Permeability Estimation Using A Hierarchical Markov Tree (HMT) Model

Jingbo WangCorporate Strategic Research, ExxonMobil Company

Phone: 908-730-2057 Email: jingbo.wang@exxonmobil.com

Nicholas ZabarasSibley School of Mechanical and Aerospace Engineering, Cornell University

Phone: 607-255-9104 Email: zabaras@cornell.edu

Outline

Overview of the problem Problem definition Challenges

Fundamentals of Bayesian statistics Markov Random Field (MRF) Markov chain Monte Carlo (MCMC) simulation Hierarchical Bayesian model for permeability estimation Examples Conclusions

Permeability estimation

• Permeability of the porous medium is a necessary input for simulation of reservoir and groundwater system• Approaches:

Local (core measurement + correlation modeling) Global (inverse modeling)

• The inverse modeling approach: Estimate permeability using flow data (pressure, concentration, …)

Heterogeneity of a porous medium Schematic of a 9-spot problem

Challenges of the inverse problem

• ill-posedness --- existence? --- uniqueness? --- continuous dependence of solutions on measurements? (stability)

identifiability

• implicit objective function (functional)

• non-linearity

• complex direct simulation

• high computation cost

• uncertainties

• Heterogeneity of permeability

• Multiscale nature of permeability

Solution procedure

optimizationobjective

minimum least-squares error

minimum total absolute error

minimum maximum absolute errord

eter

min

isti

c Newton’s methods

steepest descent gradient

conjugate gradient

(sensitivity and/or adjoint problems need to be solved)

Gra

dien

t m

etho

ds

function specification

(discretization)

Tikhonov regularization

(S) future information

iterative regularization

--- conjugate gradient

--- EM method

minimum discrepancy

principle

dete

rmin

istic

regularization

No

n-d

eter

min

isti

c

maximum Likelihood

maximum a posteriori

maximum entropy

minimum mean square error

Bayesian prior distribution regularization

greedy search

simulated annealing

genetic algorithm

evolutionary algorithmHeu

ristic

m

etho

ds

importance/rejection sampling

MCMC

Methods for inverse problems

Our approach

Fundamentals of Bayesian statistics

• Bayesian statistics

• Bayesian estimation

)|()()|( YppYp

Prior distributio

n

Likelihood Posterior

distribution

• Bayes’ formula

prior + evidence => posterior probability

)|( YθP)(

)()|(YP

θPθYP)(

),(YP

θYP

• A hierarchical formulation

)()|(),|()|,( ppYpYp

The likelihood

Y = F(θ) + ω ω ~ i.i.d. N(0, σ2)

FYθp T

2))((

21exp{)|( Y FY ))}((

• conditional probability of data (Y) on the parameter (θ)

prior posterior

Fundamentals of Bayesian statistics (cont…)

The prior

• unconditional belief of unknown before the related observations

• role of a prior distribution --- incorporate prior information --- regularize the likelihood

• may be “improper”

• techniques of prior modeling --- conjugate prior --- reference prior --- spatial statistics models

P.M. Lee, Bayesian Statistics - An Introduction, first edition, Oxford University Press, 1989.

P. Congdon, Bayesian Statistical Modeling, John Wiley \& Sons, New York, 2001.

C.P. Robert, The Bayesian Choice, From Decision-Theoretic Foundations to Computational Implementation, the second edition, Springer-Verlag New York, Inc., 2001.

Markov Random Field

}))((exp{)(~

ji jiijWp

θi

Neighbors of θ

Pair-wise Markov Random Field (MRF)

Markov process

)|(),...,,|( 1101 kkkk pp

J. Moler (editor), Spatial statistics and computational methods, Springer-Verlag New York, Inc., 2003.

J. Besag, and P.J. Green, Spatial statistics and Bayesian computation, Journal of the Royal Statistical Society, Series B, Methodological, 55:25-37, 1993.

Markov chain Monte Carlo (MCMC) simulation

1. draw an i.i.d. set of samples {x(i)} i = 1:N from a target density p(x)

2. approximate the target density with empirical point-mass function

3. approximate the integral (expectation) I(f) with tractable sums IN( f )

N

ixN

xN

xp i

1)(

1)(

N

iX

NiN dxxpxffIxf

NfI

1)()()()(

1)(

Monte Carlo PrincipleMonte Carlo Principle

J. Besag, P. Green, D. Higdon and K. Mengersen, Bayesian Computation and Stochastic Systems, Statistical Science, vol.10, pp.3-41, 1995.

P. Bremaud, Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues, Springer-Verlag, New York, 1999.

C. Andrieu, N.D. Freitas, A. Doucet and M.I. Gordan, An introduction to MCMC for machine learning, Machine Learning, vol.50, pp.5-43, 2003.

Markov chain Monte Carlo (cont…)

Markov ChainMarkov Chain

r.v. r.v. xx є X ={x є X ={x11 x x22 ..., ...,xxss }. The stochastic process }. The stochastic process xxii is called a Markov is called a Markov

chain if chain if p(xp(xii| | xxi-1i-1 ,..., ,..., xx11) = ) = p(xp(xii| | xxi-1i-1). ).

MCMC sampler

irreducible and aperiodic Markov chains that have the target distribution

as the invariant distribution.

)|()()|()(111 iiiiii

xxqxpxxqxp

Detailed balance

Metropolis-Hastings (MH) algorithm

Initialize xInitialize x00

For i=0:N-1For i=0:N-1

sample u~U(0,1)sample u~U(0,1)

sample sample xx** ~ ~ q(q(xx**|x|xii) )

if u < A(xif u < A(xii, , xx**)=min)=min{1, p(x{1, p(x**)q(x)q(xii|x|x**)/(p(x)/(p(xii)q(x)q(x**|x|xii))}))}

xxi+1i+1=x=x**

else xelse xi+1i+1=x=xii

Some properties of MH

(a) The normalizing constant is not required.

(b) Easy to simulate independent chains in parallel.

(c) The choice of proposal distribution is crucial.

Extensions of MH sampler

(a) Independent sampler: q(x*|xi) =q(x*).

(b) Metropolis algorithm: q(x*|xi) =q(xi|x*).

(c) Cycles of kernels

Initialize x0

For i=0:N-1 - sample the block xi+1

b1 according to proposal distribution q1(xi+1

b1|xi+1-b1, xi

b1) and target distribution p(xi+1

b1|xi+1-b1)

- sample the block xi+1b2 according to

proposal distribution q1(xi+1b2|xi+1

-b2, xib2) and

target distribution p(xi+1b2|xi+1

-b2) . . - sample the block xi+1

bs according to proposal distribution q1(xi+1

bs|xi+1-bs, xi

bs) and target distribution p(xi+1

bs|xi+1-bs)

(d) Gibbs sampler

)|(~ 11

ijj

ij xxpx

Initialize x0

For i = 0:N-1 For j = 1:m sample

},...,,,...,,{ )()(1

)1(1

)1(2

)1(1

)1( im

ij

ij

iiij xxxxxx

A Bayesian computational framework for inverse problems

Posterior exploration (Markov chain Monte Carlo)

Prior distribution modeling• conjugate priors• physical constraints• spatial statistical models

Likelihood computation• computational mathematics • reduced-order modeling (POD)• parallel computation

• Metropolis-Hastings sampler• symmetric sampler• independent sampler

Hierarchical Bayesian formulation

• hybrid & cyclic MCMC• sequential MCMC

)()(),|()|,( ppYpYp

Permeability estimation

Estimate permeability using dynamic well data (pressure, concentration, …)

,qu ,)()( pcxKu

,ˆ)()( qccDcutc

,0nu ,0 ncD

),()0,( 0 xcxc

),,0( Tin

),,0( Ton

in

)]}()([||||{ uEuEuID tlm

uuu

uE ||2||

1)( )()( uEIuE

44/1 ]1)[0()( cMcc

Solution to the forward problem

Masud, T. J.R. Hughes, A stabilized mixed finite element method for Darcy flow, Computer Methods in AppliedMechanics and Engineering 191 (2002) 4341-4370.R.G. Sanabria Castro, S.M.C. Malta, A.F.D. Loula, L. Landau, Numerical analysis of space-time finite elementformulations for miscible displacements, Computational Geosciences 5 (2001) 301-330.

),())(),((21

),(),(),( fqpK

ufwK

fuwpwuK

nel

e

nel

eeeeee ee

qdcwuqwdcdqccut

cwu

wdcDqcwdwdcuwdt

c

1 1

ˆˆ)(

)(

)0.1,3

min(||||2

1 e

e

P

u

h

eTe

e

uDuu

hPe 3||||

21

Hierarchical Markov Model for multiscale parameter estimation

root layer s=0

s=1

s=2

Markov chain

Markov field

A multi-layer representation of heterogeneous parameter

.

.

.

Hierarchical Markov Model for multiscale parameter estimation (cont.)

]})([])([2

1exp{)|( 2

rrTrrrr YFYFYp

)()|()|(),(),|()|,( rrssrrsrsrrrs ppYppYpYp

),(),|( ~rp

si

rsi

si i

pp

)|(),...,,|( 1021 sssss pp

)()|()...|()|()|,...,,( 1121121 1 sssssssssss pppYpYp SSSS

• the coarse scale (r) permeability distribution

• the fine scale (s) permeability distribution

• Markov assumption of multiscale models

A hybrid MCMC algorithm

Example I: a smooth permeability field

True permeability

MAP estimate on 32x32 grid with data at 24 locations

K(x,y)=exp(0.5(x-4.0)+0.5(y-4.0))

MAP estimate on 16x16 grid with data at 24 locations

MAP estimate on 8x8 grid with data at 24 locations

MAP estimate on 32x32 grid with data at 8 locations

MAP estimate on 16x16 grid with data at 8 locations

MAP estimate on 8x8 grid with data at 8 locations

Example II: A permeability field with random discontinuities

Example I: the true permeability field

the coarse scale estimate (4x4)

well distribution(pressure data)

)( 2re

A permeability field with random discontinuities (cont.)

3 realizations from the fine-scaledistribution

sample meanof the fine-scale distribution

Example II: the true permeability field |)|( re

2 realizations from the fine-scale distribution

A permeability field with random discontinuities (cont.)

Conclusions

• MRF is suitable for estimating smooth permeability field

• The hybrid MCMC algorithm is efficient in exploring the high

dimensional posterior state space

• HMT model provide flexibility to model multiscale permeability

• Sample permeability field from the posterior distribution provide

reliable basis for scenario analysis