Development of a 2-D Black Oil Reservoir Simulator with Unique Grid-Block System Harold Vance Department of Petroleum Engineering July 7, 2004

Presentation Outline Motivation Problem Definition Objectives Approach Program Validation/Evaluation Conclusions

Motivation Not to scale Diagonal Parallel

GRID ORIENTATED PARALLEL TO INJECTOR-PRODUCER PAIRS (PARALLEL RUNS) GRID ORIENTATED AT 45  TO INJECTOR-PRODUCER PAIRS (DIAGONAL RUNS)

Mobility Ratios : M = 0.5 M = 1.0 M = acres 10 acres

Saturation Distribution at PV inj =1.0 for M=0.5

Saturation Distribution at PV inj =1.0 for M=10.0

Motivation Brand, Heinemann, and Aziz (1992) – “In general, Grid Orientation Effect cannot be overcome with grid refinement.” (SPE 21228)

Motivation Todd et.al. (1972) Yanosik & McCracken (1979) Pruess & Bodvarsson (1983) Shiralkar & Stephenson (1987) Shiralkar (1990) Brand et.al. (1991) Sammon (1991) Chen & Durlofsky (1991) Ostebo & Kazemi (1992) Mattax & Dalton (1990) Wolcott et.al. (1996)

Grid orientation effect significantly affects the results of immiscible displacements in reservoir simulation Problem Definition

Objectives Developing a 2-D, 3-Phase reservoir simulator using finite difference formulation Reducing the grid orientation effects in a grid model

Approach 2-D, 3-Phase IMPES finite difference simulator with unique grid model  “Sim2D”

2D,3-Phase Initial Condition Rock/Fluid Properties Well Model Cartesian Grid IMPES HGB Grid Matrix Form Matrix Solver P n+1, So n+1, Sw n+1, Sg n+1 Program Validation Well Constraints Cutback/Saturation Control

Sim2D Demo

Sim2D VB Application

Pressure Plots

IMPES Method Finite Difference Equations Oil Water Gas

1) Calculate coefficients of the pressure equation 2) Calculate solution of the pressure equation implicitly (matrix equation) for: p n+1 3) Calculate solution of the saturation equations explicitly for: S o n+1, S w n+1, and S g n+1 IMPES Steps…

4 unknowns per block: p n+1, S o n+1, S w n+1, and S g n+1 To find the unknowns, we need one more equation per block: S o n+1 + S w n+1 + S g n+1 = 1 Assures fluid volumes fit the pore volume IMPES Method

Oil Water Gas IMPES Method

Summing up all saturation equations: IMPES Method

Approximate Vp n+1 on the right hand side using the identity: and a chord slope: IMPES Method

Then, where, Likewise, IMPES Method

Final equation can now be written as: IMPES Method

Hybrid Grid Block (HGB) System

I J I J W E N S NE SE SW NW Hybrid Grid Block (HGB) System

Grid Numbering Example: 18 Grid Blocks Numbering #1

Numbering #2 Grid Numbering Example: 18 Grid Blocks

Numbering #3 Grid Numbering Example: 25 Grid Blocks

Well Model Peaceman Well Model (1983): For square gridblock, ΔmΔm where, α = mass species; oil/water r o = effective wellbore radius

Well Model Well Model for regular polygon (Palagi, 1992): j = neighbor of wellblock i b ij = side of polygon d ij = distance between gridpoints N = number of equal sides i

Program Validation Cartesian Sim2D Cartesian Eclipse 100

Example Case: Two-Dimensional Areal Model Showing Primary Depletion of an Undersaturated Reservoir (One Producer Well, One Injector Well, Isotropic, 2-Phase, Oil/Water)

Validation with Eclipse

Application of HGB grid system to Reduce Grid Orientation Error

Saturation Distribution Map for diagonal HGB

Saturation Distribution Map for parallel HGB

Conclusions Grid orientation effect was observed in rectangular Cartesian grid models even at isotropic and homogeneous reservoir with favorable mobility ratio.

Conclusions Grid refinement can minimize the grid orientation effect in rectangular Cartesian grid models at favorable mobility ratios.

Conclusions At an unfavorable mobility ratio, neither the parallel grid, diagonal grid nor grid refinement is effective in reducing the grid orientation effect.

Conclusions HGB is able to minimize the grid orientation effect even for unfavorable mobility ratio displacement problems, with relative difference of about 6%.

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

Development of a 2-D Black Oil Reservoir Simulator with Unique Grid-Block System Harold Vance Department of Petroleum Engineering July 7, 2004