Support-Operators Method in the Identification of Permeability Tensor Orientation
- Dmitriy B. Silin (Lawrence Berkeley Natl. Laboratory) | Tad W. Patzek (U. of California at Berkeley)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2001
- Document Type
- Journal Paper
- 385 - 398
- 2001. Society of Petroleum Engineers
- 5.3.1 Flow in Porous Media, 5.5 Reservoir Simulation, 5.5.1 Simulator Development, 4.1.2 Separation and Treating, 5.1.5 Geologic Modeling, 5.6.4 Drillstem/Well Testing, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 5.1.1 Exploration, Development, Structural Geology
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The dependence of rock permeability on direction, or permeability anisotropy, is confirmed by numerous field examples. Therefore, the ability to carry out a numerical simulation of an anisotropic reservoir is very important. The support-operators method provides a conservative discretization scheme, allowing one to solve nonisotropic problems on a grid of practically arbitrary structure. Moreover, a discretization designed with the support-operators method provides a natural and convenient way of deriving and solving the adjoint system for evaluation of the gradient and second-order differential in inverse problems.
After a theoretical introduction into the support-operators method, we consider an illustrative parameter-identification problem. More specifically, we evaluate the orientation angle of a nonisotropic permeability tensor in a horizontal reservoir. We assume that the principal permeabilities near a cored or otherwise logged well are already known. To accomplish this task, we need pressure measurements in monitoring wells. We consider both rectangular and curvilinear grids. In either case, the orientation angle has been recovered with a high accuracy.
Often, when modeling fluid flow in a porous rock, it is assumed that permeability of the rock is isotropic, at least horizontally (i.e., the permeability coefficient is the same regardless of the direction of pressure gradient). In other words, the direction of flow is aligned with the pressure gradient. This simplifying assumption cannot be accepted in every case. For example, the permeability of fissured rock is substantially dependent on microfractures; therefore, the dominating orientation of the fractures can make the direction of flow different from that of the pressure anti-gradient. Occurrences of anisotropy in oil reservoirs have been well documented (e.g., Refs. 1 and 2). Romm3 provides several field examples in which measurements indicate the dependence of permeability on direction. Another field example with comprehensive well test data analysis is given in Ref. 4. The ratio of the maximum to the minimum permeability reported in the literature can be as large as 103. More references related to the studies of anisotropy of permeability in fractured reservoirs can be found in Ref. 5.
In general, anisotropy of a sedimentary formation results from the history of sedimentation. For example, an alluvial fan consists of stream-flow and debris-flow deposits and eolian sands. Strong horizontal anisotropy exists in the sediment transport direction and perpendicular to it. Thus, for realistic reservoir modeling, the ability to handle formation anisotropy is important. This requirement poses a challenge for a reservoir simulator developer, because standard numerical schemes usually assume either isotropic or diagonal tensors. In addition, if anisotropy is accompanied by heterogeneities, irregular curvilinear grids may be needed for appropriate discretization of the respective boundary-value problem. Methods for handling such situations are not broadly available in the petroleum literature. A split-tensor operator algorithm has been developed by Edwards.6-8 Ref. 8 also includes a literature survey on methods of numerical treatment of problems with full permeability tensor.
The purpose of this article is to attract attention to a powerful method, different from the methods discussed in Ref. 8, that can be used to model flow in heterogeneous nonisotropic media on irregular grids. More specifically, we present an introduction to the support-operators method.
In this article, we emphasize the potential of the method per se, rather than analyze the performance of a specific reservoir or well pattern. Therefore, the presentation has a strong theoretical bias. Inasmuch as the focus of the method is on discretization of the Laplace operator (which equals the divergence of the flux in a combination with Darcy's law), we restrict ourselves to a steady-state problem, although a transient problem incorporating time derivatives can also be considered with appropriate modifications. To avoid cumbersome calculations that may hide the essence of the proposed approach, we consider a 2D problem only, but the method can be extended into 3D in a natural way.
Besides describing the discretization procedure, we show how the support operators can be used conveniently to solve inverse problems. We consider a simple synthetic example of an inverse problem, in which support operators facilitate numerical evaluation of the gradient of the data-fitting criterion through the adjoint system. Moreover, we extend the adjoint system analysis to calculate second-order differentials as well. The latter can be used to enhance the best-fit search by incorporating a Newton-type minimization method. Theoretically, it is known that under appropriate conditions, a Newton-type method provides a high-order and high-accuracy convergence to the minimum.
Calculation of first-order differential through adjoint system of differential equations provides clues into the sensitivity of the solution with respect to lumped or distributed parameters. An example of parameter of the first kind could be the total fluid content in the reservoir. The volumetric distribution of fluid saturation in the rock is an example of parameter of the second kind. The second differential is a much more complicated object; therefore, even such a powerful tool as adjoint system can provide only a directional differential. If the number of parameters is finite, then each second-order partial derivative has to be evaluated individually. Evaluation of second-order differential remains a complex task.
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