Computational Challenges in Reservoir Modeling Sanjay...
Transcript of Computational Challenges in Reservoir Modeling Sanjay...
Computational Challenges in Reservoir Modeling
Sanjay Srinivasan The Pennsylvania State University
Well Data
3D view of well paths • Inspired by
an offshore development
• 4 platforms
• 2 vertical wells
• 2 deviated wells
• 8 horizontal wells
Sample well log • Well data include:
– horizontal location – true vertical depth – porosity – permeability – density – sonic log – lithology – layer indicator
• Well markers: true vertical depth when well intercepts a layer surface
Seismic impedance
Impedance
Case Study Summary • Reference reservoir is a fluvial reservoir • Porosity and permeability histogram show distinct bi-modality
Compartmentalization of reservoir into pay / non-pay – decision of stationarity
Freq
uenc
y
porosity
0.000 0.100 0.200 0.300 0.400 0.5000.0000
0.0100
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700All Facies Porosity
Number of Data 1916mean 0.16
std. dev. 0.10coef. of var 0.63
maximum 0.34upper quartile 0.27
median 0.14lower quartile 0.06
minimum 0.02
Porosity across facies Number of Data
mean std. Dev.
coef. Of var
1916 0.16 0.10 0.63
porosity 0.00 0.10 0.20 0.30 0.40 0.50
Facies classification basis
• Seismic impedance in channels, mudstone and border sands :
Pay Facies: channel sands Non-pay facies: mudstone + splays + levies
Freq
uenc
y
Impedance
2500. 4500. 6500. 8500. 10500. 12500.0.000
0.100
0.200
0.300
Impedance in mudstone
Number of Data 59311mean 8527.0
std. dev. 597.9coef. of var 0.07
Freq
uenc
y
Impedance
2500. 4500. 6500. 8500. 10500. 12500.0.000
0.100
0.200
0.300
Impedance in channel sand
Number of Data 51003mean 7927.7
std. dev. 532.3coef. of var 0.07
Freq
uenc
y
Impedance
2500. 4500. 6500. 8500. 10500. 12500.0.000
0.100
0.200
0.300
Impedance in border sand
Number of Data 9686mean 8251.9
std. dev. 560.6coef. of var 0.07
Impedance in mudstone
Number mean
59311 8527
Impedance in channel
Number mean
51003 7928
Impedance in border sand
Number mean
9286 8252
Impedance 12500 8500 4500
Impedance Impedance 12500 8500 4500 12500 8500 4500
Work flow • simulate multiple pay / non-pay facies models • within non-pay, simulate border facies
• simulate porosity models separately in pay and non-pay
• same for permeability • merge pay / nonpay, porosity / permeability
models using matching facies template multiple equi-probable reservoir models
multiple facies templates
Pay / Non-pay facies modeling Available Data 1) Vertical proportion curve
uncertainty in modeled surfaces
uncertain vertical proportion curve
Reference reservoir / surfaces
Well data / reference surfaces
Well data / model surfaces
Proportion of sand 0.1 0.3 0.5 0.7 0.9
Stra
tigra
phic
dep
th
0.0
0.2
0.4
0.6
0.8
1.0
Semi-Variogram
Defined as: In the spatial context, Under intrinsic stationarity – the RF Z(u) itself might not be stationary, but increments of the RF {Z(u)-Z(u+h)} are presumed stationary. Thus:
{ }2)(21 YXEXY −=γ
( ) ( ){ }2)(),(2 huuhuu +−=+ ZZEγ
( ) ( ){ } ( ) ( )}{21)(
21),( 2 huuhuuhuu +−=+−=+ ZZVarZZEγ
stationary )(hγ=
Well markers - Top surface
0. 500. 1000. 1500. 2000. 2500.
0.
1000.
2000.
3000.
200.0
300.0
400.0
Variogram Inference • Inference of the experimental
variogram requires: – Plotting h-scatterplot by
systematically varying h in particular spatial directions.
– For each scatterplot compute the corresponding moment of inertia of the scatterplot.
– Plot the moment of interia versus the lag h in specific directions
Some remarks •There must be sufficient number of pairs in each scatterplot (statistical mass) •Outliers (abnormal high or low values) can cause the moment of inertia to fluctuate wildly
γ(h)
h 0
Curve shape depends on nature
of phenomena
Facies modeling - Data • Indicator variogram - “hard” data
γ
0 . 0 0 0
0 . 1 0 0
0 . 2 0 0
0 . 3 0 0
0 . 4 0 0
γ
I n d i c a t o r v a r i o g r a m s
0 . 0 0 0
0 . 1 0 0
0 . 2 0 0
0 . 3 0 0
0 . 4 0 0
γ
0 . 0 0 0
0 . 0 5 0
0 . 1 0 0
0 . 1 5 0
0 . 2 0 0
• Seismic impedance
45o azimuth 135o azimuth
Horizontal indicator variograms
0.0 400.0 800.0 1200.0 Distance 0.0 0.4 0.8
Vertical indicator variogram
Depth
Horizontal variogram
45o azimuth 135o azimuth
0.0 400.0 800.0 1200.0 Distance
Facies modeling - variograms Horizontal borrowed from seismic Vertical from “hard” data
γ
0 . 0 0 0
0 . 0 5 0
0 . 1 0 0
0 . 1 5 0
0 . 2 0 0
γ
0 . 0 0 0
0 . 0 5 0
0 . 1 0 0
0 . 1 5 0
0 . 2 0 0
0 . 2 5 0
45o azimuth 135o azimuth
0.0 400.0 800.0 1200.0 Distance
0.0 Depth
0.4 0.8 1.2
Spatial Estimation Simple kriging (SK) of the depth at unsampled locations from marker data
nd ,..,1),( =ααx)( od x ox
)(1)(1
* ∑ ⋅+⋅
∑−=
=
nAo DmD
ααα
αα λλ xx
nCC on
,...,1,)()(1
=−=−⋅∑=
αλ αβαβ
β xxxx
SK estimate:
SK system:
Requisites
{ } AmDE =)(x average depth-data over study A covariance modeled after a spatial average over “stationary” area A
)(hC)(hAC
( )dxu ,: =Note
Indicator Paradigm
• Consider the indicator RV:
• Important property:
or better still, given n data:
Indicator basis function • Consider the expansion defined on the
basis of n indicator random variables : - Given the n indicator random variables, this
expansion is the most complete possible. - There are a total of terms in
the above expansion - There are 2 outcomes for each RV there are
outcomes possible for the function . - These outcomes define the space
Indicator Expansion
• The conditional expectation is precisely the projection of the unknown indicator event on to .
• If instead the projection function is defined on a reduced basis Lk:
Indicator Expansion
• Another way to write the expansion: basis function
Normal Equations
• The implication of the projection theorem is that:
or in terms of projections: which is a system of normal equations
Examples
- unbiasedness - reproduction of indicator
covariances
- reproduction of the 3rd order covariances
In general: requires knowledge of up to order indicator
covariances
Facies modeling - indicator approach Indicator kriging (Simple IK)
[ ]∑=
−⋅+=)(
1
* )()()(u
uuun
pIpIα
ααλ
Interpretation
{ }datapay nuProbI |)(* ∈=u
IK System
)()()( 0
)(
1α
βαββλ hhu
u
I
n
I CC =⋅∑=
Facies modeling - indicator approach Indicator kriging local conditional probability
{ }datapay nuProb |∈
• Draw ]1,0[Uniform∈rSimulation
If then else
{ }data|pay nuProbr ∈≤pay:1)( ∈= uuI nonpay:0)( ∈= uuI
• Add simulated into data set : (n) (n+1) • Visit next node
)(uI'u
Locally varying mean Requisites
• vertical proportion of pay facies in stratigraphic layer :
• Indicator variogram model,
Facies modeling - indicator approach
[ ]∑=
−⋅+=)(
1
* )()()()()(u
uuun
VV dpIdpIα
αααλ
)(dpVd
)(hIγ
Indicator Simulation - Results • Slices through the facies model
2-D Slice # 7
East
Nor
th
0.0 100.0000.0
120.000
• Variogram reproduction
2-D XZ Slice # 40
East
Nor
th
0.0 100.000500
5002-D YZ Slice # 40
East
Nor
th
0.0 120.000500
500
Nonpa
Pay
γ
0 . 0 0 0
0 . 1 0 0
0 . 2 0 0
0 . 3 0 0
XY slice #7
XZ slice #20 YZ slice #20 pay
nonpay
45o azimuth 135o azimuth
0.0 400.0 800.0 1200.0 2000.0 1600.0 Distance
Horizontal indicator variograms
Indicator Simulation - Results • Vertical proportion
Freq
uenc
y
0.300 0.350 0.400 0.450 0.5000.000
0.050
0.100
0.150
0.200
0.250Hist. of global proportion (before trans)
Number of Data 50mean 0.38
coef. of var 0.07
Freq
uenc
y
0.300 0.350 0.400 0.450 0.5000.00
0.20
0.40
0.60
0.80
Hist. of global proportion (after trans)
Number of Data 50mean 0.35
coef. of var 0.01
• Global pay proportion 0.20 0.80 0.40
Stra
tigra
phic
dep
th
0.0
0.2
0.4
0.6
0.8
1.0
0.60 Sand proportion
Number of Data mean
coef. Of var
50 0.38 0.07
before trans after trans
Number of Data mean
coef. Of var
50 0.35 0.01
simulated target
0.0 1.0
Indicator Simulation - Results • Perform a histogram transformation • Simulation after trans
2-D XY Slice # 7
East
Nor
th
0.0 100.0000.0
120.000
2-D XZ Slice # 40
East
Nor
th
0.0 100.000500
5002-D YZ Slice # 40
East
Nor
th
0.0 120.000500
500
Nonpa
Pay
XY slice #7
XZ slice #20 YZ slice #20 pay
nonpay
Facies modeling - using seismic info. • Likelihood of seismic impedance |
pay∈u{ }pay)()1( ∈≤= uu sSProbsF
Similarly,
{ }nonpay)()0( ∈≤= uu sSProbsF
• If: then impedance discriminates pay / nonpay
)0()1( sFsF ≠
Freq
uenc
y
Imp
4000. 6000. 8000. 10000. 12000.0.000
0.040
0.080
0.120
0.160
Likelihood of meas. Zimp (Non-pay)Number of Data 279
mean 8575.4std. dev. 1054.6
coef. of var 0.12maximum 9950.0
upper quartile 9450.0median 8850.0
lower quartile 7850.0minimum 5750.0
Freq
uenc
y
Imp
4000. 6000. 8000. 10000. 12000.0.000
0.100
0.200
0.300
Likelihood of meas. Zimp (Pay)Number of Data 174
mean 6540.2std. dev. 707.49
coef. of var 0.11maximum 9350.0
upper quartile 6850.0median 6350.0
lower quartile 6150.0minimum 5150.0
No. of Data mean
279 8575
Likelihood of impedance in nonpay
4000 6000 8000 10000 12000 Impedance
No. of Data mean
174 6540
Likelihood of impedance in nonpay
4000 6000 8000 10000 12000 Impedance
Facies modeling - using seismic info. • Want
• Apply Bayes’ inversion:
[ ]{ }1,)(|1)( +∈= ll sssIProb uu
{ } { }{ }
{ }AProbBProb
ABProbBAProb ⋅=||
[ ]{ }1,)(|1)( +∈= ll sssIProb uu
[ ]{ }[ ]{ }
{ }1)(,)(
1)(|,)(1
1 =⋅∈
=∈=
+
+ uu
uu IProbsssProb
IsssProbll
ll
[ ]{ })(
,)()1|()1|(1
1 uu
psssProb
sFsFll
ll ⋅∈−
=+
+
Facies modeling - Bayes’ inversion
Requisites – Prior vertical proportion, – Likelihood, – Seismic impedance distribution,
)(dp)1(sF
)(sF
[ ]{ }1,)(|1)()(' +∈== ll sssIProbp uuu
used as locally varying mean after standardization
)(),('1 dpdxpN V
xd
∑ =
Implementation
layer d pay proportion
where ),( dxu=
Bayes’ inversion - Results • Slices through facies model
Facies Slice 7 after trans -T
East
Nor
th
0.0 100.0000.0
120.000
• Vertical proportion
Facies Slice 7 before trans -
East
Nor
th
0.0 100.0000.0
120.000
pay
nonpay
XY slice #7 before trans XY slice #7 after trans
0.0 0.2 0.4
Stra
tigra
phic
dep
th
0.0
0.2
0.4
0.6
0.8
1.0
0.6 Sand proportion
target simulated
0.8 1.0
Facies modeling- object based fluvsim (Deutsch and Wang (1996)) Modeling using reversible coord. transforms
channel complex
channel
Stratigraphic coord. transformation
channel complex areal transformation
areal transformation
channel
Simulation procedure in transformed space – annealing simulation of channel complex
Facies modeling - object based
∑∑ −+−+−=n
CCn
VVGGcc IIdpdpppOz
)()(||*)()(||*| αα uu
global proportion vertical proportion well data
where if simulated 1)( =uI channel∈u
– annealing simulation of channel – perturb channel until convergence – back coord. transform to obtain reservoir model
– perturb channel complex until convergence
Fluvsim - results Facies model ( well mismatch: 11.2 % ) Vertical proportion curve
Fluvsim only vertical prop. XY Slice 6
East
Nor
th
0.0 100.0000.0
120.000
Fluvsim only vertical prop. XZ Slice 20
East
Nor
th
0.0 100.000500
500Fluvsim only vertical prop. YZ Slice 20
East
Nor
th
0.0 120.000500
500
Pay
Nonpa
XY slice #7
XZ slice #20 YZ slice #20 pay
nonpay
0.2 0.4
Stra
tigra
phic
dep
th
0.0
0.2
0.4
0.6
0.8
1.0
0.8 Sand proportion
target
simulated
0.0 0.6 1.0
Fluvsim model - using seismic info. • Average impedance vs. areal pay proportion
Cokrige areal proportion map using:
– vertically averaged proportion data – exhaustive 2-D seismic impedance – variogram of facies proportion – co-located cokriging under MMI
Impe
danc
e
Areal Proportion
Areal proportion (wells) vs. impedance
0.0 0.20 0.40 0.60 0.80 1.007000.
8000.
9000.
10000.
Number of data 19correlation -0.49
rank correlation -0.25
Impedance vs. areal proportion
Areal proportion
Impe
danc
e
No. of Data correlation
rank correlation
19 -0.49 -0.25
Fluvsim - using seismic info. • Cokriged areal proportion map
Fluvsim Slice 20 XZ with areal prop.
East
Nor
th
0.0 100.000500
500Fluvsim Slice 20 YZ with areal prop.
EastN
orth
0.0 120.000500
500
Nonpa
Pay
Cokriged areal prop. -
East
Nor
th
0.0 100.0000.0
120.000
0.3000
0.3500
0.4000
0.4500
0.5000
0.5500
0.6000
0.6500
0.7000
0.7500
0.8000
• fluvsim slices ( well mismatch - 14 %) Areal prop. on reference reservoir
East
Nor
th
0.0 100.0000.0
120.000
0.30
0.55
0.80
Cokriged areal proportion
XY slice #7
XZ slice #20 YZ slice #20 pay
nonpay
Fluvsim - using seismic info. • fluvsim vertical proportion
Areal prop. on reference reservoir
East
Nor
th
0.0 100.0000.0
120.000Areal prop. in sim. with no areal
East
Nor
th
0.0 100.0000.0
120.000
0.3000
0.3500
0.4000
0.4500
0.5000
0.5500
0.6000
0.6500
0.7000
0.7500
0.8000
• impact of areal proportion Areal proportion - reference Areal prop. - unconditional
0.30
0.55
0.80
Sand proportion 0.0 0.2 0.4
Stra
tigra
phic
dep
th
0.0
0.2
0.4
0.6
0.8
1.0
0.6
target
simulated
0.8 1.0
1.0
Areal prop. in sim. with no areal
East
Nor
th
0.0 100.0000.0
120.000
Areal proportion - reproduction
Facies modeling - Some remarks • Object based models geologically realistic but
require numerous shape parameters
• trans post-processing of indicator simulations essential
• Integration of seismic has significant impact
Indicator basis function • Consider the expansion defined on the
basis of n indicator random variables : - Given the n indicator random variables, this
expansion is the most complete possible. - There are a total of terms in
the above expansion - There are 2 outcomes for each RV there are
outcomes possible for the function . - These outcomes define the space
Indicator Expansion – Another Interpretation In general the extended equations are for all 2n realizations of the n data. If instead, the estimate corresponding to a specific realization of the indicator RVs, say , then: Unbiasedness:
Orthogonality:
Two bases 1,D – two equations to obtain two unknowns
Single Normal Equation
Calibration of geologic information
East
Nor
th
0.0 100.0000.0
150.000
Geologic model
i(u;sk)
st(u) = { s(u+hα), α = 1, … , nt }
Set of data events = training data set
mp statistics
Variogram – two-point statistics
γ h
traditional
GrowthSim components
Matching Configuration
Data Configuration
Pattern Statistics pattern observation
To condition realization to hard (static) data Sample from distribution P(A | B )
where A = Simulation pattern B = Information from conditioning data & geology
Geological / Hard information Well data
Multipoint Statistics
( , )( | )( )
P AP BAP
BB
=Use Bayes’ Rule
MP Histogram
A – single point simulation event
Strebelle (2000), Caers (1999)
A – multiple point simulation event
GROWTHSIM, Arpat (2005)
Pattern Statistics optimized template ◦ goal reduce noises reduce spurious pattern configurations
CO
RR
ELAT
ION
Training Image
(100 x 100)
Template Domain
(7 x 7)
Spatial Correlation
Pattern Statistics effects of optimized template
GrowthSim algorithm
1. gather conditioning data around some node in the simulation grid
2. look for matching pattern configurations to the conditioning data in the statistics
3. apply one of the matching pattern configuration to the simulation grid
4. repeat 1-3 around the newly simulated node, until the simulation grid is filled
GrowthSim 100x100, 2-category data TI
Multiple Grid Simulation
effects of multiple-level simulation
Calibration of geologic information
East
Nor
th
0.0 100.0000.0
150.000
Geologic model
i(u;sk)
st(u) = { s(u+hα), α = 1, … , nt }
Set of data events = training data set
mp statistics
Variogram – two-point statistics
γ h
traditional An exhaustive training image required for inference of statistics
Noisy spatial covariance (variogram) inferred using sparse data, modeled using smooth functions restrictive, unrealistic
• Develop a geostatistical simulation method that does not rely on an exhaustive training model
• Perform inference and modeling of spatial connectivity functions in the spectral domain using sparse data.
Moment Generating Function
Expanding the function eωz as a Taylor series about the origin:
Cumulant generating function of Z is the Neperian logarithm of the moments generating function M:
Moments and Cumulants
Relation between the first few moments and cumulants
Moments can be calculated from the cumulants by
Three-point moment is a measure of similarity between three spatial locations. High value implies
three locations jointly have high values of the spatial attribute Z Spatial cumulant is also a
measure of spatial connectivity.
Moments and Cumulants
Polyspectra
For a stochastic process Z(t), energy is defined as
Power Spectrum
Property of Fourier transform - if an image is rotated, its Fourier transform also rotates
changes in direction of spatial continuity rotation in Fourier transform space orientation of objects can be detected from power spectrum
Power Spectrum – Detecting orientations
Orientation calculated using eigenvectors of the inertia matrix computed by thresholding the Fourier transform
Power Spectrum – Detecting orientations
Bispectrum
Bispectrum
Bispectrum
Phase of Fourier transform is a nonlinear function of frequency, extracted by biphase
Biphase Phase of Fourier spectrum
Need to find a feature identifier from bispectrum that can distinguish object shapes within the images - invariant under translation, scaling, rotation, and amplification Define the integrated bi-spectrum: Define phase of integrated bi-spectrum
Identifier 𝑃(𝑎) is invariant to translation, scaling, rotation and amplification (Nikias and Raghuveer, 1987 and others)
Integrated Bispectrum
Detecting Shapes
So far: Higher-order spectra can provide some interesting insights into spatial characteristics of reservoir models However, FFT requires an exhaustive dataset or image (such as a training image) What about when only scattered data is available?
×106
2
6
4
0
Image reconstructed using only highest 15% of Fourier coefficients
Bispectrum from sparse data
Consider the inverse transform:
used for geostatistical simulation (Jafarpour, Goyal, McLaughlin, & Freeman, 2009) assuming a DCT unique solution – no uncertainty quantification
Compressed Sensing
Estimate Fourier transform coefficients from scattered data solving a regularization problem Optimum lambda established using jack-knife
least mixed norm (LMN) [p=2, q=1]
Band-limited coefficients
Sparse reconstruction
Log(Amplitude) Reconstruction
Conditioning data
Sparse reconstruction – some results
One idea to improve quality of reconstruction - limit the search space to low frequency coefficients
How to automatically select the search space? Apply a jackknife method.
Randomly select 10% of the conditioning data (validation data) and use
the remaining data for reconstruction
An improvement
Reference Models Sparse Conditioning Data
Can we detect shapes using sparse data?
Our goal is to use the integrated bi-spectrum computed using the Fourier transform from sparse data to classify the models reflecting different features
Can we detect shapes using sparse data?
Conclusions • Performing inference and modeling of connectivity statistics in the Fourier space
provides an avenue for modeling geological realism in a computationally efficient manner
• Computation of higher order moments, cumulant and polyspectra using sparse data appears feasible
• Detection of size, orientation and shapes of features consistent with available conditioning data is an interesting aspect of this research training image selection
• While a broad idea about reservoir structures is possible using the power spectrum, the location of objects and details of the shapes are only possible by identifying phase bispectrum, higher-order spectra