Stochastic inverse modeling under realistic prior model constraints with multiple-point geostatistics

Jef Caers

Petroleum Engineering DepartmentStanford Center for Reservoir ForecastingStanford, California, USA

AcknowledgementsI would like to acknowledge the contributions of the SCRF team, in particular Andre Journel and all graduate students who contributed to this presentation

QuoteTheory should be as simple as possible,but not simpler as possible.

Albert EINSTEIN

Overview Multiple-point geostatistics Why do we need it ? How does it work ? How do we define prior models with it ?

Data integration Integration of multiple types/scales of data Improvement on traditional Bayesian methods

Solving general inverse problems Using prior models from mp geostatistics Application to history matching

Part IMultiple-point Geostatistics

Limitations of traditional geostatisticsVariograms EW312Variograms NS2-point correlation is not enough to characterize connectivity A prior geological interpretation is required and it is NOT multi-Gaussian123

Stochastic sequential simulation Define a multi-variate (Gaussian) distribution over the random function Z(u)

Decompose the distribution as followsOr in its conditional form

Practice of sequential simulation

Multiple-point GeostatisticsReservoirmultiple-pointdata eventP ( A | B ) ?SequentialsimulationAB

Extended Normal Equationsuua

Single Normal Equation

The training image module

Training image module = standardized analog modelquantifying geo-patternsSNESIM algorithmRecognizing P(A|B) for all possible A,B

The SNESIM algorithmTraining imageData template (data search neighborhood)Search treeConstruction requires scanning training image one single time

Minimizes memory demand

Allows retrieving all training cpdfsfor the template adopted!

Probabilities from a Search Tree uSearch neighborhoodSearch treeTraining image 5 4 3 2j=1i = 1 2 3 4 5 u uLevel 0 (no CD)Level 1 (1 CD)Level 4 (4 CD)...... 1 2 3 4 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 u 1 2 3 4

Example400 sample data RealizationTrue imageTraining imageCPU

2 facies, 1 million cells

= 4 30

On 1GHz PC

Where do we get a 3D TI ?Training image requires "stationarity" Only patterns = "repeated multipoint statistics" can be reproducedValid training imageNot Valid

Modular training imageModular ? * no units * rotation-invariant * affinity-invariantTraining ImageModels generated with snesimusing the SAME training image

Properties of training imageRequired

Stationarity: patterns by definition repeat Ergodicity: to reproduce long range feature => large image Limited to 4-5 categories

Not required Univariate statistics need not be the same as actual field No conditioning to ANY data Affinity/rotation need not be the same

Part IIMultiple-point Geostatisticsand data integration

Simple question, difficult problemA geologist believes based on geological data that there is 80% chance of having a channel at location X

A geophysicist believes based on geophysical data that there is 75% chance of having a channel at location X

A petroleum engineer believes based on engineering data that there is 85% chance of having a channel at location X

What is the probability of having a channel at X ?The essential data integration problemP(A|B)P(A|C)P(A|D)P(A|B,C,D)?

Combining sources of information

Conditional independenceO = In practice not necessarilyYES

Correcting conditional independence

Permanence of ratios hypothesis

Advantages of using ratios No term P(B,C), hence McMC is not required

Work with P(A|B),P(A|C), more intuitive than P(B|A),P(C|A)

Verifies all consistency conditions by definition

It is still a form of independence, Yet dependence can be reintroduced

Simple problemP(A|B) = 0.80

P(A|C) = 0.75 => P(A|BC)

Suppose P(A) = 0.5 => P(A|BC) = 0.92Suppose P(A) = 0.3 => P(A|BC) = 0.95 = compounding of events

Lesson learned : if geologist and geophysicist agreefor almost 80%, you can be even more certain that thereis a channel !

Example reservoirP(A|C)Training imageP(A|B)Single realization

P(A|C), A = single-point !P(A|C)RealizationWhen combing P(A|B) fromgeology and P(A|C) from seismic to P(A|BC),

A is still a single point event !

Certain patterns, such aslocal rotation will be ignored

Honor seismic only as a single-point probability?

Concept of MODULAR training imageModular ? * Stationary patterns * rotation-invariant * affinity-invariant * no unitsModular Training ImageModels generated with snesimusing the SAME training image

Local rotation angle from seismicP(A|C)Local angle

Results2 realizations with anglewithout angle

Constrain to local channel featuresP(A|C)Hard data from seismicSoft data from seismic

Part IIIInverse modeling withmultiple-point geostatistics

Application to history matching

Production data does not inform geological heterogeneityaaaaaaaaaaaaaa

ApproachMethodology

Define a non-stationary Markov chain that moves a realization to match data, two properties

At each perturbation we maintain geological realism use term P(A|B)

Construct a soft data set P(A|D) such that we move the current realization as fast as possible to match the data => Optimization of the Markov chain at each step

Methodology: two faciesD = set of historic production data (pressures, flows)Some notation:Initial guess realization: Realization at iteration

Define a Markov chainDefine a transition matrix:

Transition matrix2 x 2 transition matrix describes the probability of changing facies at location u and we define it as follows

Parameter rD

Determine rDUse P(A|D) as a probability model in multiple-point geostatistics

Combine P(A|B) (from training image) with P(A|D) from production data D into P(A|B,D)

Allows generating iterations that are consistent with prior geological vision

Allows combining geological information with production data

Allows determining an optimal value for rD as follows

rD determines a perturbationrD=0.01Some initial modelrD=0.1rD=0.2rD=0.5rD=1Find rD that matches bestthe production data= one-dimensional optimization

Complete algorithm Construct a Training Image with the desired geological continuity constraint

Use snesim (P(A|B)) to generate an initial guess

Until adequate match to production data D

Define a soft data P(A|D) as function of rD

Perform snesim with P(A|D) to generate a new guess

Find the value of rD that matches best the data D

ExamplesGenerate 10 reservoir modelsthat 1. Honor the two hard data 2. Honor fractional flow 3. Have geological continuitysimilar as TIIP

Single model

rD values, single 1D optimizationrD valueObjective function

Different geology

More wells

Hierarchical matching* First choose fixed permeability per facies,perturb facies model

* Then, for a fixed facies perturb the permeability within facies(using traditional methods, ssc, gradual deformation

ExampleIP

ResultsKlow = 50Khigh = 500Klow = 50Khigh = 500Klow = 12Khigh = 729Klow = 12Khigh = 729Klow = 11Khigh = 694Klow = 150Khigh = 750

Results

More realisticReferenceInitial modelmatched model

ConclusionsWhat can multiple-point statistics provide

Large flexibility of prior models, no need for math. def.

A fast, robust sampling of the prior

A more realistic data integration approach than traditional Bayesian methods

A generic inverse solution method that honors prior information

More on conditional independenceQ? why should eB and eC be independent unless they are homoscedastic, i.e. independent of A ?Q? Is it not a mere transfer of independence hypothesis to eB and eC