Usage of X-ray CT in Dual Porosity Simulation.

Usage of X-ray CT in Dual Porosity Simulation.

Prasanna K Tellapaneni

Presentation Outline

• Motivation

• Problem Definition

• Objectives

• Approach

• Validation

• Conclusions

Dual PorosityMicro-Fractures/Fractures

Vugs

Actual Grid Block Idealized Grid Block

Primary Porosity

Secondary Porosity

Dual Porosity

))(.()( ghpB

kkS

t ppfpfpf

rpffppf

)( mf ppk Shape Factor

Shape factor is the bone of contention in dual porosity simulation.

2L

a

Motivation

• There are 23 transfer functions present in the literature – Ries and Cil (1998)

• No experimental backing

Motivation

Other assumptions

•Linear pressure gradient

•“Pseudo-Steady State” assumption of matrix blocks

•Rn = n(R1)

•Four unknowns per grid block.

Motivation

Problem Definition

Imbibition laboratory experiments are excluded in dual porosity simulation

Objectives

• Modeling imbibition experiments to obtain unique transfer function.

• Development of a dual porosity simulator with the derived empirical transfer function and its validation.

Approach

•Develop Dual Porosity Simulation Formulation

•Model Imbibition Experiments

•Derive Empirical Transfer Function

•Validate with a Commercial Dual Porosity Simulator

))(.()( ghpB

kkS

t oofofof

roffowf

Dual Porosity Flow Equations

))(.()( ghPpB

kkS

t wcfofwfwf

rwffwwf

t

Sppk wm

omofww

)(Conventional Dual Porosity

Approach

Combining Aronofsky (1958) and deSwaan (1978)

dS

eR wft

o

tD

)(

Empirical Transfer Function

Approach

R

Water Tank

Core

Weight Balance

Data acquisition system

Garg et al Experiment

Approach

Imbibition Experiments

Spontaneous Imbibition Experiments

Spontaneous Imbibition in Double Porosity Modeling

BrineCore plug

Glass funnel

Oil bubble

Oil recovered

Governing Equation

w

cwro

ow S

pfk

k)S(D

t

S

x

SSD ww

w

)(

Assumptions

Fracture submerged in water

Matrix

Fracture submerged in water

Matrix

No gravity effectOnly Pc as driving forceFluid and rock are incompressible

Spontaneous Imbibition Modeling in Single Porosity Simulation

Approach

Approach

)log(0

0

wcc

nwrwrw

SPP

Skk

Corey’s CorrelationWe need X-ray CT

X-ray CT Imbibition Experiments

Approach

X-Ray Tube

Detector Array

Ro

tati

ng

D

uri

ng

Sca

nCT brine = 0 H

CT Berea = ~1400 H

CT air = -1000 H

X-Ray Tube

Detector Array

Ro

tati

ng

D

uri

ng

Sca

nCT brine = 0 H

CT Berea = ~1400 H

CT air = -1000 H

X-Ray Emitters

X-Ray Detectors

80s 120s 160s

200s 320s 360s

80s 120s 160s

200s 320s 360s

Approach

X-ray CT Result Simulation Result

Approach

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Height

CT

Wa

ter

Sa

tura

tio

n

80 s

120 s

160 s

200 s

320 s

360 s

Krwo= 0.045 and N=8.5

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90

Squre Root of Time, Seconds

Am

ou

nt

of

Wa

ter

Imb

ibin

g i

nto

th

e C

ore

, F

rac

tio

n

Approach

Curve Fitting ParametersRinf = 1.0lamda = 0.031

Approach

))(.()( ghPpB

kkS

t wcfofwfwf

rwffwwf

))(.()( ghpB

kkS

t oofofof

roffowf

Recalling the flow equations

owf

t

o

tw d

SeR D

)(

In order to solve this non-linear equations we use Newton-Raphson’s method.

t

Spa w

o

ioEk

ioE iwn

iwk

in

in

a a S S p p - - -+

- -+ + + ( - ) - 1 1 1

11 1

1 1

Writing oil-phase equation

Discretizing using finite difference

SwSwtBV

ppappannopn

ini

noE i

ni

ni

noE i ) - (

/ - = ) - ( + ) - ( 1+1+1+

1+1+1+1+

1-1+

1-

) - ( /

- =

)]()([ - ) - ( + +

1+

01

1+1+1+

1+

SStBV

etStSRppSSaa

nwiw

nop

n

j

n

jk

tjwfjwf

ni

ni

kwi

nw ioE i

koEi

i

k

Expanding

Approach

Approach

The equations can be written as

BXA

RBXA

X

RXXRXXR

)()(

Writing the Taylor Series Expansion

X

X

R

XR

)(

X

R

XRXX oldnew

)(

Newton Raphson’s Solution

Validation

• Kazemi et al (SPE 5719)

• 2 – D Kazemi Grid (Extension of SPE 5719)

•Comparison with Sub Domain Method

Validation

Kazemi et al (SPE 5719)

1800

1810

1820

1830

1840

1 3 5 7 9

Node Number

Pre

ss

ure

(p

si)

0

0.04

0.08

0.12

Wa

ter

Sa

tura

tio

n (

Fra

cti

on

)

Pressure Kazemi Pressure Saturation Kazemi Saturation

Kazemi et al 2-D grid

Validation

Pressure profile along a line parallel to X axis

Validation

1960

1961

1962

1963

1964

1965

0 2 4 6 8 10

X-Direction Node Number

Pre

ss

ure

(p

si)

Simulated Pressure ECLIPSE pressure

Well

Dual Porosity Sub Domain Method

Fracture Fracture

Matrix

Fracture

MatrixMatrix

Validation

Sub Domain Method

Validation

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000

Time (Days)

Ma

trix

Wa

ter

Sa

tura

tio

n (

Fra

cti

on

)

SubDomain Dual Poro Using 10 Grid Blocks Using 15 gridBlock

Limitations

• Stability

• Material Balance

Empirical transfer functions are proposed for dual porosity simulation

X-ray CT is used for modeling imbibition experiments

Dual porosity simulator is developed and validated with test cases.

Recap

ConventionalDual Porosity This Study

•Four unknowns per grid block.

•Two unknowns per grid block.

•No experimental backing – based on Darcy’s Law

•Honors experiments – derived from experiments.

•Non-Standard Formulation.

•Standard Formulation

Conclusions

Conclusions

•Too many “Best Guess” values

•Fewer “Best Guess” values

•Linear pressure gradient.

•Pressure difference is not used.

•Rn = nR1 •Rn = sum(Ri)

•Psuedo-steady state assumption

•Even transient state is modeled.

This StudyConventionalDual Porosity

• Standardized formulation of dual porosity simulation

• Reduction in simulation time and computation efficiency

• Better reservoir management by accurate fluid flow simulation

Value to the industry

Acknowledgement

• Dr. Schechter, Texas A&M University

• Dr. Erwin Putra, Texas A&M University

• Department of Energy

Usage of X-ray CT in Dual Porosity Simulation.

Usage of X-ray CT in Dual Porosity Simulation.

Prasanna K Tellapaneni