Three-Dimensional Reservoir Description From Multiwell Pressure Data and Prior Information
- Nanqun He (Chevron Petroleum Technology Company) | Albert C. Reynolds (U. of Tulsa) | Dean S. Oliver (Chevron Petroleum Technology Company)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 1997
- Document Type
- Journal Paper
- 312 - 327
- 1997. Society of Petroleum Engineers
- 4.3.4 Scale, 5.6.4 Drillstem/Well Testing, 5.1.5 Geologic Modeling, 5.5.8 History Matching, 2.2.2 Perforating, 5.5 Reservoir Simulation, 1.2.3 Rock properties, 3.3.1 Production Logging, 3.3.6 Integrated Modeling, 5.1 Reservoir Characterisation
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Bayesian estimation techniques are applied to generate three-dimensional permeability and porosity fields conditioned to prior means, variograms, hard data and/or multiwell pressure data. A posteriori variances are generated to obtain a measure of the uncertainty in the rock property fields, or equivalently, a measure of the variability in realizations of the rock property fields. These a posteriori variances can also be used to quantify the value of data, i.e., to determine the reduction in uncertainty achieved by adding a particular type of data. A key ingredient of our methodology is the development and implementation of an efficient procedure to estimate sensitivity coefficients for three-dimensional single-phase flow problems.
In previous work,1-4 we presented procedures based on inverse problem theory5 to generate realizations of rock property fields conditioned to prior means, variograms, hard data and multiwell pressure data for one and two-dimensional single-phase flow problems where the rock property fields are represented by simulator gridblock values of porosity and permeability. The objective of this work is to extend these techniques to three-dimensional single-phase flow problems.
The solution of the inverse problem requires the derivation of the a posteriori probability density function for the rock property fields. This probability density function (pdf) is based on all available information and data that will be used as conditioning data when generating realizations of the rock property fields. In a sense,5 this pdf represents the solution of the inverse problem. In practice, however, we wish to generate a set of realizations of the rock property fields which represent a sampling of the a posteriori probability density function. As discussed in Refs. 1 - 4, we can accomplish the latter task in two steps. First, we estimate the most probable model (maximum a posteriori estimate) by using the Gauss-Newton method to minimize an objective function associated with the a posteriori probability density function. This is simply Bayesian estimation. Realizations of the rock property fields can then be generated from a Cholesky decomposition of the a posteriori covariance matrix.1-4 The diagonal elements of the a posteriori covariance matrix are particularly important as they represent a posteriori variances of the model parameters and give a measure of the uncertainty in the rock property fields, or equivalently, a measure of the variability in realizations of the gridblock values of porosity and log-permeability. As discussed previously,1-4 due to nonlinearity, the set of realizations generated from the Cholesky decomposition of the a posteriori matrix is only an approximate sampling. Procedures to obtain a more accurate sampling of the a posteriori pdf are given in Refs. 6-9.
As noted in Refs. 1-4, many of the ideas we apply have existed for some time both in general theoretical form5 and in the language of specific disciplines. In the reservoir engineering field, the work of Gavalas et al.10 and Shah et al.11 is of historical importance. For one-dimensional single-phase flow problems, they applied the same type of Bayesian estimation procedure used here, but did not consider the problem of generating realizations.
|File Size||5 MB||Number of Pages||16|