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Computational Geosciences

, Volume 21, Issue 2, pp 267–287 | Cite as

Gaussian Processes for history-matching: application to an unconventional gas reservoir

  • Hamidreza Hamdi
  • Ivo Couckuyt
  • Mario Costa Sousa
  • Tom Dhaene
Original Paper

Abstract

The process of reservoir history-matching is a costly task. Many available history-matching algorithms either fail to perform such a task or they require a large number of simulation runs. To overcome such struggles, we apply the Gaussian Process (GP) modeling technique to approximate the costly objective functions and to expedite finding the global optima. A GP model is a proxy, which is employed to model the input-output relationships by assuming a multi-Gaussian distribution on the output values. An infill criterion is used in conjunction with a GP model to help sequentially add the samples with potentially lower outputs. The IC fault model is used to compare the efficiency of GP-based optimization method with other typical optimization methods for minimizing the objective function. In this paper, we present the applicability of using a GP modeling approach for reservoir history-matching problems, which is exemplified by numerical analysis of production data from a horizontal multi-stage fractured tight gas condensate well. The results for the case that is studied here show a quick convergence to the lowest objective values in less than 100 simulations for this 20-dimensional problem. This amounts to an almost 10 times faster performance compared to the Differential Evolution (DE) algorithm that is also known to be a powerful optimization technique. The sensitivities are conducted to explain the performance of the GP-based optimization technique with various correlation functions.

Keywords

Gaussian process Expected improvement Tight formation History-matching 

Nomenclature

1D

One-dimensional

A

Non-dominated region

Bar sign “¯”

Average value

CCE

Constant Composition Experiment

c(x,x’)

Kernel or covariance function between two location x and x’

CGR

Condensate Gas Gatio

d

The dimension of problem

det[C]

Determinant of covariance matrix C

Dn

Training data set with n samples

Dn+1

Augmented data set with n + 1 samples

DE

Differential Evolution

DFIT

Diagnostic Fracture Injection Test

EI

Expected Improvement

EnKF

Ensemble Kalman Filter

ES

Ensemble Smoother

ES-MDA

Ensemble Smoother for Multiple Data Assimilation

F

Degree Fahrenheit

f

The output of the truth function

f

A vector containing the output of the truth function in several locations

F

The output of the truth function as a random variable

F

The output of the truth function in several location as a random vector

GP

Gaussian Process

h

Reservoir thickness, ft

k × w

Fracture conductivity, md ft

kf

Current fracture permeability, md

ki

Original (initial) fracture permeability, md

KB

Kelly bushing

l

Lateral length, ft

Ln(L)

Negative concentrated log-likelihood

M

Misfit function

n

The number of available samples

(simulations)

p

Current pressure, psi

pi

Initial pressure, psi

pr

Probability distribution

PR-EOS

Peng-Robinson Equation of State

q

Production flow rate, bbl/day (liquid) or MMscf (gas)

Qwr

Remaining water in the reservoir after injection, ft 3

r

The correlation vector between sample

the x* and the data D

R

Covariance matrix = σ 2 C

S2(x*)

The variance of predicted value y* corresponding to sample x* by GP

SRV

Stimulated Reservoir Volume

Swinit

Initial water saturation in the model

SwSRV

Initial water saturation in the SRV

SRV

Stimulated Reservoir Volume

TVD

True Vertical Depth

xi

A sample i

x f

Fracture half length, ft

Y

The posterior distribution of the modeled objective

ŷ

The predictive mean of the predictive GP

w

Fracture width, ft

WSRV

The width of a 1D SRV,

Subscripts

b

The current best member

n+1

The augmented training data set by adding

a new sample

o,g,w

oil, gas, water

obs

Observed data

sim

Simulation data

Greek letters

γ

Fracture reduction factor

δ

Molar composition of components in

oil or gas

𝜃

GP hyperparameters (length scales) of dimension d

λ

An anisotropic distance measure

\(\hat {\mu }\)

The estimated mean of the GP model

knowing data

μ

The prior mean of the GP model

ν

A constant used in defining Matérn

correlation function

\(\hat {\sigma }^{\mathrm {2}}\)

The estimated variance of the GP model knowing the data

φ

Porosity

ψ

The normal cumulative distribution function

ψs

The standard normal cumulative distribution function

ϕ

The normal probability density function

ϕs

The standard normal probability density function

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Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Hamidreza Hamdi
    • 1
  • Ivo Couckuyt
    • 2
  • Mario Costa Sousa
    • 1
  • Tom Dhaene
    • 2
  1. 1.Department of Computer ScienceUniversity of CalgaryCalgaryCanada
  2. 2.Department of Information Technology (INTEC)Ghent University-iMindsGhentBelgium

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