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

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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


Presentation outline
Presentation Outline

  • Motivation

  • Problem Definition

  • Objectives

  • Approach

  • Program Validation/Evaluation

  • Conclusions


Motivation
Motivation

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


Saturation distribution at pv inj 1 0 for m 0 5
Saturation Distribution at PVinj =1.0 for M=0.5


Saturation distribution at pv inj 1 0 for m 10 0
Saturation Distribution at PVinj =1.0 for M=10.0


Motivation1
Motivation

Brand, Heinemann, and Aziz (1992) –

“In general, Grid Orientation Effect

cannot be overcome with

grid refinement.”

(SPE 21228)


Motivation2
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)

Pruess & Bodvarsson (1983)

Todd et.al. (1972)


Problem definition
Problem Definition

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


Objectives
Objectives

  • Developing a 2-D, 3-Phase reservoir simulator using finite difference formulation

    • Reducing the grid orientation effects in a grid model


Approach
Approach

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





Impes method finite difference equations
IMPES MethodFinite Difference Equations

  • 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:



Hybrid grid block hgb system1

I

I

J

J

N

W

E

S

Hybrid Grid Block (HGB) System

NW

NE

SE

SW



Grid numbering
Grid Numbering

Numbering #1

Example: 18 Grid Blocks


Grid numbering1
Grid Numbering

Numbering #2

Example: 18 Grid Blocks


Grid numbering2
Grid Numbering

Numbering #3

Example: 25 Grid Blocks


Well model
Well Model

Peaceman Well Model

(1983):

Δm

For square gridblock,

where,

α = mass species; oil/water

ro = effective wellbore radius


Well model1

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

Well Model

i


Program validation

Cartesian Sim2D

Cartesian Eclipse 100

HGB Sim2D

Program Validation


Example case
Example Case:

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




Conclusions
Conclusions

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


Conclusions1
Conclusions

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


Conclusions2
Conclusions

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


Conclusions3
Conclusions

  • 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


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