Simulation of Gas/Oil Drainage and Water/Oil imbibition in Naturally Fractured Reservoirs
- R.H. Rossen (Exxon Production Research Co.) | E.I.C. Shen (Exxon Production Research Co.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1989
- Document Type
- Journal Paper
- 464 - 470
- 1989. Society of Petroleum Engineers
- 5.8.6 Naturally Fractured Reservoir, 5.3.2 Multiphase Flow, 5.5.8 History Matching, 4.1.2 Separation and Treating, 5.2.1 Phase Behavior and PVT Measurements, 1.2.3 Rock properties, 5.5 Reservoir Simulation, 5.2 Reservoir Fluid Dynamics
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Much attention has been given to the simulation of naturally fractured reservoirs in the recent literature. The most prevalent approach is a dual-porosity (or dual-porosity/dual-permeability) formulation with computation blocks that may represent several individual matrix blocks. In models of this type, the processes of gas/oil drainage and water/oil imbibition have caused particular difficulties. Some authors have attempted to represent the correct behavior through a gravity term that assumes a simplified fluid distribution in the matrix. Others have used pseudo-capillary-pressure functions for the matrix, the fracture, or both. These functions also assume a simplified matrix fluid distribution or are obtained by "history matching" with a fine-grid model of a single matrix block. Still others have introduced refinement of the matrix into multiple blocks.
In this paper, the authors examine the mechanisms involved in gas/oil drainage and water/oil imbibition and propose a simple way to represent that behavior in a dual-porosity simulator. Basically. the formulation uses pseudo-capillary-pressure curves for both the matrix and fracture. The fracture curve can be determined directly from rock properties and matrix-block dimensions, while the matrix curve can be obtained from the results of a single simulation of a fine-grid model of a single matrix block. The approach is less costly than matrix subdomain and history-matching alternatives and is often more accurate than solutions that rely on a simplified gravity term.
Predicting the behavior of naturally fractured reservoirs has presented a challenge for reservoir engineers for many years. Use of reservoir simulation as a predictive tool has been complicated by the vast discrepancies in properties between the matrix rock and the interconnected fracture system. The bulk of the fluid is contained in the low-porosity, low-permeability matrix while fluid flow occurs mainly in the fractures.
The first approaches to full-field modeling of fractured reservoirs were described by Kazemi and Rossen. Rossen represented the contribution of matrix blocks as source and sink terms in a simulation of the fracture system. The source/sink terms were determined by use of fine-grid simulations of single matrix blocks and were dependent on pressure, fluid content in the fracture and the elapsed time since that fracture content changed. The source/sink terms were treated semi-implicitly for stability.
Kazemi first proposed a dual-porosity representation where the matrix and fracture were modeled as two overlapping continua. Flow could occur between matrix and the corresponding fracture blocks or between adjacent fracture blocks but not between matrix blocks. Flow between matrix and fracture was proportional to a "shape factor," generally represented as a (F, in this paper), and the driving force was the pressure difference between a matrix block and its surrounding fracture.
The dual-porosity approach has been extended by several authors. Most of these subsequent papers addressed the questions of gravity treatment or the definition of the shape factor. The primary drawback of the original proposal was the treatment of the gravity term. Because each matrix block was assumed to be at the same depth as its corresponding fracture block, gravity had no explicit impact on the exchange between matrix and fracture. In 1983, Thomas et al. introduced the use of a "pseudo" capillary pressure for the matrix. The pseudo was calculated from vertical equilibrium calculations. Their method included a gravity effect and could reproduce the correct equilibrium matrix saturation but had no guarantee of accurately calculating the rate of imbibition or drainage in approaching that saturation.
Gilman and Kazemi added a gravity term to the dual-porosity exchange. Fluid contact heights were calculated for the matrix and for the fracture, and the difference was the driving force for a gravity term. Sonier et al. and Litvak used similar approaches, but added irreducible saturation to the calculation of contact depths. Their treatments do not include the time-dependent nature of gravity segregation and, as illustrated by Gilman and Kazemi, they can lead to optimistic results.
In 1983, Saidi recognized the inherent difficulties in the dual-porosity approach and proposed that the matrix be subdivided into subdomains to represent pressure and saturation variations with position more accurately. Gilman, Lee and Tan, Chen et al., and Gilman and Kazemi used subdomain approaches to improve the matrix representation. While more accurate than the preceding dual-porosity, single-domain formulations. the use of subdomains also increases the computer time and storage required to run a full-field study.
The preceding discussion deals with representation of systems in which the matrix blocks are smaller than the computation blocks and for which fluid transfer between computation blocks in the matrix can be ignored. If the matrix blocks are larger than the computation blocks, however, such transfer must be considered in the flow equations. Hill and Thomas and Blaskovich et al. described dual-porosity/dual-permeability formulations to handle this situation. In this paper, we confine our investigation to the dual-porosity case.
Dean and Lo recently demonstrated that the effect of gravity drainage could be included in pseudo-capillary-pressure terms for both the matrix and fracture without the need to include explicitly a gravity term. They determined these pseudo-capillary-pressure terms by history matching with a fine-grid model of a single matrix block. When a combination of pseudos was found to give an acceptable match of the single-block results, it was assumed to provide a reasonable representation of the system behavior. The drawback to their approach appears to be the time and effort that might be required in history matching to generate the pseudos.
The other aspect of the dual-porosity approach that has received attention is the shape factor. Kazemi's first proposal included the following factor, adjusted for anisotropic permeabilities:
Some authors continued to use this term while others recognized that
is more appropriate for gas/oil drainage processes. Considerable debate still exists in the literature about the correct shape factor, and some authors have proposed using it as a matching parameter.
In this paper, we present a method for modeling naturally fractured reservoirs that uses pseudo-capillary-pressure curves for both the matrix and fracture that are similar to those derived by Dean and Lo. Unlike Dean and Lo, however, we describe a procedure for generating the pseudo curves from a single fine-grid simulation without history matching, which might require many such simulations. We also suggest a unified treatment of the shape function for gas/oil drainage and for water/oil imbibition.
Our approach is currently implemented in a three-phase simulator that includes mass transfer between the phases.
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