Computational Challenges in Reservoir Modeling. Sanjay Srinivasan The Pennsylvania State University

Transcription

1 Computational Challenges in Reservoir Modeling Sanjay Srinivasan The Pennsylvania State University

2 Well Data 3D view of well paths Inspired by an offshore development 4 platforms 2 vertical wells 2 deviated wells 8 horizontal wells

3 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

4 Seismic impedance Impedance

5 Case Study Summary Reference reservoir is a fluvial reservoir Porosity and permeability histogram show distinct bi-modality Frequency Porosity across facies All Facies Porosity Number of Data Number of Data 1916 mean 0.16 std. mean dev coef. of var 0.63 std. Dev. coef. Of var maximum 0.34 upper quartile 0.27 median 0.14 lower quartile 0.06 minimum porosity Compartmentalization of reservoir into pay / non-pay decision of stationarity

6 Facies classification basis Seismic impedance in channels, mudstone and border sands : Impedance in mudstone in mudstone Impedance in channel in sand channel Impedance in border in sand border sand Frequency Number Number of Data mean mean std. dev coef. of var 0.07 Frequency Number Number of Data mean std. dev coef. of var 0.07 Frequency Number 9286 Number of Data 9686 mean mean std. dev coef. of var Impedance Impedance Pay Facies: Non-pay facies: Impedance Impedance Impedance Impedance channel sands mudstone + splays + levies

7 Work flow simulate multiple pay / non-pay facies models within non-pay, simulate border facies multiple facies templates 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

8 Pay / Non-pay facies modeling Available Data 1) Vertical proportion curve Stratigraphic depth Reference reservoir / surfaces Well data / reference surfaces Well data / model surfaces uncertainty in modeled surfaces Proportion of sand uncertain vertical proportion curve

9 Semi-Variogram Defined as: γ XY = In the spatial context, 1 2 E { 2 } ( X Y ) { ( ) ( } ( Z u u ) 2γ ( u, u + h) = E Z + h Well markers - Top surface 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 ( ) ( } 1 ( Z u Z u + h ) = Var{ Z( u) ( u h)} 1 γ ( u, u + h) = E Z = γ (h) stationary

10 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) 0 Curve shape depends on nature of phenomena h

11 Facies modeling - Data Indicator variogram - hard data Horizontal indicator variograms Vertical indicator variogram I n d i c a t o r v a r i o g r a m s γ o azimuth 135 o azimuth γ Distance Seismic impedance Horizontal variogram Depth γ o azimuth 135 o azimuth Distance

12 Facies modeling - variograms Horizontal borrowed from seismic γ o azimuth 135 o azimuth Vertical from hard data Distance γ Depth

13 Spatial Estimation Simple kriging (SK) of the depth d( x o ) at unsampled locations from marker data d ( ), α = 1,.., n SK estimate: SK system: ( ) Note: u = x, d Requisites { D )} ma D n * β = 1 ( x λ o β ) x α = 1 λ m C( x α + A α α λ α α = 1 x β ) = C( x α n x o D( x xo α ) ), α = 1,..., n E (x = average depth-data over study A C(h) covariance modeled after a spatial average C A (h) over stationary area A

14 Indicator Paradigm Consider the indicator RV: Important property: or better still, given n data:

15 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

16 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 L k :

17 Indicator Expansion Another way to write the expansion: basis function

18 Normal Equations The implication of the projection theorem is that: or in terms of projections: which is a system of normal equations

19 Examples - unbiasedness - reproduction of indicator covariances In general: - reproduction of the 3 rd order covariances requires knowledge of up to order covariances indicator

20 Facies modeling - indicator approach Indicator kriging (Simple IK) [ ] = + = ) ( 1 * ) ( ) ( ) ( u u u u n p I p I α α α λ Interpretation { } data pay n u Prob I ) ( * = u IK System ) ( ) ( ) ( 0 ) ( 1 α β αβ β λ h h u u I n I C C = =

21 Facies modeling - indicator approach Indicator kriging Simulation Draw r Uniform[0,1] If then r Prob u pay n data I ( u) = 1: u pay else I local conditional probability Prob u pay n data { } { } ( u) = 0: u nonpay Add simulated I (u) into data set : (n) (n+1) Visit next node u'

22 Facies modeling - indicator approach Locally varying mean I * Requisites ( u) = p V ( d) + n( u) [ ] vertical proportion of pay facies in stratigraphic layer d : (d) α = 1 λ( u α p V ) I( u α ) p V ( d α ) Indicator variogram model, γ I (h)

23 Indicator Simulation - Results Slices through the facies model XY slice #7 2-D Slice # XZ slice #20 YZ slice #20 2-D XZ Slice # D YZ Slice # 40 Pay pay North North North East East Nonpa nonpay East Variogram reproduction Horizontal indicator variograms γ o azimuth 135 o azimuth Distance

24 Indicator Simulation - Results Vertical proportion Stratigraphic depth target simulated Global pay proportion Frequency before trans Hist. of global proportion (before trans) Number of Data Number of Data coef. of mean var 0.07 coef. Of var Sand proportion Frequency after trans Hist. of global proportion (after trans) Number of Data Number of Data 50 mean 0.35 coef. mean of var 0.01 coef. Of var

25 Indicator Simulation - Results Perform a histogram transformation Simulation after trans XY slice #7 2-D XY Slice # XZ slice #20 YZ slice #20 2-D XZ Slice # D YZ Slice # 40 Pay pay North North North East East Nonpa nonpay East

26 Facies modeling - using seismic info. Likelihood of seismic impedance u { S( u) s pay} F( s 1) = Prob u pay Similarly, { S( u) s nonpay} F( s 0) = Prob u If: F( s 1) F( s 0) then impedance discriminates pay / nonpay Frequency Likelihood of meas. of Zimp impedance (Non-pay) in nonpay Number of Data 279 mean std. dev coef. of var 0.12 maximum mean upper quartile median lower quartile minimum No. of Data Frequency Likelihood of impedance in nonpay Likelihood of meas. Zimp (Pay) No. of Data mean Number of Data 174 mean std. dev coef. of var 0.11 maximum upper quartile median lower quartile minimum Impedance Impedance

27 Facies modeling - using seismic info. Want Prob { I u) 1 s( u) [ s, ] l s } ( = l+ 1 Apply Bayes inversion: Prob B A Prob{ A B } = Prob B Prob { } { } { I u) 1 s( u) [ s, ] l s } ( = l+ 1 Prob{ A} { s( u) [ s ] l, sl I( u) = 1} { s( u) [ s, s ] } Prob = Prob l F( sl + 1) F( sl 1) = Prob s( u) s, s l+ 1 { [ ] } { I( ) 1} + 1 u = l+ 1 p( 1 u l ) Prob

28 Facies modeling - Bayes inversion Requisites Prior vertical proportion, p(d) Likelihood, F(s 1) Seismic impedance distribution, F(s) Implementation p ( ) { I( u) = 1 s( u) [ s, ] l s } ' u = Prob l+ 1 used as locally varying mean after standardization 1 p'( x, d) = pv ( d) where u= ( x, d) N x d layer d pay proportion

29 Bayes inversion - Results Slices through facies model XY slice #7 before trans Facies Slice 7 before trans XY slice #7 after trans Facies Slice 7 after trans -T pay North North nonpay East Vertical proportion Stratigraphic depth East target simulated Sand proportion

30 Facies modeling- object based fluvsim (Deutsch and Wang (1996)) Modeling using reversible coord. transforms channel complex channel Stratigraphic coord. transformation channel complex areal transformation channel areal transformation

31 Facies modeling - object based Simulation procedure in transformed space annealing simulation of channel complex O cc = p p * + p ( d) p ( d)* + I( ) I ( α G where G n z V V n u α u global proportion vertical proportion well data I( u) = 1 if simulated u channel perturb channel complex until convergence annealing simulation of channel perturb channel until convergence back coord. transform to obtain reservoir model CC )

32 Fluvsim - results Facies model ( well mismatch: 11.2 % ) XY slice #7 Fluvsim only vertical prop. XY Slice XZ slice #20 YZ slice #20 Fluvsim only vertical prop. XZ Slice Fluvsim only vertical prop. YZ Slice 20 pay Nonpa North North North East East nonpay Pay East Vertical proportion curve Stratigraphic depth simulated target Sand proportion

33 Fluvsim model - using seismic info. Average impedance vs. areal pay proportion Areal proportion (wells) vs. impedance Impedance vs. areal proportion Impedance Impedance No. of Data Number of data 19 correlation rank correlation rank correlation Areal Proportion Areal proportion Cokrige areal proportion map using: vertically averaged proportion data exhaustive 2-D seismic impedance variogram of facies proportion co-located cokriging under MMI

34 Fluvsim - using seismic info. Cokriged areal proportion map Cokriged areal proportion Cokriged areal prop North East fluvsim slices ( well mismatch - 14 %) XY slice #7 Areal prop. on reference reservoir XZ slice #20 Fluvsim Slice 20 XZ with areal prop. Fluvsim YZ Slice slice 20 YZ with #20 areal prop. pay 500 Pay North North East North East nonpay Nonpa East

35 Fluvsim - using seismic info. fluvsim vertical proportion Stratigraphic depth target simulated impact of areal proportion Areal proportion - reference Areal prop. on reference reservoir Sand proportion Areal proportion - reproduction Areal prop. in sim. with no areal Areal prop. - unconditional Areal prop. in sim. with no areal North North North East East East

36 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

37 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

38 Indicator Expansion Another Interpretation In general the extended equations are for all 2 n realizations of the n data. If instead, the estimate corresponding to a specific realization of the indicator RVs, say Unbiasedness: Orthogonality:, then: Two bases 1,D two equations to obtain two unknowns Single Normal Equation

39 Calibration of geologic information Geologic model Variogram two-point statistics traditional γ h North East i(u;sk) mp statistics s t (u) = { s(u+h α ), α = 1,, n t } Set of data events = training data set

40 GrowthSim components Matching Configuration Data Configuration

41 Pattern Statistics pattern observation

42 Multipoint Statistics To condition realization to hard (static) data Sample from distribution P(A B ) Geological / Hard information Well data Use Bayes Rule P( A B) = PA (, B) P( B) MP Histogram where A = Simulation pattern B = Information from conditioning data & geology A single point simulation event Strebelle (2000), Caers (1999) A multiple point simulation event GROWTHSIM, Arpat (2005)

43 Pattern Statistics optimized template goal reduce noises reduce spurious pattern configurations Training Image (100 x 100) Template Domain (7 x 7) CORRELATION Spatial Correlation

44 Pattern Statistics effects of optimized template

45 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

46 GrowthSim 100x100, 2-category data TI

47 Multiple Grid Simulation effects of multiple-level simulation

48 Calibration of geologic information Geologic model Variogram two-point statistics traditional γ An exhaustive training image required for inference of statistics h North East mp statistics i(u;sk) s t (u) = { s(u+h α ), α = 1,, n t } Set of data events = training data set

49 Noisy spatial covariance (variogram) inferred using sparse data, modeled using smooth functions restrictive, unrealistic

50 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.

51 Moments and Cumulants Moment Generating Function Expanding the function e ωz as a Taylor series about the origin: Cumulant generating function of Z is the Neperian logarithm of the moments generating function M:

52 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.

53 Polyspectra

54 Power Spectrum For a stochastic process Z(t), energy is defined as

55 Power Spectrum Detecting orientations 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

56 Power Spectrum Detecting orientations Orientation calculated using eigenvectors of the inertia matrix computed by thresholding the Fourier transform

57 Bispectrum Bispectrum Bispectrum

58 Integrated 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)

59 Detecting Shapes

60 Bispectrum from sparse data 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? Consider the inverse transform: Image reconstructed using only highest 15% of Fourier coefficients

61 Compressed Sensing used for geostatistical simulation (Jafarpour, Goyal, McLaughlin, & Freeman, 2009) assuming a DCT unique solution no uncertainty quantification

62 Sparse reconstruction Estimate Fourier transform coefficients from scattered data solving a regularization problem least mixed norm (LMN) [p=2, q=1] Optimum lambda established using jack-knife Bandlimited coefficie nts

63 Sparse reconstruction some results Conditioning data Reconstruction Log(Amplitude)

64 An improvement 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

65 Can we detect shapes using sparse data? Reference Models Sparse Conditioning Data

66 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

67 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