Development of a 2-D Black Oil Reservoir Simulator with Unique Grid-Block System

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Development of a 2-D Black Oil Reservoir Simulator with Unique Grid-Block System

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Development of a 2-D Black Oil Reservoir Simulator with Unique Grid-Block System

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Development of a

2-D Black Oil

Reservoir Simulator with

Unique Grid-Block System

Emeline Chong

July 7, 2004

Harold Vance Department of Petroleum Engineering

- Motivation
- Problem Definition
- Objectives
- Approach
- Program Validation/Evaluation
- Conclusions

Diagonal

Parallel

Not to scale

GRID ORIENTATED PARALLEL

TO INJECTOR-PRODUCER PAIRS (PARALLEL RUNS)

GRID ORIENTATED AT 45

TO INJECTOR-PRODUCER PAIRS (DIAGONAL RUNS)

20 acres

Mobility Ratios :

M = 0.5

M = 1.0

M = 10.0

10 acres

Brand, Heinemann, and Aziz (1992) –

“In general, Grid Orientation Effect

cannot be overcome with

grid refinement.”

(SPE 21228)

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)

Pruess & Bodvarsson (1983)

Todd et.al. (1972)

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

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

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

2D,3-Phase

Cartesian Grid

- Initial Condition
- Rock/Fluid Properties

Well Model

Well Constraints

HGB Grid

IMPES

Matrix

Form

Matrix

Solver

Pn+1, Son+1, Swn+1, Sgn+1

Cutback/Saturation Control

Program Validation

- Oil
- Water
- Gas

IMPES Steps…

- Calculate coefficients of the pressure equation
- Calculate solution of the pressure equation implicitly (matrix equation) for: pn+1
- Calculate solution of the saturation equations explicitly for: Son+1, Swn+1, and Sgn+1

IMPES Method

- 4 unknowns per block:
pn+1, Son+1, Swn+1, and Sgn+1

- To find the unknowns, we need one more equation per block: Son+1 + Swn+1 + Sgn+1 = 1
- Assures fluid volumes fit the pore volume

IMPES Method

- Oil
- Water
- Gas

IMPES Method

Summing up all saturation equations:

IMPES Method

Approximate Vpn+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:

I

I

J

J

N

W

E

S

NW

NE

SE

SW

Numbering #1

Example: 18 Grid Blocks

Numbering #2

Example: 18 Grid Blocks

Numbering #3

Example: 25 Grid Blocks

Peaceman Well Model

(1983):

Δm

For square gridblock,

where,

α = mass species; oil/water

ro = effective wellbore radius

Well Model for regular

polygon (Palagi, 1992):

j = neighbor of wellblock i

bij= side of polygon

dij = distance between gridpoints

N = number of equal sides

i

Cartesian Sim2D

Cartesian Eclipse 100

HGB Sim2D

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

Application ofHGB grid system to Reduce Grid Orientation Error

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

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

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

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

THANK YOU

Acknowledgement

- Dr. David Schechter
- Dr. Erwin Putra
- U.S Department of Energy

Development of a

2-D Black Oil

Reservoir Simulator with

Unique Grid-Block System

Emeline Chong

July 7, 2004

Harold Vance Department of Petroleum Engineering