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


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.


Gaussian process Expected improvement Tight formation History-matching 





Non-dominated region

Bar sign “¯”

Average value


Constant Composition Experiment


Kernel or covariance function between two location x and x’


Condensate Gas Gatio


The dimension of problem


Determinant of covariance matrix C


Training data set with n samples


Augmented data set with n + 1 samples


Differential Evolution


Diagnostic Fracture Injection Test


Expected Improvement


Ensemble Kalman Filter


Ensemble Smoother


Ensemble Smoother for Multiple Data Assimilation


Degree Fahrenheit


The output of the truth function


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


The output of the truth function as a random variable


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


Gaussian Process


Reservoir thickness, ft

k × w

Fracture conductivity, md ft


Current fracture permeability, md


Original (initial) fracture permeability, md


Kelly bushing


Lateral length, ft


Negative concentrated log-likelihood


Misfit function


The number of available samples



Current pressure, psi


Initial pressure, psi


Probability distribution


Peng-Robinson Equation of State


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


Remaining water in the reservoir after injection, ft 3


The correlation vector between sample

the x* and the data D


Covariance matrix = σ 2 C


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


Stimulated Reservoir Volume


Initial water saturation in the model


Initial water saturation in the SRV


Stimulated Reservoir Volume


True Vertical Depth


A sample i

x f

Fracture half length, ft


The posterior distribution of the modeled objective


The predictive mean of the predictive GP


Fracture width, ft


The width of a 1D SRV,



The current best member


The augmented training data set by adding

a new sample


oil, gas, water


Observed data


Simulation data

Greek letters


Fracture reduction factor


Molar composition of components in

oil or gas


GP hyperparameters (length scales) of dimension d


An anisotropic distance measure

\(\hat {\mu }\)

The estimated mean of the GP model

knowing data


The prior mean of the GP model


A constant used in defining Matérn

correlation function

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

The estimated variance of the GP model knowing the data




The normal cumulative distribution function


The standard normal cumulative distribution function


The normal probability density function


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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