this identity can be confirmed by substituting 
Eqs. VI·4 and VI-27 in Eq. VI-32. In brief, Eq. VI-32 is the 
relationship between constant terminal pressure and constant 
terminal rate cases. If the Laplace transformation for one is 
known, the transform for the other is established. This inter-
change can only take place in the transformations and the 
final solution must be by inversion. 
Computation of PIt) and Q(O 
To plot P(t) and Q(t) as work-curves, it is necessary to de-
termine numerically the value for the integrals shown in Eqs. 
VI-24 and VI-28. In treating the infinite integrals for PIt) and 
Q,t), the only difficult part is in establishing the integrals for 
~mall values of u. For larger values of u the integrands con-
verge fairly rapidly, and Simpson's rule for numerical integra-
tion has proved sufficiently accurate. 
To determine the integration for Q(t) in the region of the 
origin, Eq. VI-28 can be expres;;ed as 
-u't 
4 .0 (1- e ) du 
Qo(t) =-;;) u'[Jo'(u) +Yo'(u)J (VI-33) 
, 
where the value for 0 is taken such that 1 _ e-u t ,...., u't, 
whieh is true fgr u't equal to or less than 0.02, or 0 = \' 0.02/ t 
and the simplification for Eq. VI-33 becomes 
4t Ii du 
Qo(t) = ... ,)' u[Jo'(u) + Yo'(u)J 
For n less than 0.02, J 0 (u) = 1, and 
2 u 2 
Yo (u) ,...., - i log - + "Y ~ = - i log u - 0.11593 ~ 
... 2 ... 
As the logarithmic term is most predominant in the denom. 
inator for small values of u, this eqnation simplifies to 
a du t 
Q (t) = t f -=-:--------=-::-
,1 o· u [log u - 0.11593], [0.11593 -log 6J 
(VI-34} 
The integration for P, t) 
4 0=0.02 
close to the origin is expressed by 
-u't 
(l-e ) du 
P (t) =, J a ... 0 (VI-35 ) u'[J,'(u) + y,'(u)] 
For u equal to or less than 0.02, J,(u) = 0, and Y,(u) = 
2/ ... u so that Eq. VI-35 reduces to 
a 
P (t) = J a n 
-u't 
(1- e ) 
----- du 
u 
If we let n = u't 
-n 
1 .o't (1 - e ) 
P (t) = -- j ----- dl! 
" 2 o· n 
Further, 
.O't (l - e -n) dn 
.J 
o n 
-n 
/ (1- e ) dn 
. n 
Since Euler's constant "Y is equal to 
Substitution of this relation in Eq. VI-38 gives 
.o't(l-e-n)dn ~e-" ldn 
J --- ="Y + , j - dn - , J -
o n "·t n o-t n 
(VI-36) 
(VI-37) 
( VI·38) 
and sinee the seeond term on the right is the Ei·funetion al-
ready discussed in the earlier part of this work, Eq. VI-37 
reduces to 
P (t) a 
1 
[ 'Y - Ei (- o't) + log 6't J 
2 
(VI-39) 
TABLE III - Constant Terminal Rate Case Radial Flow - Limited Reservoirs 
R , _ 1.5 R _ 2.0 R _ 2.5 
I 
R _ 3.0 R _ 3.5 
I 
R _ 4 R = 4:5~--
f3, = 6.3225 f3, = 3.1965 f3, = 2.1564 f3, = 1.6358 f3, = 1.3218 f3, = 1.1120 fJ, = 0.9609 
fJ, = 11.924 f3, = 6.3118 f3, = 4.2230 fJ, = 3.1787 f3, = 2.5526 fJ, = 2.1342 fJ, = 1.8356 
t PIt) t I PIt) t PIt) t PIt) t P(t) t P t P(t) (t) 
-.----------------~-
6.0(10)-' 0.251 22(10)-1 0.443 4.0(10)-1 0.565 5.2(10) 0.627 1.0 0.802 1.5 0.927 2.0 1.023 
8.0 " 0.288 2.4 " 0.459 4.2 " 0.576 5.4 " 0.636 1.1 0.830 1.6 0.948 2.1 1.040 
1.0(10)-1 0.322 2.6 H 0.476 4.4 " 0.587 5.6 " 0.645 1.2 0.857 1.7 0.968 2.2 1.056 
1.2 " 0.355 2.8 " 0.492 4.6 " 0.598 6.0 It 0.662 1.3 0.882 1.8 0.988 2.3 1.072 
1.4 " 0.387 3.0 " 0.507 4.8 " 0.608 6.5 u 0.683 1.4 0.906 1.9 1.007 2.4 1.087 
1.6 " 0.420 3.2 H 0.522 5.0 If 0.618 7.0 If 0.703 1.5 0.929 2.0 1.025 2.5 1.102 
1.8 " 0.452 3.4 u 0.536 5.2 H 0.628 7.5 If 0.721 1.6 0.951 l.2 1.059 2.6 1.116 
2.0 H 0.484 3.6 " 0.551 5.4 .. 0.638 8.0 " 0.740 1.7 0.973 2.4 1.092 2.7 1.130 
2.2 " 0.516 3.8 " 0.565 5.6 " 0.647 8.5 " 0.758 1.8 0.994 2.6 1.123 2.8 1.144 
2.4 " 0.548 4.0 " 0.579 5.8 " 0.657 9.0 " 0.776 1.9 1.014 2.8 1.154 2.9 1.158 
2.6 u 0.580 4.2 " 0.593 6.0 /4 0.666 9.5 " 0.791 2.0 1.034 3.0 1.184 ~.O 1.171 
2.8 u 0.612 4.4 " 0.607 6.5 " 0.688 1.0 0.806 2.25 1.083 3.5 1.255 3.2 1.197 
3.0 " 0.644 4.6 " 0.621 7.0 It 0.710 1.2 0.865 2.50 1.130 4.0 1.324 i 1.222 
3.5 " 0.724 4.8 " 0.634 7.5 " 0.731 1.4 0.920 2.75 1.176 4.5 1.392 3.6 1.246 
4.0 " 0.804 5.0 " 0.648 8.0 " 0.752 1.6 0.973 3.0 1.221 5.0 1.460 .8 1.269 
4.5 " 0.884 6.0 " 0.715 8.5 " 0.772 2.0 1.076 4.0 1.401 5.5 1.527 4.0 1.292 
5.0 " 0.964 7.0 " 0.782 9.0 .. 0.792 3.0 1.328 5.0 1.579 6.0 1.594 4.5 1.349 
5.5 " 1.044 8.0 u 0.849 9.5 " 0.812 4.0 1.578 6.0 1.757 6.5 1.660 5.0 1.403 
6.0 " 1.124 9.0 " 0.915 1.0 0.832 5.0 1.828 7.0 1.727 5.5 1.457 
1.0 0.982 2.0 1.215 8.0 1.861 6.0 1.510 
2.0 1.649 3.0 1.596 9.0 1.994 7.0 1.615 
3.0 2.316 4.0 1.977 10.0 2.127 8.0 1. 719 
5.0 3.649 5.0 2.358 9.0 1.823 
I 
.0'0 1.927 
11.0 2.031 
12.0 2.135 
13.0 2.239 
14.0 2.343 
15.0 2.447 
316 PETROLEUM TRANSACTIONS, AIME December, 1949 
A. F. VAN EVERDINGEN AND W. HURST T.P. 2732 
The values for the integrands for Eqs. VI·24 and VI·23 
have been calculated from Bessel Tables for or greater than 
()'o2 as given in W.B.F., pp. 666·697. The calculations have 
been somewhat simplified by using the square of the modulus 
of 
IHo(l) (u) l=iJo(u) +i Yo(u) I and iH,<l)(u) i=iJ,(u) +i Y,(u) I 
which are the Bessel functions of the third kind or the Hankel 
functions. 
Table I shows the calculated values for Q(t) and p(t) to 
three significant figures, starting at t = 0.01, the point where 
linear flow. and radial flow start deviating. P (t) is calculated 
only to t = 1,000 since beyond this range the point source 
solution of Eq. VI-IS applies. The values for Q,t) are given 
lip to t = 10". 
The reader may reproduce these data as he sees fit; Fig. 4 
is an illustrative plot for Q(t), and Fig. 7 is a semi-logarithmic 
relationship for P't1-
LIMITED RESERVOIRS 
As already mentioned, tIte solutions for limited reservoirs 
of radial symmetry have been developed by the Fourier-Bessel 
type of expansion.""" Their introduction here is not only to 
show how the solutions may be arrived at by the Laplace 
transformation, but also to furnish data for P(1l and Q(t) 
curves when such cases are encountered in practice. 
No Fluid Flow Across Exterior Boundary 
The first exam pIe considered is the constant terminal pres-
sure case for radial flow of limited extent. The boundary con-
ditions are such that at the well bore or field's edge, r = 1, 
the cumulative pressure drop is unity, and at some distance 
removed from this boundary at a point in the reservoir r = R, 
there exists a restriction such that no fluid can flow past this 
barrier so that at that point ( aP_) = O. 
'Or r=R 
The general solution of Eq. VI-l still applies, but to fulfill 
the boundary conditions it is necessary to re-determine values 
for constants A and B. The transformation of the boundary 
condition at r = I is expres!'ed as 
1 
- = AI" (\1 p ) + BKo (\1 p ) 
p 
(VII-I) 
and at r = R the condition is 
(VII-2) 
since it is shown in W.B.F., p. 79, that Ko' (z) = - K, (z), and 
10' (z) = I, (z). The solutions for A and B from these two 
,imultaneous algebraic expressions are 
A=K,(YpR)/p[K,(V-pR) I..(yp)+K.(Yp) I,(YpR)] 
and 
B=I, (Yp R)/p[K,( Yp R) Io( Vp) +Ko( '/p) 1,( Yp R)] 
By substituting these constants in Eq. VI-I, the general solu-
tion for the transform of the pressure dropis expressed by 
p[K,(ypR) Io(Y-p) +I,(YpR) Ko(YPlJ 
(VII-3) 
To find Q(t) the cumulative fluid produced for unit pres-
sure drop, then the transform for the pressure gradient at 
r = I is obtained as follows: 
-(..Q~)r~ [I,(yp_R)K'(Y~ -Kl(Y~R)I,(Y~)J 
a p'l' [K, ( Y p R) 10 ( Y p ) + I, ( Y p R) Ko ( Y p ) ] 
where the negative sign is introduced in order to make Q (t) 
T ABLE III - Continued 
R - 5 R _ 6.0 It _ 7.0 R _ 8.0 
I 
R _ 9.0 R - 10 
fJ, = 0.8472 fJ, = 0.6864 fJ, = 0.5782 fJ, = 0.4999 fJ, = 0.4406 fJ, = 0.3940 
fJ, = 1.6112 fJ, = 1.2963 fJ, = 1.0860 fJ, = 0.9352 fJ, = 0.8216 fJ, = 0.7333 
I 
I 
t PIt) t p(t) t P(t) t P 
I t P t P(t) ,t) I (1) -------------------------------
3.0 1.167 4.0 1.275 6.0 1.436 8.0 1.556 10.0 1.651 12.0 1. 732 
3.1 1.180 4.5 1.322 6.5 1.470 8.5 1.582 10.5 1.673 12.5 1. 750 
3.2 1.192 5.0 1.364 7.0 1.501 9.0 1.607 11.0 1.693 13.0 1. 768 
3.3 1.204 5.5 1.404 7.5 1.531 9.5 1.631 11.5 1. 713 13.5 1. 784 
3.4 1.215 6.0 1.441 8.0 1.559 10.0 1. 653 12.0 1. 732 14.0 1.801 
3.5 1.227 6.5 1.477 8.5 1.586 10.5 1.675 12.5 1. 750 14.5 1.817 
3.6 1.238 7.0 1.511 9.0 1.613 11.0 1.697 13.0 1. 768 15.0 1.832 
3.7 1.249 7.5 1.544 9.5 1.638 11.5 1. 717 13.5 1. 786 15.5 1.847 
3.8 1.259 8.0 1.576 10.0 1.663 12.0 1. 737 14.0 1.803 16.0 1.862 
3.9 1.270 8.5 1.607 11.0 1.711 12.5 1. 757 14.5 1.819 17.0 1.890 
4.0 1.281 9.0 1.638 12.0 1. 757 13.0 1. 776 15.0 1.835 18.0 1.917 
4.2 1.301 9.5 1.668 13.0 1.801 13.5 1. 795 15.5 1.851 19.0 1.943 I 4.4 1.321 10.0 1.698 14.0 1.845 14.0 1.813 16.0 1.867 20.0 1. 968 
4.6 1.340 11.0 1. 757 15.0 1.888 14.5 1.831 17.0 1.897 22.0 ~.011 
4.8 1.360 12.0 1.815 16.0 1. 931 15.0 1.849 18.0 1.926 24.0 2.063 
5.0 1.378 13.0 1.873 17.0 1. 974 17.0 1. 919 19.0 1.955 26.0 2.108 I 5.5 1.424 14.0 1.931 18.0 2.016 19.0 1.986 20.0 1.983 28.0 2.151 B.O 1.469 15.0 1.988 19.0 2.058 21.0 2.051 22.0 2.037 30.0 2.194 
6.5 1.513 16.0 2.045 20.0 2.100 23.0 2.1J6 24.0 2.090 32.0 2.236 
7.0 1.556 17.0 2.103 22.0 2.184 25.0 2.180 26.0 2.142 34.0 2.278 
7.5 1.598 18.0 2.160 24.0 2267 30.0 2.340 28.0 2.193 36.0 2.319 
8.0 1.641 19.0 2.217 26.0 2.351 35.0 2.499 30.0 2.244 38.0 2.360 
9.0 1.725 20.0 2.274 28.0 2.434 40.0 2.658 34.0 2.345 40.0 2.401 
10.0 1.808 25.0 2.560 30.0 2.517 45.0 2.817 38.0 2.446 50.0 2.604 
11.0 1.892 30.0 2.846 40.0 2.496 60.0 2.806 
12.0 1. 975 
I 
45.0 2.621 70.0 3.008 
13.0 2.059 
I 
50.0 2.746 
14.0 2.142 
15.0 ~.225 
---
December, 1949 PETROlEUM TRANSACTIONS, AIME 317 
T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS 
IN RESERVOIRS 
pOSJtJve. Theorem B shows that the integration with respect 
to time introduC'es an additional operator p in the denomi· 
nator to give 
which indicate the poles. Since the modified Bessel functions 
for positive real arguments are either increasing or decreas· 
ing, the bracketed term in the denominator does not indicate 
any poles for positive real values for p. At the origin of the 
plane of Fig. 2 a pole exists and this pole we shall have to 
investigate first. Thus, the substitution of small and real 
values for z (Eqs. VI.12 and VI.13) in Eq. VII·4, gives 
[I,(Vp R) K,(Vp] -K,(Vp R) 1,(Vp )] 
p'I'[K,(V p R) lo(Vp ) + L(Vp R) K.,(V p l] 
(VII.4) 
In order to apply Mellin's inversion formula, the first con· 
,ideration is the roots of the denominator of this equation 
- (R'-l) 
Q(,,)=-~ 
p~O 
TABLE IV - Constant Terminal Rate Case Radial Flow 
Pressure at Exterior Radius Constant 
R == 1.5 I~ == 2.0 R == 2.5 I R == 3.0 R == 3.5 
A, == 3.4029 A, == 1.7940 A, == 1.2426 A, == 0.9596 A, == 0.7852 
A, == 1.9624 ... ____ A_,_==_9---,._52_0_7 _________ A_,_==---,-4_._80_2_1 ___ I _____ Ac_. _=--,' :-3_.2_2_65 ____ 1 ____ A_._== 2.4372 
Pit) t I 
5.0(10)-' 
5.5 " 
6.0 'I 
7.0 1/ 
8.0 " 
9.0 " 
1.0(10)-1 
1.2 " 
1.4 u 
1.6 " 
1.8 " 
2.0 " 
2.2 It 
2.4 It 
2.6 " 
2.8 " 
3.0 " I' 3.5 1/ 
4.0 " 
4.5 Ie 
5.0 II 
6.0 " 
7.0 u 
8.0 II 
0.230 
0.240 
0.249 
0.266 
0.282 
0.292 
0.307 
0.328 
0.344 
0.356 
0.367 
0.375 
0.381 
0.386 
0.390 
0.393 
0.396 
0.400 
0.402 
0.404 
0.405 
0.405 
0.405 
0.405 
2.0(10)-1 
2.2 " 
2.4 " 
2.6 It 
2.8 " 
3.0 II 
3.5 " 
4.0 (( 
4.5 " 
5.0 II 
5.5 " 
6.0 II 
6.5 It 
7.0 H 
7.5 H 
8.0 " 
8.5 " g.o II 
9.5 " 
1.0 
1.2 
1.4 
1.6 
1.8 
2.0 
2.5 
3.0 
0.424 
0.441 
0.457 
0.472 
0.485 
0.498 
0.527 
0.552 
0.573 
0.591 
0.606 
0.619 
0.630 
0.639 
0.647 
0.654 
0.660 
0.665 
0.669 
0.673 
0.682 
0.688 
0.690 
0.692 
0.692 
0.693 
0.693 
t 
3.0(10)-1 
3.5 " 
4.0 H 
4.5 'f 
5.0 " 
5.5 " 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
1.0 
L~ I 
1.6 
1.8 
2.0 
2.2 
2.4 
2.6 
2.8 
3.0 
3.5 
4.0 
4.5 
5.0 
5.5 
6.0 
0.502 
0.535 
0.564 
0.591 
0.616 
0.638 
0.659 
0.696 
0.728 
0.755 
0.778 
0.815 
0.842 
0.861 
0.876 
0.887 
0.895 
0.990 
0.905 
0.908 
0.910 
0.913 
0.915 
0.916 
0.916 
0.916 
0.916 
5.0(10)-1 
5.5 " 
6.0 It 
7.0 " 
8.0 I' 
9.0 H 
1.0 
1.2 
1.4 
1.6 
1.8 
2.0 
2.2 
2.4 
2.6 
2.8 
3.0 
3.5 
4.0 
4.5 
5.0 
5.5 
6.0 
6.5 
7.0 
8.0 
10.0 
-0-.6-17-- - 50(10)--;- --0-.6-20-
0.640 6.0 " 0.665 
0.662 7.0 " 0.705 
0.702 8.0 " 0.741 
0.738 9.0 " 0.774 
0.770 1.0 0.804 
0.799 1.2 0.858 
o 850 1.4 0.904 
0.892 1.6 0.945 
0.927 1.8 0.981 
0.955 2.0 1.013 
0.980 2.2 1.041 
1.000 2.4 1.065 
1.016 2.6 1.087 
1.030 2.8 1.106 
1.042 3.0 1.123 
1.051 3.5 1.158 
1.069 4.0 1.183 
1.080 5.0 1.215 
1.087 6.0 1.282 
1.091 7.0 1.242 
1.094 8.0 1.247 
1.096 9.0 1.250 
1.097 10.0 1.251 
1. 097 12.0 1.252 
1.098 14.0 1.253 
1.099 16.0 1.253 I 
----------------------------~------~--------~---------------------------~------~---------
318 
1.0 
1.2 
1.4 
1.6 
1.8 
'J.O 
2.2 
2.4 
2.6 
2.8 
3.0 
3.4 
3.8 
4.5 
5.0 
5.5 
6.0 
7.0 
8.0 
9.0 
10.0 
12.0 
14.0 
16.0 
18.0 
R == 4.0 
A, == 0.6670 
A, == 1.6450 
0.802 
0.857 
0.905 
0.947 
0.986 
1.020 
1.052 
1.080 
1.106 
1.130 
1.152 
1.190 
1.222 
1.266 
1.290 
1.309 
1.325 
1.347 
1.361 
1.370 
1.376 
1.382 
1.385 
1.386 
1.386 
4.0 
4.5 
5.0 
5.5 
6.0 
6.5 
7.0 
7.5 
8.0 
8.5 
9.0 
10.0 
12.0 
14.0 
16.0 
18.0 
20.0 
22.0 
24.0 
26.0 
28.0 
30.0 
35.0 
40.0 
50.0 
R == 6.0 
A, == 0.4205 
A, == 1.0059 
1.275 
1.320 
1.361 
1.398 
1.432 
1.462 
1.490 
1.516 
1.539 
1.561 
1.580 
1.615 
1.667 
1.704 
1. 730 
1.749 
1. 762 
1.771 
1.777 
I. 781 
1.784 
1. 787 
1.789 
1. 791 
1. 792 
TABLE IV -- Continued 
7.0 
7.5 
8.0 
8.5 
9.0 
9.5 
10.0 
12.0 
14.0 
16.0 
18.0 
20.0 
22.0 
24.0 
26.0 
28.0 
30.0 
35.0 
40.0 
45.0 
50.0 
60.0 
70.0 
80.0 
R == 8.0 
A, == 0.3090 
A, == 0.7286 
1.499 
1.527 
1.554 
1.580 
1.604 
1.627 
1.648 
1. 724 
1.786 
1.837 
1.879 
1.914 
1.943 
1.967 
1.986 
2.002 
2.016 
2.040 
2.055 
2.064 
2.070 
2.076 
2.078 
2.079 
n == 10 
A, == 0.2448 
A, == 0.5726 
10.0 
12.0 
14.0 
16.0 
18.0 
20.0 
25.0 
30.0 
35.0 
40.0 
45.0 
50.0 
55.0 
60.0 
65.0 
70.0 
75.0 
80.0 
90.0 
10.0(10)1 
11.0 " 
12.0 H 
13.0 " 
14.0 " 
16.0 I( 
1.651 
1. 730 
I. 798 
1.856 
1.907 
1.952 
2.043 
2.111 
2.160 
2.197 
2.224 
2.245 
2.260 
2.271 
2.279 
2.285 
2.290 
2.293 
2.297 
2.300 
2.301 
2.302 
2.302 
2.302 
2.303 
PETROlEUM TRANSACTIONS, AIME 
R == 15 
A, = 0.1616 
A, == 0.3745 
20.0 
22.0 
24.0 
26.0 
28.0 
30.0 
35.0 
40.0 
45.0 
50.0 
60.0 
70.0 
SO.O 
90.0 
10.0(10)' 
12.0 If 
14.0 " 
16.0 " 
18.0 H 
20.0 H 
22.0 u 
24.0 H 
26.0 " 
28.0 H 
30.0 " 
1.960 
2.003 
2.043 
2.080 
2.114 
2.146 
2.218 
2.279 
2.332 
2.379 
2.455 
2.513 
2.558 
2.592 
2.619 
2.655 
2.677 
2.689 
2.697 
2.701 
2.704 
2.706 
2.707 
2.707 
2.708 
December, 1949 
A. F. VAN EVERDINGEN AND W. HURST T.P. 2732 
and by the application of Mellin's inversion formula applied 
at the origin, then 
lished by the Mellin's inven;ion formula by letting A = u'ehr ; 
then by Eqs. VI·IS 
1 (R'-l) R'-l 
----dA=--
A 2 
(VII·5) 
I f At-- e Q(A)d A = 
27ri 
A" A" etc. 
-u't 
du I fe [.I,(uR) Y,(u) - Y,(uR) .I,(u)] 
7ri u'[J,(uR) Yo(u) - Y,(uR) .In(U)] 
(VII,6) 
a1 , a'2, etc. 
where at, a" and an are the roots of 
An investigation of the integration along the negative real 
"cut" both for the upper and lower portions, Fig. 2,reveals 
that Eq. VIl-4 is an e\'en function for which the integration 
along the paths is zero. However, poles are indicated along 
the negative real axis and the~e residuals together with Eq. 
VII·S make up the wlution for the constant terminal pre '~ure 
("acc for the limited radial sy~tem. The re,iduaI, are e,tab· 
[.L(anR) Yn(an) - Yt(anR) .In(a.,)] = 0 (VII.7) 
and the pole, are represented on the negatiye real axis 
by An = - an', Fig. 3. The residuals of Eq. VII·6 are the series 
expansion 
TABLE IV - Continued 
A, = 0.1208 A, = 0.0964S A = 0.08032 
-----'----,-
R = 40 
A, = 0.06019 
A, = 0.1384 
R = 50 
A = 0.04~13 
A: -- 0.110 
R := 20 R' = 25 n = 30 I 
A, = 0.2788 A, = 0.~223 , A: = 0.1849 
t P tiP' t P' t P'tl t P't) (t) (t) ttl 
-----------------~ ---- --------------- ------- -------- --------
30.0 2.,48 50.0 2.389 70.0 2.551 120(10)1 2.813 20.0(10)1 3.064 
~gZ ~~~g ~gg I ~:m ~gg ~m ltg :: ~~~~ ~U :: ~lU 
45.0 2.338 65.0 1 2.514 10.0(10)1 2.723 18.0 " 3.011 26.0 " 3.193 
50.0 2.388 70.0 2.550 12.0 " 2.812 20.0 " 3063 28.0 " 3.229 
60.0 2.475 75.0 2.583 14.0 " 2.886 22.0 " 3.1O~ 30.0 " 3.263 
70.0 2.547 80.0 2.614 16.0 " 2.950 24.0 " 3.152 35.0 " 3.339 
80.0 2.609 85.0 2.643 &5 " 2.965 26.0 " 3.191 40.0 " 3.405 
90.0 2.658 90.0 2.671 17.0 " 2.979 28.0 " 3.226 45.0 " 3.461 
10.0(10)1 2.707 95.0 2.697 17.5 " 2.992 30.0 " 3.259 50.0 " 3.512 
10.5 " 2.728 10.0(10)1 2.721 18.0 " 3.006 35.0 " 3.331 55.0 " 3.55~ 
11.0 " 2.747 12.0 " 2.807 20.0 " 3.054 40.0 " 3.391 60.0 " 3.591> 
11.5 " 2.764 14.0 " 2.878 25.0 " 3.150 45.0 " 3.440 65.0 " 3.630 
12.0 " 2.781 16.0 " 2.936 30.0 " 3.219 50.0 " 3.482 70.0 " 3.661 
12.5 " 2.796 18.0 " 2.984 35.0 " 3.269 55.0 " 3.516 75.0 " 3.688· 
13.0 " 2.810 20.0 " 3.024 40.0 " 3.306 60.0 " 3.545 80.0 " 3.713 
13.5 " 2.823 22.0 " 3.057 45.0 " 3.332 65.0 " 3.568 85.0 " 3,731> 
14.0 " 2.835 24.0 " 3.085 50.0 " 3.351 70.0 " 3.588 90.0 " 3.754 
14.5 " 2.846 26.0 " 3.107 60.0 " 3.375 80.0 " 3.619 95.0 " 3.771 
15.0 " 2.857 28.0 " 3.126 70.0 " 3.387 90.0 " 3.640 10.0(10)' 3.787 
16.0 " 2.876 30.0 " 3.142 80.0 " 3.394 10.0(10)' 3.655 12.0 " 3.833 
18.0 " 2.906 35.0 " 3.171 90.0 " 3.397 12.0 " 3.672 14.0 " 3.862 
20.0 " 2.929 40.0 " 3.189 10.0(10)' 3.399 14.0 " 3.681 16.0 " 3.881 
24.0 " 2.958 45.0 " 3.200 12.0 " 3.401 16.0 " 3.685 18.0 " 3.892 
28.0 " 2.975 50.0 " 3.207 14.0 " 3.401 18.0 " 3.687 20.0 " 3.900 
30.0 " 2.980 60.0 " 3.214 20.0 " 3.688 22.0 " 3.904 
40.0 " 2.992 70.0 " 3.217 25.0 " 3.689 24.0 " 3.907 
50.0 " 2.995 80.0 " 3.218 26.0 " 3.90!) 
90.0 " 3.219 I 28.0 " 3.9W 
TABLE IV -Continued 
----------'---'-'---"------,-------------;---- -,-- "-"-------,-----------
It = 60 R = 70 R = 80 R = 90 R = 100 
t t P," 
------1------1·------------ ---------.-- --------1------ ----------,-
3.0(10)2 
4.0 " 
5.0 " 
6.0 " 
7.0 /I 
8.0 " 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
16.0 " 
18.0 " 
20.0 " 
25.0 " 
30.0 " 
35.0 H 
40.0 " 
45.0 " 
50.0 " 
55.0 " 
December, 1949 
3.2m 
3.401 
3.512 
3.602 
3.676 
3.739 
3.792 
3.832 
3.908 
3.959 
3.996 
4.023 
4.043 
4.071 
4084 
4.090 
4.092 
4.093 
4.094 
4.094 
50(10)' 
6.0 " 
7.0 .• 
8.0 " 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
16.0 " 
18.0 " 
20.0 " 
2.1.0 " 
30.0 " 
35.0 " 
40.0 " 
45.0 " 
50.0 " 
55.0 " 
60.0 " 
65.0 " 
70.0 " 
75.0 " 
80.0 " 
3.512 
3.603 
3.689 
3.746 
3.803 
3.854 
3.937 
4.003 
4.054 
4.095 
4.127 
4.181 
4.211 
4.228 
4.237 
4.242 
4.245 
4.247 
4.247 
4.248 
4.248 
4.248 
4.248 
60(10)' 
7.0 " 
8.0 .. 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
15.0 If 
16.0 " 
18.0 " 
20.0 " 
25.0 (j 
30.0 " 
35.0 " 
40.0 (( 
45.0 " 
50.0 (( 
60.0 " 
70.0 (( 
80.0 " 
90.0 H 
10.0(10)' 
11.0 " 
3.603 
3.689 
3.747 
3.805 
3.857 
3.946 
4.019 
4.051 
4.080 
4.130 
4.171 
4.248 
4.297 
4.328 
4.347 
4.360 
4.308 
4.3i6 
4.380 
4.381 
4.382 
4.382 
4.382 
80(10)' 
9.0 " 
1.0(1,)' 
1.2 " 
1.3 " 
1.4 " 
1.5 " 
1.8 " 
2.0 " 
2.5 " 
3.0 " 
3.5 " 
4.0 H 
4.5 " 
5.0 " 
6.0 " 
7.0 " 
8.0 " 
9,0 H 
10.0 " 
11.0 " 
12.0 " 
14.0 " 
PETROlEUM TRANSACTIONS, AIME 
3.747 
3.803 
3.858 
3.949 
3.988 
4.025 
4.058 
4.144 
4.192 
4.285 
4.349 
4.394 
4.426 
4.448 
4.404 
4.482 
4.491 
4.496 
4.498 
4.499 
4.499 
4.500 
4.500 
1. 0(10)3 
1.2 " 
1.4 " 
1.6 " 
1.8 " 
2.0 " 
2.5 " 
3.0 II 
3.5 " 
4.0 H 
4.5 " 
5.0 " 
5.5 u 
6.0 " 
6.5 " 
7.0 II 
7.5 " 
8.0 II 
9.0 " 
10.0 " 
12.5 " 
15.0 " 
3.859 
3.949 
4.026 
4.092 
4.150 
4.200 
4.303 
4.379 
4.434 
4.478 
4.510 
4.534 
4.552 
4.565 
4.579 
4.583 
4.588 
4.593 
4.598 
4.601 
4.604 
4.605 
319 
T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS 
IN RESERVOIRS 
e-an't[l,(a oR) Y,(an) -Y,(anR) ll(an)] 
d . (VII.8) 
ao'limd)J,(uR) Yo(u) -Y, (uR) Jo(u)] 
u~an 
etc. 
Therefore, the solution for Q (t) is expres>'ed by 
ll' -. 1 (jJ 
Q(I, = .---- 2 ~ 
2 a,. a, 
-fln't 
e ],'(floll) 
-----
fl,,'[lo'(fln) -],')fl"R)] 
(VII.lO) 
since etc. 
J,'(z) = L(z) -J,(Z)/Z (VII·9) 
and 
J.'(z) =-J,(z) 
which are recurrence formulae for both first and ~econd kind 
of Bessel functions, W.B.F., p. 45 and p. 66, then by the iden· 
tities of Eqs. VII·7 and VI.23, the relation VII·8 reduces to 
This is essentially the solution developed in an earlier work: 
hut Eq. VII·lO is more rapidly convergent than the solution 
previously reT,orted. 
The values of Q(,) for the constant terminal pressure case 
for a limited reservoir have been calculated from Eq. VII·IO 
for R = 1.5 to 10 and are tabulated in Table 2. A reproduction 
of a portion of these data is given in Fig. 5. As Eq. VII·IO is 
rapidly convergent for t greater than a given value, only two 
(jJ e-a,,'t J,' (aoR) 
- 2 :!: 
aha, fl,,'[J..'(a n ) -J,'(a"R)] 
etc. 
T ABLE IV - Continued 
--------_._-_ .. _------_. __ . __ ._---------
t R = 200 p(,,_' ___ _ t R 300 Pet) _ __ t He = 1
400 p(~ __ I ____ P._'_=_5_00 __ 
p
(t) 
--------.--- - - --- - - ----- -- -- ---- - -- ---- - - ----- --- 1 
1.5(10)' 4.061 60(10)' 4.754 1.5(10)' I 5.212 
2.0 " 4.205 8.0 " 4.898 2.0 " 5.356 
2.51" 4.317 10.0 " 5.010 3.0 " 5.556 
3.0 " 4.40S 12.0 " 5.101 4.0 " 5.689 
a.5~" 4.485 14.0 " 5.177 5.0 " 5.781 
4.01" 4.552 16.0 " 5.242 6.0 " 5.845 
5.01" 4.663 18.0 " 5.299 7.0 " 5.889 
6.0 " 4.754 20.0 " .1.348 8.0 " 5.920 
7.0 " 4.829 24.0 " 5.429 9.0 " 5.942 
8.0 " 4.894 28.0 " 5.491 10.0 " 5.957 
9.0 " 4.949 30.0 " 5.517 11.0 " 5.967 
10.0 " 4.996 40.0 " 5.606 12.0 " 5.975 
12.0 " 5.072 50.0 " 5.652 12.5 " 5.977 
14 0 " 5.129 60.0 " 5.676 13.0 " 5.980 
16.0 " 5.171 700 " 5.690 14.0 " 5.983 
18.0 ., 5203 80.0 " .5.696 16.0 " 5.988 
20.0 " 5.237 90.0 " 5.700 18.0 " 5.990 
2.5.0 " 5.264 10.0(10)' 5.702 20.0 " 5.991 
30.0 " 5.282 12.0 " 5.703 24.0 " 5.991 
35.0 " 5.290 14.0 " 5.704 26.0 " 5.991 
40.0 " 5294 15.0 " 5.704 
. __ ._-_._-'----
TABLE IV - Continued 
R = 700 R = 800 R = 900 
t 
2.0(10)' 
2.5 " 
3.0 " 
3.5 It 
4.0 " 
4.5 " 
5,0 " 
6.0 u 
7.0 " 
8.0 " 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
16.0 " 
18.0 " 
20.0 " 
25.0 H 
30.0 " 
35.0 " 
40.0 " 
t 
R 1000 
5.356 
5.468 
5.559 
5.636 
5.702 
5.759 
5.810 
5.894 
5.960 
6.013 
6.055 
6.088 
6.135 
6.164 
6.183 
6.195 
6.202 
6.211 
6.213 
6.214 
6.214 
R = 600 
40(10)' 
4.5 H 
5.0 " 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
111.0 " 
18.0 " 
20.0 " 
25.0 " 
30.0 " 
35.0 " 
40.0 " 
.~~g :: 
R 
t 
1200 
5.703 
5.762 
5.814 
5.904 
5.979 
6.041 
6.094 
6.139 
6.210 
6.262 
6.299 
6.326 
6.345 
6.374 
6.387 
6.392 
6.395 
6.397 
6.397 
p(t) 
---------·----1----- ·----1----·--1----- ------ ---.-.-.---- .. ----
5.0(10)' 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
16.0 " 
18.0 " 
20.0 " 
25.0 " 
30.0 " 
35.0 " 
40.0 " 
45.0 " 
50.0· " 
60.0 " 
70.0 t( 
80.0 H 
5.814 
5.905 
5.982 
6.048 
6.105 
6.156 
6.239 
6.305 
6.357 
6.398 
6.430 
6.484 
6.514 
6.530 
6.540 
6.545 
6.548 
6.550 
6.551 
6.551 
7.0(10)' 
8.0 " 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
16.0 " 
18.0 " 
20.0 " 
25.0 " 
30.0 " 
35.0 H 
40.0 " 
45.0 " 
50.0 " 
55.0 " 
60.0 " 
70.0 " 
80.0 " 
100.0 " 
5.983 
6.049 
6.108 
6.160 
6.249 
6.322 
6.382 
6.432 
6.474 
6.551 
6.599 
6.630 
6.650 
6.663 
6.11716.676 
6.6i9 
6.682 
6.684 
6.684 
8.0(10)' 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
16.0 " 
18.0 " 
20.0 " 
25.0 " 
30.0 " 
40.0 " 
45.0 " 
50.0 " 
55.0 " 
60.0 " 
70.0 H 
80.0 " 
90.0 " 
10.0(10)' 
6.049 
6.108 
6.161 
6.251 
6.327 
6.392 
6.447 
6.494 
6.587 
6.652 
6.729 
6.751 
6.766 
6.777 
6. i85 
6.794 
6.798 
6.800 
6.801 
1.0(10)' 
1.2 " 
1.4 " 
.1.6 " 
1.8 " 
2.0 " 
2.5 " 
3.0 " 
3.5 " 
4.0 " 
4.5 " 
5.0 " 
5.5 " 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
16.0 " 
6.161 
6.252 
6.329 
6.395 
6.452 
6.503 
6.605 
6.681 
6.738 
6.781 
6.813 
6.837 
6.854 
6.868 
6.885 
6.895 
6.901 
6.904 
6.907 
6.907 
6.908 
-------'--_._._------- ------'-------'--------'-------'-------'-----
320 PETROLEUM TRANSACTIONS, AIME 
2.0(10)' 
3.0 " 
4.0 " 
5.0 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
10.0 " 
12.0 " 
14.0 " 
16.0 " 
18.0 
19.0 " 
20.0 " 
21.0 " 
22.0 " 
23.0 " 
24.0 " 
6.507 
6.704 
6.833 
6.918 
6.975 
7.013 
7.038 
7.056 
7.067 
7.080 
7.085 
7.088 
7.089 
7.089 
7.090 
7.090 
7.090 
7.090 
7.090 
December, 1949 
A. F. VAN EVERDINGEN AND W. HURST T.P. 2732 
terms of the expansion are necessary to give the accuracy 
needed in the calculations. 
Likewise froll'l the foregoing work it can be easily shown 
that the transform of the pressure drop at any point in the 
formation in a limited reservoir for the constant terminal rate 
case, is expressed by 
[K,(V p R) Io(Y' p r) +I, (Y p R) Ko(Y p r)] 
p'J' [I, (Y p R) K, ( Y p ) - K, (Y p R) I, \' p )] 
(VII.ll ) 
An examination of the denominator of Eq. VII·ll indicate, 
that there are no roots for positive values of p. However, a 
<Iouble pole exists at p = O. This can be determined by ex· 
panding K,. (z) and K, (z) to Eecond degree expansions for 
small values of z and third degree expansions for I.. (z) and 
I, (z). and substituting in Eq. VII·ll. It is fonnd for small 
values of p. Eq. VII·ll reduce, to 
1 
P(r,P) ==-
p 
p~O 
H' It (Il' -. r') It' log It 
log - - ---- + ---
(R'-I) r 2(lt'-I) (R'-I)' 
(R' + 1) 1 2 
- 4(R'_I),f +-;; (R'-I) (VII.I2) 
This equation now indicates both a single and double pole at 
the origin, and it can be shown from tables or by applying 
Cauchy's theorem to the Mellin's formula that the inversion (If 
Eq. VII·I2 is 
p, = __ ~. [r' + t ] (VII.13 \ 
,.1) (R'-l) 4 
R' r3R'-4R' log R-2R'-Il 
----loo-r-
(R'-l) .., 4(R' _1)' 
which holds when the time, t, is large 
As in the preceding case, there are poles along the negative 
real axis, Fig. 3, and the residuals are determined as before 
by letting A = u' eiTr, and Eqs. VI·IS give 
TABLE IV - Continued 
----------c---- ---------;----------'C.---------- ......... ---.--.--. ---
R:= 1400 R:= 
t P(t> t 
2.0(10)5 6.507 2.5(10)5 
2.5 H 6.619 3.0 H 
3.0 j( 6.709 3.5 " 
3.5 " 6.785 4.0 H 
4.0 " 6.849 5.0 " 
0.0 " 6.950 6.0 " 
6.0 H 7.026 7.0 " 
7.0 H 7.082 8.0 " 
8.0 " 7.123 9.0 H 
9.0 H 7.154 10.0 " 
10.0 " 7.177 15.0 " 
15.0 " 7.229 20.0 H 
20.0 " 7.241 25.0 H 
25.0 u 7.243 30.0 " 
30.0 " 7.244 35.0 H 
31.0 " 7.244 40.0 " 
32.0 H 7.244 42.0 " 
33.0 " 7.244 44.0 " 
I 
1600 
P(t> 
6.619 
6.710 
6.787 
6.853 
6.962 
7.046 
7.114 
7.167 
7.210 
7.244 
7.334 
7.364 
7.373 
7.376 
7.377 
7.378 
7.378 
7.378 
R := 1800 
3.0(10)' 
4.0 " 
5.0 H 
6.0 " 
7.0 " 
8.0 " 
9.0 " 
10.0 " 
15.0 H 
20.0 " 
30.0 H 
40.0 " 
50.0 " 
51.0 " 
52.0 " 
53.0 " 
54.0 " 
56.0 " 
6.710 
6.854 
6.965 
7.054 
7.120 
7.188 
7.238 
7.280 
7.407 
7.459 
7.489 
7.495 
7.495 
7.495 
7.495 
7.495 
7.495 
7.495 
TABLE IV - Continued 
R = 2000 
4.0(10)' 
5.0 " 
6.0 " 
7.0 " 
8.0 II 
9.0 " 
10.0 " 
12.0 " 
14.0 u 
16.0 " 
18.0 
20.0 " 
25.0 " 
30.0 " 
35.0 " 
40.0 H 
50.0 " 
60.0 " 
64.0 " 
6.854 
6.966 
7.056 
7.132 
7.196 
7.251 
7.298 
7.374 
7.431 
7.474 
7.506 
7.530 
7.566 
7.584 
7.593 
7.597 
7.600 
7.601 
7.601 
R := 2200 
5.0(10)' 
5.5 .. 
6.0 H 
6.5 " 
7.0 H 
7.5 " 
8.0 I' 
8.5 " 
9.0 H 
10.0 " 
12.0 " 
16.0 " 
20.0 " 
25.0 " 
30.0 " 
35.0 " 
40.0 " 
50.0 " 
60.0 " 
70.0 H 
SO.O .. 
i 
6.966 
7.013 
7.057 
7.097 
7.133 
7.167 
7.199 
7.229 
7.256 
7.307 
7.390 
7.507 
7.579 
7.631 
7.6ft1 
7.677 
7.686 
7.693 
7.69& 
7.69& 
7.696 
-- -----~-------- -
-----------;----------------------------_._ .. - - _._-- .... -. __ ._-----
I R := 2800 I R := 3000 I R := 2400 R := 2690 
-c----~ -~__;____-,------,--------. 
t PIt) t PIt> 
---------[------- -_._---[------- -------[---_.-- -----[------- ------ .-. ----
6.0(10)' 
7.0 " 
8.0 " 
9.0 " 
10.0 " 
12.0 " 
16.0 " 
20.0 H 
24.0 " 
28.0 " 
30.0 " 
35.0 " 
40.0 .. 
50.0 " 
60.0 " 
70.0 " 
SO.O H 
90.0 " 
95.0 " 
December, 1949 
7.057 
7.134 
7.200 
7.259 
7.310 
7.398 
7.526 
7.611 
7.668 
7.706 
7.720 
7.745 
7.760 
7.770 
7.7SO 
7.782 
7.783 
7.783 
7.783 
7.0(10)5 
8.0 " 
9.0 
10.0 " 
12.0 " 
14.0 " 
16.0 " 
18.0 " 
20.0 " 
24.0 " 
28.0 " 
30.0 " 
35.0 " 
40.0 " 
50.0 " 
60.0 H 
70.0 " 
SO.O " 
90.0 " 
10.0(10)' 
7.134 
7.201 
7.259 
7.312 
7.401 
7.475 
7.536 
7.588 
7.631 
7.699 
7.746 
7.765 
7.799 
7.821 
7.845 
7.856 
7.860 
7.862 
7.863 
7.863 
8.0(10)' 
9.0 H 
10.0 " 
12.0 " 
16.0 " 
20.0 " 
24.0 " 
28.0 " 
30.0 " 
35.0 " 
40.0 " 
50.0 H 
60.0 " 
70.0 " 
80.0 " 
90.0 II 
10.0(10)' 
12.0 " 
13.0 " 
7.201 
7.260 
7.312 
7.403 
7.542 
7.644 
7.719 
7.775 
7.797 
7.840 
7.870 
7.905 
7.922 
7.930 
7.934 
7.936 
7.937 
7.937 
7.937 
1.0(10)' 
1.2 " 
1.4 " 
1.6 " 
1.8 " 
2.0 jl 
2.4 
2.8 " 
3.0 " 
3.5 " 
4.0 " 
4.5 " 
5.0 " 
6.0 " 
7.0 " 
8.0 jl 
9.0 " 
10.0 " 
12.0 " 
15.0 .. 
PETROLEUM TRANSACTIONS, AIME 
7.312 
7.403 
7.4SO 
7.545 
7.602 
7.651 
7.732 
7.794 
7.320 
7.871 
7.908 
7.935 
7.955 
7.979 
7.992 
7.999 
8.002 
8.004 
8.006 
8.006 
321 
T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS 
IN RESERVOIRS 
1 f 
A" A" etc. 
= ~feU2t [J~R) Yo(ur) - Y, (uR) lo(ur)] du (VII-l4) 
lI"i u'[J,(uR) Y, (u) -J,(U) Y, (uR)] 
f3" f3" etc. 
where f3" f3" etc., are roots of 
[J,(f3"R) Y, (f3n) -J,(f3n) Y,(f3nR)] = 0 . (VII-lS) 
with An = -f3,,'. The residuals at the poles in Eq. VII-I4 give 
the series 
00 
2 ~ 
fll,f3" etc. 
-f3o t ] e [J, (f3nR) Yo (f3nr) - Y, (f3nR) Jo (f3n) 
d 
I1n' lim.- [JI(uR) Y, (u) -J, (u) YI(uR)] 
du 
u~f3n (VII-16) 
By the recurrence formulae Eqs. VII-9, the identity VII-IS, 
and Eq. VI-23, this series simplifies to 
00 e-f3n't JI(f3"R) [J,(f3n) Y,,(f3nr) - Y, ([3,.) J o (f3nr )] 
11" ~ ---------------::--,--::------
f31' f3" etc. f3,,[Jt'([3,R) -Jt'(fln)] 
(VII-17) 
Therefore. the sum of all residuals, Eqs. VII-I3 and VII-17 is 
the solution for the cumulative pressure drop at any point in 
the formation for the constant terminal rate case in a limited 
reservoir. or 
2 (r' ) R' (3R<-4R' log R-2R'-I) 
P ---- -+t - ---logr - -'--------,--,---
(r,t)- (R'-I) 4 (R'-I) 4(R'-I)' 
-f3:t e II(f3"R) [J,(fln) Yo (f3nr) - Y, (f3n) Io(flnr )] 
fln [I,' (flnR) -1,' (f3n) ] 
(VII-IS) 
which is essentially the solution given by Muskat; now de· 
veloped by the Laplace Transformation. Finally, for the cumu· 
lative pressure drop for a unit rate of production at the well 
hore, r = 1, this relation simplifies to 
P = __ 2 ____ (~-1- t) __ (3R'-4R'logR-2R'-1) 
(t) (R' _ I) 4 4(R' _ I)' 
00 c -(:I:t J,' (f3nR) 
+2 ~ 
f3,. f3, f3"'[J,'(f3,,R) - J,'(f3n)] 
(VII.19 
The calculations for the constant terminal rate case for a 
reservoir of limited radial extent have been determined from 
Eq. VII-19. The summary data for R = 1.5 to 10 are given in 
Table .3. An illustrative graph is shown in Fig. 6. The effect 
of the limited reservoir is quite pronounced as it is shown 
that producing the reservoir at a unit rate increases the pres-
sure drop at the well bore much faster than if the reservoir 
were infinite, as the constant withdrawal of fluid is reflected 
very soon in the productive life by the constant rate of drop 
in pressure with time. 
Pressure Fixed at Exterior Boundary 
As a variation on the condition that ( dP = 0 ) 
dr r=R 
we 
may assume that the pressure at r = R is constant. In effect, 
this assumption helps to explain approximately the pressure