connection facilities (CHESHIRE et al (1980))made it practical to incorporate radial refinements in a cartesian global grid. (Figure 3). The use of non-neighbour ...
LOCAL GRID REFINEMENT I.M. Cheshire (~cploration Consultants Ltd) and A. Renriquez (Statoil)
1.
INTRODUCTION
Local grid refinement is a technique which aims to provide improved accuracy in sub-regions of a reservoir simulation study. HEINEMANN et al (1983) discuss dynamic grid refinement to improve the resolution at flood fronts. However, most papers on the subject published since MROSOVSKY et al (1973) focus on enhanced accuracy near wells. The objective is to develop efficient techniques for field scale studies with full resolution of awkward coning effects near troublesome production wells. Figure 1 illustrates typical radial and Cartesian refinements required by reservoir engineers. Growing interest in the application of horizontal wells gives added impetus to the technology. Recent work in local grid refinement has been reviewed by EWING et al (1989). The conventional method for enhanced accuracy in sub-regions is to specify small cells in the region of interest. Because grid lines must extend to the reservoir boundary this leads to a large number of cells (Figure 2), particularly in three dimensions, and computing costs are prohibitive. The introduction of efficient fully implicit simulators with non-neighbour connection facilities (CHESHIRE et al (1980))made it practical to incorporate radial refinements in a cartesian global grid (Figure 3). The use of non-neighbour connections for cartesian refinements (Figure 4) is less satisfactory, because, as pointed out by QUANDALLE et al (1985) it may be necessary to interpolate pressures (and depths) within the global grid in order to compute accurate flows between the global and local grid systems (Figure 5). PEDROSA et al (1985) described a hybrid method which combines IMPES in the global Cartesian grid with the fully implicit method in the local grids surrounding production wells. This technique has considerable merit and is investigated further in the present paper. Because the global and local systems are
338
CHESHIRE P~ND HENRIQUEZ
decoupled at a high level they can be solved independently with obvious potential for parallelization. Computing effort can be concentrated on the difficult local problems without holding up the progress of the overall simulation. To increase the global timesteps we have extended the hybrid method by applying the fully implicit method to the global grid and by the use of smaller local timesteps in the local grids. A material balance correction is introduced at the end of each global timestep to aid stability. The method permits a straight forward application of the pressure and depth interpolation of QUANDALLE et al (1985). Vertical equilibrium may be used in the global grid with dispersed flows in the local grids.
Figure 1:
Typical local grid refinements
LOCAL GRID REFINEMENT
Figure 2:
1
The conventional method for generating local grids near wells
2
4
7
Figure
3:
339
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CHESHIRE
AND
HENRIQUEZ
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4:
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11
12
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LOCAL GRID
Global cell number 5 is replaced by the local grid. The non-neighbour connections are represented by I and u
i~i:::
:::zz~i Figure
5:
The local grid equivalent of Figure 2. Flows between the local and global grids should use interpolated pressures and depths at the points indicated in Figure 2.
LOCAL GRIT) REFINEMENT
2.
THE LINEAR EQUATIONS
With the exception of PEDROSA et al (1985), papers on local grid refinement treat the solution of the global and local systems simultaneously. The linear equations take the form; A.x L.x
+ +
U.y B.y
= =
Rg Ri
(1) (2)
where A is the usual banded matrix for the global system, B is the set of banded matrices for the local systems .and L and U are the coupling terms connecting the local and global systems. The elements of A corresponding to refined cells are either eliminated, since they are inactive, or retained to enhance vectorization but decoupled from other cells in the global system. The latter option is usually chosen since it is then possible to switch the local refinement off dynamically by reactivating the coupling between global neighbouring cells. Hg and Rl are the residuals in the global and local cells respectively and x and y are the solution increments (pressures and saturations) in the global and local systems. The matrix B may be regarded as a set of uncoupled matrices Bi, B2, B3.. .one for each local grid system. Equations 1 and 2 may be solved in a variety of ways with various levels of efficiency for parallel and vector processors.
3.
THE SIMULTANEOUS METHOD
The most natural solution procedure is to apply a powerful iterative method, such as nested factorization (APPLEYARD et al (1983)), to the combined set of equations 1 and 2. Here the L and U terms are simply non-neighbour connections which can be handled by well established techniques. Because the local grids are coupled only to the global grid and not to each other, the solution procedure can be parallelized if the cells are numbered with the global cells first followed by the cells in the first local grid followed by the cells in the second local grid etc. This method converges slowly because the off band coupling terms, L and U, are far from the diagonal. To reduce the number of iterations required to solve the combined equations we must bring L and U closer to the diagonal. This can be done by numbering the cells with the local grids in place. We begin by numbering the cells in the global grid in the usual way until we reach a refined global cell. The next cell is then the first cell in the local grid. The numbering continues until all the local cells within the refined global cell are numbered. The numbering then reverts to the global grid and continues until the next refined global cell is
342
CHESHIRE AND HENRIQUEZ
reached etc. This method is probably the most powerful procedure on conventional machines. There is some loss of efficiency in vector processors because the preconditioned vector runs terminate at globally refined cells but this effect is small in commercial simulators. The main defect of the method is its loss of potential parallelization. The cell numbering system is illustrated in Figure 6.
Ii L L~
2
3
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22
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GLOBAL GRID
6
7
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9
10
11
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13
14
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LOCAL GRID
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Figure
4.
6:
Local cells numbered in place
SEQUENTIAL METHODS In a sequential method we first solve for x in the global grid If the local
and then solve equation 2 for y in the local grids.
LOCAL GRID REFINEMENT
solution variables, y, are formally eliminated from equations 1 and 2 we obtain (A
-
U.B~.L).x
=
Rg
-
U.B~.R1
=
(3)
Rt
To formulate an iterative method for the solution of equation - U.B~.L to serve as the pre-conditioning matrix. If the column sum of the pre conditioning matrix is the same as the column sum of A -U.B’-.L then material will be conserved exactly at each iteration and the number of iterations required to solve the equation will be halved. This suggests that we replace U.B~.L by the diagonal matrix colsum(U.B’.L) and then compute a mass conserving pre conditioning matrix based on an approximation to A colsum(U.B~-.L).
3, we must construct a good approximation to A
At each iteration we must update the residual calculating r
=
Rt
-
(A
-
(error) by
(4)
U.B~.L).x
This calculation must be performed precisely. If an approximation to B is used at this step the wrong equation will be solved. It follows therefore that a precise solution of each local problem must be computed at each iteration of the global problem and the method becomes unattractive. If the local grids are sufficiently small to permit a direct solution method to be used then the inverse of B need only be computed once and the calculation of r becomes less expensive. A more practical decomposition
iterative
C
L
U
D
=
procedure
C U
xl
is
based
on
the
C.L
E
I
(5)
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=
D
-
colsum(U.C.L)
(6)
At each iteration, the solution sequence is C.z E.x C.y
= = =
Rl Rg - U.z R - L.x
(7) (8) (9)
344
CHESHIRE ~.ND HENRIQUEZ
and the residuals are rl rg
= =
Ri Rg
-
B.y U.y
-
L.x A.x
(10) (11)
A method analogous to that outlined above has been proposed by EWING et al (1989). The technique requires two inversions of the approximate local grid matrix C at each iteration. However, since the local grids are not coupled to each other this calculation may be parallelized.
5.
THE HYBRID METhOD
PEDROSA et al (1985) proposed a hybrid method in which, at each timestep, a solution is first obtained in the global grid using IMPES, as if the local grid did not exist. Wells in the local grids are replaced by pseudo wells in the global grid. The pressures and saturations in the global grid at the end of each timestep are then used as boundary conditions for the fully implicit solution within the local grids. This technique is highly efficient and clearly maintains full vectorization and parallelization potential. It has a strong appeal to common sense. IMPES is normally applicable to field scale studies with large grid cells and is unsuitable for the study of detailed coning effects near wells. By using the fully implicit method in the local grids we appear to have the best of both worlds. The local grids may be either radial or cartesian. The automatic dimensioning feature of ECLIPSE makes it relatively easy for us to implement the hybrid method. The simulator is effectively converted to N+1 simulators where N is the number of local grids. In an attempt to overcome the IMPES timestep limitations in the global grid we have also applied the fully implicit method to the solution of the global system. In practice however we have also found that it is necessary to restrict the timestep in the global grid due to the fundamentally explicit nature of the method. To retain material balance and to prevent the development of explosive instabilities we have found it necessary to apply careful material balance corrections at the end of each global timestep. Material in the local cells is summed and substituted for the erroneous material in the corresponding pseudo global cell. Material in global cells adjacent to refined cells is corrected by accounting for the difference between the flows from
LOCAL GRID REFINE~4ENT
345
global to local cells and the flows from global to pseudo global cells. A distinct advantage of the hybrid method is that smaller timesteps may be used in the local grids than those used in the global grid. This possibility was first investigated by QUANDALLE et al (1985). To obtain boundary conditions for the local grid problems the pressures and saturations in the global grid are assumed to vary linearly with time over a global timestep. Thus detailed coning effects near troublesome wells can be resolved without holding up the progress of the full field simulation. Switching refinements on and off at various times during a simulation is a useful option which is relatively simple to implement in the hybrid method. When the refinement is switched off the pseudo cells in the global system are treated as normal global cells. In the loop over refined grids we simply skip any calculations for the refinements which have been switched off. When a refinement is switched on then the material in the global pseudo cell is initially distributed uniformly within the local grid. It is therefore advisable to switch the refinement on several timesteps before the local wells are activated to obtain a better local fluid distribution. Another advantage of the hybrid method is that vertical equilibrium may be used in the global grid with dispersed flow in the local grids. This option is particularly effective in ECLIPSE which contains a collapsed VE option in which a 3D global grid is treated as a 2D areal system while retaining full vertical variation of rock properties. The main defect of the hybrid method is that the pseudo cells in the global grid do not fully represent the local grids. The material balance corrections referred to above certainly help to maintain stability but it may also be necessary to develop special pseudo functions for the pseudo global cells to help mimic the behaviour of the local systems they represent. Pseudo wells in the global pseudo cells must be used to represent real wells in the local grids. At the end of each global timestep we compute phase dependent mobility multipliers for each pseudo well to align production rates between the pseudo and local wells. These multipliers are used at the next global timestep. Phase dependent transmissibility multipliers may also be required for the pseudo cells before the hybrid method becomes sufficiently reliable for routine use in a commercial simulator. This requirement is the focus of our current research and will be reported more fully in future.
346 6.
CHESHIRE ~ND HENRIQUEZ
ThE USER INTERFACE
Static local grid refinements can be modelled in any standard implicit simulator with a non-neighbour connection facility. The user sets up separate grid systems for the global and local models together with a set of non-neighbour transmissibilities connecting the local grids to the global grid. The refined cells of the global grid must be disconnected from other cells in the global grid or simply blanked out. The disadvantages of this approach are that it is extremely cumbersome for the user and there is some loss of accuracy because pressures are not interpolated in the global grid when calculating flows between the global and local systems. A practical local grid refinement option should make its application as easy as possible for the user. The calculation of transmissibilities between the global and local grids must be computed automatically by the simulator. Data preparation should be as simple as possible. Output reports of pressures and saturations etc., should be reported separately for global and local grids. 7.
PRACTICAL APPLICATIONS
Some field scale applications of local grid refinement are discussed below. These illustrate horizontal well modelling, gas and water coning, condensate dropout and well interference. All CPU times quoted are for an IBM 3090 with vector facility. Horizontal wells may be represented in a simulation by modifying Peaceman’s formula or by the inclusion of very small blocks of unit porosity and high horizontal permeability (KOSSAK et al (1987)). To test the local grid refinement option in ECLIPSE we used the high permeability method adding an additional 60 blocks to represent the well in a 1440 block 3-phase simulation. The problem was solved first using non-neighbour connections (NNCs) to attach the local well model to the global grid and secondly using the hybrid method with small local timesteps. The CPU times for the three cases were; Modified Peaceman in the global grid High permeability local grid with NNCs Hybrid method with small local timesteps
510 seconds 372 seconds 213 seconds
This test convinced us of the clear advantages of small local timesteps and that the hybrid method was the only practical way of including large numbers of local refinements within a full field study.
LOCAL GRID REFINEMENT
Gas and water coning are characterized by large saturation changes in short times in the vicinity of production wells. Vertical spatial resolution of under one metre is often required. Large timesteps may be used once a stable cone has developed but short timesteps are required during the development of the cone and at gas or water breakthrough. Timesteps as short as one minute are not uncommon and these may occur at different times in different local grids. Spatial refinement alone is therefore insufficient to avoid prohibitive computing costs and the small timesteps required for coning calculations must be isolated to the local grid regions. The hybrid method with local timesteps was therefore used in the remaining examples. Figure 7 shows an areal view of the simulation grid with 25*23*19 cells covering an area of 6km by 8km. This grid illustrates the need for cell amalgamation in the peripheral regions.
—
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7:
—
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—
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.
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~
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—
—
Horizontal well and coning, full field simulation grid. The well is located in the middle of the finer grid section.
348
CHESHIRE AND HENRIQtJEZ
Figure 8 shows an XZ cross section with the fluid contacts indicated by dotted lines. Figure 9 shows the refinement needed to capture the coning behaviour giving an additional 3780 cells. Figures 10 and 11 show water and gas coning in a cross section perpendicular to the horizontal well and for a cross section along the length of the well. Figure 12 shows the water cut with and without local grid refinement. As expected, breakthrough occurs earlier when local grid refinement is used. CPU time for the unrefined simulation with 10925 cells was 853 seconds compared with 1723 seconds for the local grid refinement run with the additional 3780 cells.
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8:
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LOCM~ GRID PEFINEMENT
Figure
9:
Figure 10:
349
Horizontal well arid coning, grid refinement for simulating the horizontal well.
Horizontal well and coning, YZ cross section of cone: cross section perpendicular to the well showing the hatched blocks where there is no reduction in the oil saturation.
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LOCAL GRID
REFINEMENT
Figure 13 shows a 3D full field grid of a gas condensate reservoir with the location of several 5*5 refinements. Figure 14 shows a cross section with two of the refinements. Figure 15 shows the oil saturation against distance from a well for runs with and without local grid refinement. The case for local grid refinement is clearly illustrated by the improved resolution of condensate dropout near the well.
Figure 13:
Condensate dropout, 3D view of simulation grid: the location of the wells are marked with black grid cells.
CHESHIRE AND HENRIOtJEZ
Figure 14:
Condensate dropout, grid cross section: the refinement at two well locations is shown in a vertical cross section.
0.50 0
~
a
—•—
0.40
‘-.
Global grid cells Localgridcells
LL~
j
S
0.20 .. —
•. —
0.10 0.30 0.00
I
0
20
I
40
60
80
100
120
140
160
180
200
Distance from well (Metres)
Figure 15:
Condensate dropout, resolution of dropout: liquid saturation for a coarse well block simulation and for a refined well block simulation as a function of the distance to the well.
LOCAL GRID REFINEMENT
The global grid has 2699 active cells and 1750 local cells. The study was run without refinement, with refinement using NNCs and with refinement using the hybrid method. CPU times were 262, 638 and 377 seconds respectively which once again illustrates the relative efficiency of the hybrid method with short local timesteps. In our final example we test the validity of PEACEMAN (1988) approximate method for representing the interference effect when several wells are included in the same grid cell. Our case had 24 wells in one block of 1600 by 1600m. Using local grid refinement the wells were resolved areally within an 11 by 11 local grid. Five new layers were also introduced to accurately represent the depths at which the wells are perforated. The global grid is shown in Figure 16. The areal and vertical refinements are shown in Figures 17 and 18 respectively. Figure 19 shows the production profile and Figure 20 compares the pressure in the refined block using Peaceman’s correction with that obtained using local grid refinement. The results show the validity of Peaceman’s approximation in this case. For cases where the aspect ratio of the block does not warrant the use of Peaceman’s approximation, or where several phases are present it is expected that grid refinement will give results which are numerically more correct.
Figure
16:
Well interference, simulation grid for full field: the block with 5 effective wells in the gas region is marked.
CHESHIRE ~ND HENRIQUEZ
Figure 17:
Well interference, XY cross section of the grid with refined blocks: the refinement has llxllx5 blocks.
Figure 18:
Well interference, YZ cross section of the grid with refined blocks: the refinement has 5 extra layers in the second layer of the global grid. The wells are perforated in the refined layers 1 to 4.
355
LOCAL GRID REFINE~NT
Locally refined grid Normal grid and well correction from Peaceman
interference
w z
0 C) D 0
0 0. Ci)
0
TIME
Figure 19:
Well interference, gas production for the centre. Locally refined grid Normal grid and well correction from Peaceman
5 well
interference
U C’) Cl) U 0.
TIME
Figure 20:
Well interference, pressure in the refined block.
356
8. 1. 2.
3. 4. 5. 6.
9.
CHESHIRE AND HENRIQUEZ
CONCLUSIONS The hybrid method offers a practical solution to the local grid refinement problem in field scale applications. Local time stepping is easily incorporated into the hybrid method arid has proved to be more efficient than alternative methods. Material balance corrections are necessary in the practical application of the hybrid method. Global timesteps are smaller using the hybrid method than for comparable runs without local grid refinement. Well pseudos have proved to be a useful extension to the hybrid method. The automatic dynamic generation of pseudos for globally refined cells may improve the technique further and remove the timestep restriction on the global solution. ACKNOWLEDGEMENTS
The authors would like to thank Oystein Lie of Statoil who performed the simulation runs and Jon Holmes of ECL who coded the well model in the ECLIPSE local grid option. This work has been supported financially by Statoil, Norsk Hydro and Saga and the permission of these companies to publish the results is gratefully acknowledged. 10.
REFERENCES
1.
HEINEMANN Z.E., GERMIN G., VON HANDELMAN G., (November 1983), Using Local Grid Refinement in a Multiple Application Reservoir Simulator, SPE12255, presented at the Seventh SPE Symposium on Reservoir Simulation, San Francisco California.
2.
MROSOVSKY I. and RIDINGS R.L., (January 1973), TwoDimensional Radial Treatment of Wells within a ThreeDimensional Reservoir Model, SPE4286, presented at the Third Numerical Simulation Conference in Houston Texas
3.
EWING R.E., BOYErI’ B.A., BABU D.K., HEINEMANN R.F., (February 1989). Efficient Use of Locally Refined Grids for Multipurpose Reservoir Simulation, SPE18413, presented at the Tenth SPE Symposium on Reservoir Simulation, Houston, Texas.
4.
CHESHIRE I.M., APPLEYARD J.R., BANKS D., CROZIER R.J., and HOLMES J.H., (October 1980). An Efficient Fully Implicit Simulator, EUR179, presented at the European Offshore Petroleum Conference, London.
LOCAL GRID REFINE?Th~NT
QUANDALLE P., arid BESSET P., (February 1985).
Reduction of Grid Effects due to Local Sub-Gridding in Simulations using a Composite Grid, SPE13527, presented at the Eighth SPE Symposium on Reservoir Simulation, Dallas Texas. PEDROSA O.A., AZIZ K., (February 1985). Use of Hybrid Grid In Reservoir Simulation, 5PE13507, presented at the EighthSPESyinposiumonReservoirSimulation, DallasTexas. APPLEYARD J.R., CHESHIRE I.M., (November 1983). Nested Factorization, 5PE12264 presented at the Seventh SPE Symposium on Reservoir Simulation, San~ Francisco California. QUANDALLE Flexible 5PE12239, Reservoir
P., BESSET P., (November 1983). The Use of Gridding for Improved Reservoir Modeling, presented at the Seventh SPE Symposium on Simulation, San Francisco California.
K055A1( C.A. and KLEPPE J. Oil Production from the Troll Field: A Comparison of Horizontal and Vertical Wells, SPE16869 PEACEMAN D.W,, (September 1988). Near Singularities of Pressure and Concentration of the Welibore in Reservoir Simulation, presented at the First International Forum on Reservoir Simulation, Alpbach, Austria.
