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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
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
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
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
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