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

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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.

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Abbreviations

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

D n :

Training data set with n samples

D n+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

k f :

Current fracture permeability, md

k i :

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

p i :

Initial pressure, psi

pr:

Probability distribution

PR-EOS:

Peng-Robinson Equation of State

q :

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

Q wr :

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

S 2(x*):

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

SRV:

Stimulated Reservoir Volume

S winit :

Initial water saturation in the model

S wSRV :

Initial water saturation in the SRV

SRV:

Stimulated Reservoir Volume

TVD:

True Vertical Depth

x i :

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

W SRV :

The width of a 1D SRV,

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

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

ψ s :

The standard normal cumulative distribution function

ϕ :

The normal probability density function

ϕ s :

The standard normal probability density function

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Authors and Affiliations

  1. Department of Computer Science, University of Calgary, Calgary, Alberta, Canada

    Hamidreza Hamdi & Mario Costa Sousa

  2. Department of Information Technology (INTEC), Ghent University-iMinds, Ghent, Belgium

    Ivo Couckuyt & Tom Dhaene

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  1. Hamidreza Hamdi
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Hamdi, H., Couckuyt, I., Sousa, M.C. et al. Gaussian Processes for history-matching: application to an unconventional gas reservoir. Comput Geosci 21, 267–287 (2017). https://doi.org/10.1007/s10596-016-9611-2

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