1. Development of a Reservoir Simulator with Unique Grid-Block System
Emeline Chong
Master Division
Student Paper Contest 2004
Harold Vance Department of Petroleum Engineering

2. Presentation Outline Motivation Problem Definition Objectives Approach
Results
Conclusions
My presentation outline consists of ….

3. “Homogeneous Reservoir”
Motivation
“Homogeneous Reservoir”
Diagonal
Parallel
Finite difference solutions of 2D frontal displacement problems can be strongly influenced by the orientation of the of the underlying grid.
To simulate a symmetry element, say a 5-spot problem, 2 orientations of a square grid can be chosen naturally.
These grids are called the diagonal or the parallel grid, as straight lines connecting the injectors and producers diagonal or parallel to the grid lines, respectively.
Whether we like it or not, GOE is a serious problem in numerical simulation.
To investigate the severity of this problem, we conducted 2 case studies, where the diagonal & parallel grid orientation were applied to both a homogeneous & a heterogeneous reservoir model.
This sketch here shows the homogeneous reservoir model
With the black dot represents an injection well and the white dot a production well, and the red lines showing the fluid paths, we can see that the fluid moves preferentially along the grid lines.
Not to scale

4. Homogeneous Reservoir
Now, let us look at the results for the homogeneous reservoir model:-
Supposedly, these 2 models would give the same results since we use exactly the same grid dimension and data but the results show that the recovery and the water cut are different for the 2 models.
The water would breakthrough fastest in the parallel grid since the water move in a direct path to the producer.

5. “Heterogeneous Reservoir”
Motivation
“Heterogeneous Reservoir”
Diagonal
Parallel
The permeability field is isotropic. Two different permeability values - 1mD and
1000mD - are considered in a square domain (Figure 4.4). High permeability areas are
defined by a set of square regions arranged along the diagonal of the field. As before,
water is injected uniformly on the left-hand side; oil is produced on the right-hand side.
In the heterogeneous model, the only changes to the homogeneous model was introducing a permeability anisotropy to the direction as shown to both grids.
Mobility = The ratio of permeability to viscosity.
Not to scale K1 K2
K1 >> K2

6. Heterogeneous Reservoir
In this case, the differences are even more pronounced, especially in the water cut performance.

7. Motivation “In general, Grid Orientation Effect
Brand, Heinemann, and Aziz (1992) –
“In general, Grid Orientation Effect
cannot be overcome with
grid refinement.”
To add matters worse, Brand, Heinemann, and Aziz concluded in their studies that in general, Grid Orientation Effect cannot be overcome with grid refinement.”
(SPE 21228)

8. Shiralkar & Stephenson (1987)
Motivation
Sammon (1991)
Chen & Durlofsky (1991)
Mattax & Dalton (1990)
Wolcott et.al. (1996)
Shiralkar (1990)
Brand et.al. (1991)
Ostebo & Kazemi (1992)
Yanosik &
McCracken
(1979)
Shiralkar & Stephenson (1987)
Even though the GOE is widely known & has been reported by many authors since early 70s, more than 30 yrs later….it still remains a problem many of us choose to ignore.
Todd et.al. (1972)
Pruess & Bodvarsson (1983)

9. Fractures with Multiple Joint Sets
Motivation
Fractures with Multiple Joint Sets
2ndly, a conventional grid model is not able to model permeability anisotropy realistically. For example, features like fractures with multiple joint sets create a high permeability anisotropy & it is quite impossible for a conventional grid to model this.
Courtesy of Imperial College
Fractures create high permeability anisotropy in rock masses!

10. Motivation General Darcy’s Law: Permeability Tensor: Simplified:
If we have a directional (anisotropic) permeability, the Darcy’s Law would look like this:
The permeability becomes a tensor which looks like a matrix:-
However, most reservoir simulators assumed a diagonal permeability tensor, where the off-diagonal terms are set to zero. This is done by orienting the coordinates of the flow system along the axes of permeability.
These axes are orthogonal and will be lined up with the maximum and minimum permeabilities. When the coordinates are oriented in this manner, then each direction has its own permeability and the tensor has only 3 non-zero values.
Kxy = flow in the x-direction due to the pressure drop in the y-direction

11. Problem Definition
Grid orientation and heterogeneity significantly affects the results of reservoir simulation
Therefore, since the grid orientation and heterogeneity ….., we need a more flexible grid model

12. Problem Definition
In these situations, the off-diagonal terms of permeability tensor have a strong impact on fluid flow and therefore should be considered in the calculations.

13. a full tensor representation must be considered!
Problem Definition
We need a grid model that can incorporate permeability anisotropy in multiple directions –
a full tensor representation must be considered!
In these situations, the off-diagonal terms of permeability tensor have a strong impact on fluid flow and therefore should be considered in the calculations.

14. Objectives
Developing a 2-D, 3-Phase reservoir simulator using finite difference formulation
Reducing the grid orientation effects in a grid model
Creating a grid model that can be used to simulate multiple permeability directions
This brings me to the objectives of my study, which include:-

15. Approach
2-D, 3-Phase IMPES finite difference simulator using VBA with unique grid model
To start off, let’s look at the model design and implementation.
A 2D-3P IMPES finite difference simulator is built using VBA with unique grid model which I’ll show in a bit
This simulator assumed no Pc and gravity effects….
Additional features which have been incorporated in this model include the cutback…. And several well constraints as shown.
No flow at the boundaries are assigned by giving the respective transmissibility a zero value at that point.

16. Rock/Fluid Properties Transmissibility Terms
2D,3-Phase
Initial Condition
Rock/Fluid Properties
HGB Model
Well Model
Transmissibility Terms
Well Constraints
Grid Numbering
IMPES
Matrix
Form
Matrix
Solver
A 2D-3P simulator was developed from scratch.
The schematic diagram here simplifies how this is done.
Pn+1, Son+1, Swn+1, Sgn+1
Program Validation

17. Hybrid Grid Block (HGB) SystemJ I J 3 4 5 1 2 I J 1 2 N NW NE W E
As we can see here, now the fluid will flow to 8 directions in the octagon, the rectangle 4, the corner blocks consist of wall triangle and corner triangle with 3 directions each.
Our main challenge here was to solve the connectivity issue for the entire grid system. SW SE S

18. Hybrid Grid Block (HGB) SystemJ 1 2 3 4 5
Example Grid:
5 x 4
Total Number
of Grid Blocks = 61
This unique grid block assignment help to reduce (if not eliminate) the grid orientation effects and it can handle more than 2 directions of permeability anisotropy.

19. IMPES Method Finite Difference Equations
Oil
Water
Gas
The general IMPES formulation was used to solve the finite difference equations. The main changes here are the addition of the transmissibility terms.
In IMPES, flow coefficients are evaluated at the n time level. Pressure equations are solved for implicitly at each time step. Then, the saturations are solved explicitly.
IMPES is fast & accurate for many applications.

20. Grid Numbering #1 & Matrix Form
1 of 3
Grid Numbering #1 & Matrix Form
Example: 3x2 1 2 3 4 5 6 7 8 15 9 10 16 13 14 17 11 12 18
Using this unique grid model, several numbering systems were experimented.
Grid ordering is important as it will affect the order in which we write our linear equations.
Considering a simple 3x2 case, this figure here shows an example grid ordering & the corresponding Jacobian matrix structures.
We can see that matrix formed is spare & irregular.
Numbering system for the 2D grid and the corresponding non-zero co-efficients in the matrix equation for AP = B

21. Grid Numbering #2 & Matrix Form
2 of 3
Grid Numbering #2 & Matrix Form
Example: 3x2 2 4 6 9 11 13 10 12 1 3 5 8 14 15 16 17 18
Numbering system for the 2D grid and the corresponding non-zero co-efficients in the matrix equation for AP = B 7

22. Grid Numbering #3 & Matrix Form
3 of 3
Grid Numbering #3 & Matrix Form
Example: 3x3 10 10 17 17 22 22 25 25 11 11 18 18 1 23 23 24 24 5 5 12 12 19 19 6 6 13 13 20 20
…but currently, we chose this grid numbering as it provides the smallest matrix bandwidth, which means less computational time to solve the matrix equation.
This type of grid numbering also gives us a diagonally dominant matrix form, which we can solve using many common matrix solvers.
This bandwidth gives us a computing advantage requiring less arithmetic to solve the matrix equation.
Gauss Elimination:-
Forward elimination process, we transform matrix A to an upper triangular matrix U.
Backward substitution
Advantage: Gives exact solutions, but subject to round-off error
Numbering system for the 2D grid and the corresponding non-zero co-efficients in the matrix equation for AP = B 2 2 7 7 14 14 21 21 8 8 15 15 3 3 1 1 4 4 9 9 16 16

23. Well Model Peaceman Well Model (1983): For square gridblock, Δm where,
Eq. for boundary-dominated flow
where,
ro = effective wellbore
radius

24. Well Model Well Model for regular polygon (after Palagi,1992): bij dij
In these 4 cases, using the Peaceman Well Model, 1 injector and 2 producers were used.
j = neighbor of wellblock i
bij = side of polygon
dij = distance between gridpoints
Θij = angle open to flow

25. Maximum Material Balance Error = 4.2602E-05%
Results: Case#1
Inj
Prod
Model Dimension:
640 ft x 640 ft x 10 ft
Permeability: 100mD
Porosity: 20%
Well Constraints:-
Const. Qinj
Const. Qo
Maximum Material Balance Error = E-05%

26. Results: Case#1
Contour Map
10 days
20 days
40 days

27. Maximum Material Balance Error = 2.587E-03%
Results: Case #2
Inj
Prod2
Prod1
Same dataset, except:
1 permeability direction
Kmax = 500 mD
Kmin = 100 mD
Maximum Material Balance Error = 2.587E-03%

28. Results: Case #2
Contour Map
10 days
20 days
40 days

29. Results: Case#2

30. Maximum Material Balance Error = 5.2654E-03%
Results: Case #3
Inj
Prod2
Prod1
Same dataset, except:
3 permeability directions
Kmax = 500 mD
Kmin = 100 mD
Maximum Material Balance Error = E-03%

31. Results: Case #3
Contour Map
10 days
20 days
40 days

32. Maximum Material Balance Error = 2.9696E-03%
Results: Case #4
Homogeneous reservoir
1 injector
4 producers
Maximum Material Balance Error = E-03%

33. Pressure Distribution Chart
Results: Case #4
Pressure Distribution Chart

34. Conclusions
Grid orientation and heterogeneity affects significantly the results of reservoir simulation (ie. water breakthrough times & recovery)
A full tensor representation must be considered if reservoir flow performance is to be predicted accurately
As a result of this study, the following general conclusions can be drawn:-

35. Conclusions Proposed HGB model is able to
reduce the grid orientation effects
model different sets of permeability anisotropy
As a result of this study, the following general conclusions can be drawn:-

36. Future Application Local Grid Refinement
We all know that LGR (which involves using a fine grid inside a coarse-based grid) are useful when greater detail is desired in the region of interest.
HGB model can be used as a type of LGR, especially when we have a reservoir with permeability anisotropy in a few directions. HGB will then help to reduce the number of gridblocks necessary to model this reservoir as well as to improve the accuracy of solution in those desired regions.

37. THANK YOU Acknowledgement Dr. Erwin Putra U.S Department of Energy
Dr. David Schechter
Dr. Erwin Putra
U.S Department of Energy

38. Development of a Reservoir Simulator with Unique Grid-Block System
Emeline Chong
Master Division
Student Paper Contest 2004
Harold Vance Department of Petroleum Engineering