Reparameterization Techniques for Generating Reservoir Descriptions Conditioned to Variograms and Well-Test Pressure Data
- A.C. Reynolds (University of Tulsa) | Nanqun He (University of Tulsa) | Lifu Chu (University of Tulsa) | D.S. Oliver (Chevron Petroleum Technology Company)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 1996
- Document Type
- Journal Paper
- 413 - 426
- 1996. Society of Petroleum Engineers
- 4.3.4 Scale, 5.1.8 Seismic Modelling, 4.6 Natural Gas, 5.3.2 Multiphase Flow, 5.5 Reservoir Simulation, 1.6.9 Coring, Fishing, 5.1.5 Geologic Modeling, 5.6.4 Drillstem/Well Testing, 5.5.3 Scaling Methods, 5.1 Reservoir Characterisation, 5.1.6 Near-Well and Vertical Seismic Profiles, 5.5.8 History Matching
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Recently, we have shown that reservoir descriptions conditioned to multiwell pressure data and univariate and bivariate statistics for permeability and porosity can be obtained by techniques developed from inverse problem theory. The techniques yield estimates of well skin factors and porosity and permeability fields which honor both the spatial statistics and the pressure data. Imbedded in the methodology is the application of the Gauss-Newton method to construct the maximum a posteriori estimate of the reservoir parameters. If one wishes to determine permeability and porosity values at thousands of grid-blocks for use in a reservoir simulator, then inversion of the Hessian matrix-at each iteration of the Gauss-Newton procedure becomes computationally expensive. In this work, we present two methods to reparameterize the reservoir model to improve the computational efficiency. The first method uses spectral (eigenvalue/eigenvector) decomposition of the prior model. The second method uses a subspace method to reduce the size of the matrix problem that must be solved at each iteration of the Gauss-Newton method. It is shown that proper implementation of the reparameterization techniques significantly decreases the computational time required to generate realizations of the reservoir model, i.e., the porosity and permeability fields and well skin factors, conditioned to prior information on porosity and permeability and multiwell pressure data.
Proper integration of static data (core, log, seismic, and geologic information) with dynamic data (production and well tests) is critical for reservoir characterization. It is known that ignoring prior information obtained from static data when history matching production data yields nonunique solutions, i.e., widely different estimates of the set of reservoir parameters may all yield an acceptable match of the production history. As early as 1976, Gavalas et al. recognized that incorporating prior data when history matching production data would reduce the variation in the estimates of gridblock values of porosity and permeability.
Inverse problem theory provides a methodology to incorporate prior information when history matching production data. The standard application of inverse problem theory depends on the assumption that prior information on the model (set of reservoir parameters to be estimated) satisfies a multinormal distribution and that measurement errors in production data can be considered as Gaussian random variables with zero mean and known variance. Under these assumptions, the most probable model (the maximum a posteriori estimate) conditioned to both prior information and production data can be obtained by minimizing an objective function derived directly from the a posteriori probability density function. Since the a posteriori probability density function is derived from Bayes's theorem, this approach is often referred to as Bayesian estimation. It is convenient to minimize the objective function by a gradient method to obtain an approximation to the most probable model which is referred to as the maximum a posteriori estimate.
Gavalas et al. used Gaussian type expressions for the covariance functions of porosity and permeability, the cross covariance between them, and the prior estimates of the means of porosity and permeability to incorporate prior information in the objective function when history matching multiwell pressure data obtained in a synthetic one-dimensional reservoir under single-phase flow conditions. They showed that incorporating the prior information reduced the errors in the estimates of permeability and porosity and also improved the convergence properties of the minimization algorithms considered.
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