SPE-25233-PA

Prévia do material em texto

Reservoir Simulation Grids: 
Opportunities and Problems 
Summary 
Grid selection is one of the most difficult 
and time-consuming tasks in the simulation 
of geologically complex reservoirs. 
A grid is imposed on a reservoir so that 
we can solve the nonlinear flow equations 
that predict the reservoir's response to 
changes at wells or other boundaries. 
Therefore, I first discuss the role of grid 
geometry in the evaluation of each term in 
the flow equations. Then I describe several 
conventional and some new gridding tech-
niques, as well as their advantages and dis-
advantages. Next I make recommendations 
based on my own experience and discussions 
with colleagues. My assessment is that grids 
should be as close to orthogonal as possi-
ble. Also, local grid refinement should be 
used with care; its inappropriate use can be 
counterproductive. Considerable progress 
has been made recently on the use of flexi-
ble grids, but efficient use of unstructured 
locally orthogonal grids requires further re: 
search. Techniques that automatically align 
the grid with major reservoir features (e.g., 
faults) are being developed and are expect-
ed to simplify grid generation. 
Introduction 
A reservoir simulator predicts reservoir per-
formance by solving flow equations on a dis-
crete grid chosen by the simulation engineer 
to represent the reservoir. The grid normally 
is selected with one or more of the follow-
ing considerations in mind. 
1. Reservoir geology and size and the data 
available for reservoir description. 
2. Type of fluid displacement or depletion 
process to be modeled. 
3. Past and anticipated field development 
(location and type of wells). 
4. Numerical accuracy desired. 
5. Available software options. 
6. Objectives of the simulation study. 
7. Competence of the simulation engineer 
or team. 
8. Available computer resources, time 
constraints, or project budget. 
In the early days of reservoir simulation, 
the last consideration often determined the 
number of gridblocks, and the available soft-
ware limited the choice of grid type, usually 
to block-centered Cartesian or cylindrical 
grid. Research in reservoir simulation and 
hardware developments, especially over the 
past 10 years, has greatly extended the grid-
ding options available. Modern commercial 
Copyright 1993 Society of Petroleum Engineers 
658 
Khalid Aziz, SPE, Stanford U. 
Khalid Aziz is the Otto N. Miller professor of Earth Sciences 
and Professor of chemical and petroleum engineering 
at the U. of Calgary and manager of the Computer 
Modelling Group. He has held several other academic 
and industrial positions in North America. In Pakistan, he 
~orke? for t~e Karachi Gas Co. in various capacities, 
including chief engineer. At Stanford, Aziz has served as 
Assoc. Dean for Research (School of Earth Sciences) and chairman of 
the Petroleum Engineering Dept. Aziz received his engineering 
e?ucation at the U. of Michigan, U. of Alberta, and Rice U. A 
Distinguished Member, Aziz has received the Cedric K. Ferguson 
Medal, and th~ Re~ervoir Engineering, Lester C. Uren, and Distinguished 
Petroleum Engineering Faculty awards. His main research interests are 
reser.voir si.mulation, .natural gas engineering, multi phase flow in porous 
media, flUid properties, ond multi phase flow in pipes. 
simulators typically offer local grid refine-
ment, hybrid grid, curvilinear (stream-tube) 
grid, Voronoi or perpendicular bisector grid 
(generalization of point-distributed grid), 
corner-point geometry, dynamic grid, or 
automatic grid generation. Some research 
simulators also offer elastic grid-adjustment 
methods, control-volume finite-element 
(CVFE) methods, and free Lagrangian 
methods. 
While this abundance of options provides 
flexibility, it can also make choosing the ap-
propriate grid bewildering for the simulator 
user. This paper provides a short description 
of some common gridding techniques avail-
able in commercial and research simulators. 
Where possible, experience with various 
gridding techniques also is discussed. The 
fluid flow equations are considered before 
gridding techniques are discussed so that the 
advantages and disadvantages of various 
gridding techniques can be understood 
better. 
Flow Equations 
The conservation of mass for Component c 
(for black-oil models, c=oil, gas, and water 
at standard conditions) combined with Dar-
cy's law yields the following set of flow 
equations: 
Nj Np 
E E TC,Pi)cf>P,j-cf>p,i) 
} p 
Vb,; 
=-[(M .)n+1-(M .)n]+q . At C,l C,l C,l' 
.................. (1) 
where the transmissibility between Nodes i 
andj is 
TC,Pi.j = (fcAkkrpwc,pILJ.tp);,j' .... (2) 
For each block, one equation of this type 
is written for each component or pseudo-
component, c, in the system. The required 
geometric properties of the gridblock are 
block volume, Vb; the area of each block 
face, A; and the distance between i and j for 
each connection, L. This method of writing 
flow equations, the control-volume finite-
difference (CVFD) method, reduces to the 
standard finite-difference method for Carte-
sian grids. An important characteristic of 
this method is that in Eq. 1 the flow across 
the gridblock face between i andj depends 
only on the component of the potential gra-
dient in the i-j direction. For nonorthogonal 
coordinate systems, the flow calculation 
across a block face would depend on all 
components ofthe gradient of the potential 
on the surface. Hence, an error in flow cal-
culations results if only the component of 
the gradient in the i-j direction is considered 
in a nonorthogonal grid. 
The terms in Eq. 1 represent net flow into 
Block i, accumulation of mass in Block i, 
and flow from wells within the block, re-
spectively. The indexj is for blocks connect-
ed to Block i, the block for which the mass 
balance is written. Note that the connections 
of Block i need not be neighbors of this 
block. The gridblock shape and gridpoint lo-
cation within the block influence the evalu-
ation of each term. The flow between blocks 
is calculated by multiplying the interblock 
transmissibility with the difference in poten-
tial between the blocks (Fig. 1). 
This flow term depends on both the grid 
geometry and the gridpoint location in the 
block. The gridpoints should be selected so 
that the finite-difference approximation of 
July 1993 • JPT 
Fig. 1-Gridblock ; and its connections. 
pressure gradient is as accurate as possible. 
In other words, the difference in potential 
between the two nodes on either side of a 
boundary divided by the distance between 
the nodes should be a good approximation 
at the boundary for the average potential gra-
dient normal to the boundary. The accumu-
lation term uses the gridblock volume and 
the pressure at the node to calculate the mass 
in the block at different times. For this pur-
pose, the gridpoint should be as close to the 
mass center of the block as possible. Finally, 
for calculation of well flow, a well model 
is required to relate the wellblock pressure 
to the well pressure. The well model de-
pends on the grid type, and for certain kinds 
of grid, the well model must be adjusted as 
the flow field in the reservoir changes. 1 
Simulation Grids 
Globally Orthogonal Grids. Most com-
monly used grids are constructed by aligning 
y 
~ - - - -
ij+l 
-~ - - - -
I I 
I 1 
. 1.6- - - - - -6-:-.- - - -1- J 1 1 1J 
I I 
1 1 
0-
- - - - -0- -ij-l - - -
a) Block-centered grid 
-
i .. , .. ,f ",!",,~.,,: · .. ,: .... ,r ::., •. : ... ,! '.:i ... ,l.· :·.·1., .. ,~ ir it :j1 it ·1 
.
'.:.!.. .'.i.t".:::,! .:.:.~., ,: ". ',: :. ":; ,:; :: ':: ::' ::: t :: 
the gridblocks along orthogonal coordinate 
directions and then distorting the grid where 
necessary to fit majorreservoir features 
(e.g., dip). Fig. 2 shows examples of these 
kinds of grids, the standard Cartesian block-
centered and point-distributed grids. 
The block-centered grid is advantageous 
for calculating accumulation terms, while 
the point-distributed grid is more accurate 
for calculating flow between blocks. When 
the grid is almost uniform, the differences 
between these two types of grids are insig-
nificant. For highly irregular grids, how-
ever, the r~sults obtained are strongly 
influenced by grid type. Even though Settari 
and Aziz2 demonstrated the superior ac-
curacy of the point-distributed grid in 1972, 
its use in commercial models has been rather 
limited. The primary reasons probably are 
tradition and the fact that most engineers find 
it easier to think of dividing the reservoir 
into blocks rather than choosing gridpoints 
that automatically generate blocks accord-
- r--~-
"A reservoir simulator 
predicts reservoir 
performance by 
solving flow equations 
on a discrete grid 
chosen ... to represent 
the reservoir." 
ing to some rules, as in point-distributed 
grids. Nacul and Aziz3 investigated the use 
of these two grids and four other related 
grids that try to take advantage of the best 
features of the block-centered and point-
distributed grids. They also provided a prac-
tical approach to constructing point-
distributed grids. The idea is first to divide 
the reservoir into blocks and then to adjust 
the grid to meet the requirements of the 
point-distributed grid. Nacul and Aziz 3 
showed that for some problems increasing 
the number of gridblocks by subdividing 
blocks in some regions for block-centered 
grids can actually yield worse results than 
if a uniform grid with fewer blocks (without 
subdivision) is used. It is interesting that, 
in the examples that Nacul and Aziz tried, 
point-distributed grid always produced im-
proved results with grid refinement. 
Local Grid Refinement. For large reservoir 
simulation problems, fine grid is needed 
- -
r--- -~ -- - -y 
~ b- - ~-b---- ~- --b- - - - - 6 1 lij+l 1 1 , 
I I I 
1 b-
- ~-b-- - - 1--- - b- - - - -6 
- .6. l' 1 i-lj I ij 1 i+lj 1 11+ J 1 I I I 
I 1 I I I 
I 
-
-0 
1 1 I 1 
1 I I 1 
b-
-
- ~j-~ -
-
1- __ 
- b- - - - -6 ... 
b) Point-distributed grid x 
x 
Fig. 2-Block-centered and point-distributed grids (from Settari and Aziz2). 
JPT • July 1993 659 
"The need to satisfy in 
field-scale applications 
a variety of constraints 
that often are 
conflicting makes grid 
generation difficult and 
time-consuming." 
I(®) I~ 
I'). ~ 'X! X 
I.'" 
"" 
/' ~ 
Fig. 3-Hybrid grid (from Pedrosa and 
t-
(a) Cartesian (b) locally refined Cartesian 
(c) Cylindrical (d) Hexagonal (e) Curvilinear 
M 
v.a 
I--J 
(f} Hybrid-Cartesian (g) Hybrid-hexagonal 
AZiz 8 ). Fig. 4-Examples of Voronoi grids (from Palagi and Aziz11). 
only in the parts of the reservoir where satu-
rations or pressures change rapidly. Using 
the standard irregular grid leads to unwanted 
small blocks in some parts of the reservoir. 
While Cartesian refinement within a Carte-
sian grid appears to be attractive,4-7 it does 
not always improve the solution. 3 Here, 
the problem is accurate calculation of flow 
between blocks at the intersection of coarse 
and fine grids. The problem is less severe 
when hybrid grids (cylindrical or other cur-
vilinear grids) are used in the region of one 
or more Cartesian blocks, as Pedrosa and 
Aziz 8 proposed, for greater accuracy 
around wells. Even for hybrid grids, cer-
tain assumptions must be made to evaluate 
transmissibilities for flow to and from the 
irregular blocks between the two types of 
grids (Fig. 3). Eikrann9 developed similar 
grids that provide a smooth transition from 
an almost cylindrical grid to the surrounding 
Cartesian grid. Hybrid grids, however, are 
useful for accurate calculation of WOR and 
GOR. 
Locally Orthogonal Grids. Voronoi grid, 
defined in 1908, is extremely flexible and 
locally orthogonal. It has been used exten-
sively in many branches of science and en-
gineering. lO A Voronoi block is defined as 
the region of space closer to its gridpoint 
than to any other gridpoint. Consequently, 
a line that joins gridpoints of any two con-
nected gridblocks is perpendicular to the 
660 
gridblock boundary between these two grid-
points and is bisected into two equal parts 
by that boundary. Voronoi grid can be 
viewed as a generalization of the point-
distributed grid. Heinemann and his col-
leagues,S-7 who pioneered its use for petro-
leum reservoir simulation, called it the PEBI 
grid. PalagilO and Palagi and Aziz ll-13 re-
solved some ofthe problems with its use in 
heterogeneous reservoirs. Fig. 4 shows ex-
amples of Voronoi grids. 
Voronoi grid provides a natural way to 
construct hybrid grids, grids aligned with 
wells and major geological features, and lo-
cally refined grids. It can be used easily by 
constructing and combining modules (Fig. 
5). These modules can be moved, scaled, 
rotated, and placed anywhere in the domain 
of interest. Furthermore, the geometric fac-
tor in transmissibilities, which depends on 
rock permeability and grid geometry, can 
be calculated automatically for any 
grid. 9•10 Because the flow across a bound-
ary is assumed to be proportional to the pres-
sure difference between the gridpoints on 
either side of the boundary, flow calculation 
is most accurate when the line joining these 
gridpoints is bisected at its midpoint by the 
boundary, as in Voronoi grid. Only a limited 
amount of work has been done to establish 
practical guidelines for using various grids 
that are special cases of Voronoi grid. The 
flexibility provided by Voronoi grid is par-
ticularly useful for modeling coning phe-
nomenon in vertical and horizontal wells. 
Our preliminary results show that for homo-
geneous reservoirs the number of blocks in 
refined regions, not the block shape, has the 
most influence on results. 14 Also, to obtain 
both the magnitude and shape of the WOR 
curve correctly, refinement in the aerial 
plane has to be balanced carefully with ver-
tical refinement in the region of the 
producer. Other advantages are that 
(1) these grids provide the possibility of very 
accurate computations for the simulation of 
well tests in complex reservoirs and (2) they 
reduce the grid-orientation effect. II One 
drawback is that they result in much more 
complex Jacobian matrices than standard 
grids. Efficient solution techniques for 
sparse linear systems generated by unstruc-
tured grids are needed. Fig. 6 demonstrates 
the great flexibility of this grid. 
Even greater accuracy and the same flex-
ibility are possible with finite-element and 
CVFE methods, 15-17 but the computational 
cost resulting from the additional complexity 
of the flow equations probably is not justi-
fied for general field-scale applications. 
Corner-Point Geometry. Complex reser-
voir geometries can be represented accurate-
ly by specifying the corners of each 
gridblock. This is known as corner-point ge-
ometry. 18 While the calculations are more 
involved than in standard Cartesian grids, 
all the geometric quantities in Eq. 1 can be 
July 1993. JPT 
(b) Cartesian 
(d) cylindrical 
~ ~ 
( a) hexagonal 
(c) irregular 
Fig. 5-Modules for constructing Voronoi grids (from Palagi and Fig. 6-Flexibility of Voronoi grid (from Palagi and Aziz 11). 
Azizl1). 
calculated. The real problem with this type 
of grid is that flow across a block face now 
depends on more than two pressures on 
either side of that face. This complicates the 
flow term in Eq. 1. The reason is that, when 
the grid is skewed, connections between 
blocks are no longer orthogonal to the block 
faces . Unless all components of the poten-tial gradient at the block face are considered, 
this kind of nonorthogonal grid can lead to 
serious errors in the calculation of interblock 
flow. Almost the same flexibility can be 
achieved with Voronoi grid , which always 
satisfies the condition of local orthogonality. 
Orthogonal and Almost-Orthogonal Cur-
vilinear Grids. Curvilinear grids have been 
used to simulate flow in elements of sym-
metry of pattern floods. 19-22 The grid is 
constructed by solving the potential flow 
equation for streamlines and equipotential 
lines , which are mutually orthogonal. All 
geometric factors in Eq. I can be calculated 
by transforming the flow equations to the 
curvilinear coordinates. As long as the or-
thogonality condition is satisfied , no addi-
tional connections beyond those for 
Cartesian grids are introduced. Because the 
potential solution is used only to construct 
the grid, flow across streamlines is allowed. 
This is the main difference between using 
curvilinear grids and using streamtube 
models that do not allow flow along equi-
potential lines. Sharp and Anderson23 •24 
developed an intriguing method of generat-
ing curvilinear grids that conform to ar-
bitrary internal and external boundaries . 
They solved- a set of quasilinear parabolic 
partial-differential equations (as opposed to 
the elliptic potential equation for standard 
curvilinear grid). This method generates 
nearly orthogonal grids even for complicated 
reservoir problems. But in trying to satisfy 
the orthogonality condition, dense grids may 
be generated in regions where they are not 
needed. Once this kind of grid is generated 
and appropriate geometric factors are cal-
culated, a standard simulator can be used. 
Fig. 7 shows a grid generated by this tech-
nique for a field-scale problem. 
- This algorithm may produce a skewed 
(nonorthogonal) grid when the orthogonality 
condition cannot be satisfied , so it has some 
of the same problems as the corner-point ge-
ometry in terms of flow dependence on all 
components of the gradient of the potential. 
The main advantage of Sharp and Ander-
son's method is that it tries to make the grid 
as orthogonal as possible. 
Automatic Grid Generation With Homo-
geneous Blocks. In the transmissibility cal-
culation (Eq. 2), an effective value of 
permeability between Gridpoints i and j is 
needed. This calculation is simplified if the 
blocks are homogeneous. Techniques are 
Fig. 7-Nearly orthogonal grid for a field (from Sharpe and 
Anderson 23). 
Fig. a-Blocks as homogeneous as possible (from Garcia et 
al. 25 ). 
JPT • July 1993 661 
available for automatically generating grids 
that are as homogeneous as possible. Garcia 
et ai. 25 developed a grid-adjustment meth-
od that associates an elastic band with each 
block edge of all blocks. The potential ener-
gy of an edge is assumed to be proportional 
to the square of the length of the edge and 
a coefficient of elasticity. This coefficient 
of elasticity is made a function of heteroge-
neity of the gridblocks adjacent to the edge. 
Starting with an initial regular grid, a grid 
that is as homogeneous as possible is gener-
ated by minimizing the potential energy of 
the overall grid system. The reSUlting grid 
is not constrained to any orthogonality con-
dition. Fig. 8 shows an example of this 
method. 
Farmer et ai. 26 used another approach to 
achieve the same objective. Methods of this 
type have yet to be applied to field-scale 
reservoir problems. Such grid-adjustment 
methods could be the first step in the gener-
ation of locally orthogonal Voronoi grid. 
Dynamic Grid. In principle, all the static 
grid-generation techniques can be combined 
with dynamic block addition and removal. 
Questions of accuracy and computational ef-
ficiency must be resolved for grids that 
change with time. Several authors5,27,28 
have discussed dynamic Cartesian grids. 
The most practical approach seems to be to 
use a base grid that is fixed and allow dy-
namic refinement or coarsening of some 
blocks within the base grid. Hydrodynamic 
models for 3D atmospheric flows have been 
developed using dynamic Voronoi 
grid. 29 ,30 Conservation equations are 
solved explicitly by moving particles of 
fixed mass. This is called the "free Lagran-
gian method." Some of the gridding tech-
niques developed for other fluid flow 
problems may prove useful in reservoir 
simulation. 
Practical Use 
The need to satisfy in field-scale applications 
a variety of constraints that often are con-
flicting makes grid generation difficult and 
time-consuming. This is particularly true of 
geologically complex fields like the Gullfaks 
in the Norwegian sector of the North 
Sea. 31 To represent the complex system of 
faults in this field, several months are re-
quired to develop an appropriate grid. For 
such reservoirs, comer-point geometry and 
Voronoi grids provide the required flexibil-
ity, but the simulation engineer normally 
will select the one that is easier to use. As 
Petterson31 pointed out, tools that can help 
the reservoir engineer to build a grid quickly 
that accurately computes flow in the reser-
voir are much more important than small 
savings in computer time during simulation 
runs. Data integration tools and 3D visuali-
zation tools are indispensable for grid gener-
ation and analysis of results. Such tools are 
just starting to appear on the market. Even-
tually, expert systems should be available 
to help the engineer integrate reservoir 
description, generate grids, and make 
history matches. In the meantime, selecting 
a type of grid and the number of blocks re-
662 
mains an art. Understanding flow equations 
(Eq. 1) will help minimize errors caused by 
the use of inappropriate grids. To select the 
best grid for modeling a heterogeneous 
reservoir, one must decide on the scales of 
heterogeneities that should be represented 
implicitly through effective parameters 32 
and the method for calculation of the con-
nection transmissibilities. 33 
Concluding Remarks 
The simulation engineer has the opportunity 
to use many kinds of flexible grids. How-
ever, from a practical standpoint, the most 
important problem is selecting a grid for a 
specific problem. As mentioned, an increase 
in the number of gridblocks does not auto-
matically translate into increased accuracy. 
Here are some comments and guidelines. 
1. Local grid refinement (Cartesian or 
hybrid) improves WOR and GOR predic-
tion when sharp saturation gradients exist 
near wells, as in coning problems. The re-
fined region should be large enough to in-
clude the extent of the reservoir with sharp 
saturation gradients. 
2. Unless there are compelling reasons, 
the grid should be orthogonal, at least lo-
cally, and as uniform as possible. Large 
blocks next to small blocks should be avoid-
ed. Irregular Voronoi grid (generalization 
of point-distributed grid) usually is more 
reliable than block-centered grid. 
3. So far, dynamic grids have proved to 
be of limited value in field applications. The 
ability to add or remove blocks efficiently 
with opening or shutting wells, however, is 
useful. 
4. Unstructured Voronoi grids require 
special matrix solution techniques. Domain 
decomposition is useful for taking advantage 
of the grid structure. A large problem can 
be reduced into several smaller problems. 
This also provides a natural approach for 
parallel computations and local time-
stepping. 
5. Expert systems are needed for interac-
tive grid generation to take advantage of 
available gridding possibilities and to reduce 
the time involved in building grids for geo-
logically complex reservoirs. 
6. Analysis of results requires powerful 
3D flow visualization on complex grids. 
Software for visualizing flow over irregu-
lar grids is needed. 
NomenclatureA = block-face area, L 
Ie = transmissibility correction 
factor 
k = absolute permeability, L2 
kr = relative permeability, L2 
L = distance between gridpoints, L 
Me = mass of c in block, M 
Nj = number of connected blocks 
N p = number of phases 
p = phase index 
qe = flow rate from well, L3 It 
T = transmissibility 
f1t = timestep, t 
Vb = block volume, L3 
<l> = potential 
we,p = concentration of c in Phase p 
Subscripts 
i = block for which equation is 
written 
j = block connected to i 
Superscript 
n = timestep 
Acknowledgment 
Reservoir simulation research at Stanford U. 
is supported by an international consortium 
of organizations through the Stanford U. 
Reservoir Simulation Industrial Affiliates 
Program (SUPRI-B). 
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Anaheim, Feb. 17-20. 
28. Biterge, M.B. and Ertekin, T.: "Develop-
ment and Testing of a Static/Dynamic Local 
Grid-Refmement Technique," JPT (April 
1992) 487-95. 
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Lagrangian Hydrodynamics," Los Alamos 
Nati. Laboratory, LA-UR-89-11-79, Albu-
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32. Wattenbarger, C., Orr, F.M., and Aziz, K.: 
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of Flow in Heterogeneous Reservoirs With 
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Provenance 
Original SPE manuscript, Reservoir Simu-
lation Grids: Opportunities and Prob-
lems, received for review Feb. 28, 1993. 
Revised manuscript received May 11, 1993. 
Paper accepted for publication May 11, 
1993. Paper (SPE 25233) first presented at 
the 1993 SPE Symposium on Reservoir Sim-
ulation in New Orleans, Feb. 28-March 3. 
JPT 
663