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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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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DOI: https://doi.org/10.1007/s10596-016-9611-2