Comparison of 2D and 3D Fractal Distributions in Reservoir Simulations
- R.A. Beier (Conoco Inc.) | H.H. Hardy (Conoco Inc.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- August 1994
- Document Type
- Journal Paper
- 195 - 200
- 1994. Society of Petroleum Engineers
- 5.3.1 Flow in Porous Media, 5.8.7 Carbonate Reservoir, 4.6 Natural Gas, 5.6.1 Open hole/cased hole log analysis, 1.6 Drilling Operations, 5.4.1 Waterflooding, 1.6.9 Coring, Fishing, 5.1.5 Geologic Modeling, 5.1 Reservoir Characterisation, 4.3.4 Scale, 5.7.2 Recovery Factors, 5.5 Reservoir Simulation
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Fractal geostatistics can be used to generate distributions of such reservoir properties as porosity and permeability. We compare three different methods that use these distributions in reservoir simulations. The first method is a hybrid finite-difference/streamtube method that starts with 2D vertical cross sections with relatively fine grids, the second uses pseudo-relative permeabilities and also is based on 2D cross sections, and the third uses 3D distributions. A mature waterflood in a carbonate reservoir serves as a field test case.
Past studies1-3 have used fractal geostatistics techniques to generate porosity and permeability distributions for vertical cross sections. The generated fractal distributions were 2D because 3D distributions with the same fine gridding have many more grids than a reservoir flow simulator can handle.
Approaches based on 2D distributions require two successive flow simulations: one on the 2D vertical cross section and a second areal flow simulation. That is, the results from flow simulations on the 2D cross sections must be scaled up to an areal model to represent a field project. Chevron's hybrid finite-difference/streamtube method4 is one method to scale up the cross-sectional flow results. A second procedure5,6 is to generate pseudo-relative-permeability curves from the fine-scale vertical cross section. Then an areal finite-difference simulation uses the pseudo-relative-permeability curves to simulate fluid displacement in the field project.
Two-dimensional methods for fluid displacement cannot capture the main flow mechanisms that occur in some reservoirs. For example, 2D methods cannot handle gravity override in steeply dipping beds unless the well pattern (injectors and producers) aligns the dominant flow direction along the dip axis. These methods also cannot handle a strong bottomwater drive. In both cases, simulated fluid displacements on 2D cross sections cannot capture 3D flow patterns in the reservoirs.
A third approach is to generate 3D distributions for subsequent reservoir flow simulations. The distributions must start at sufficient1y fine scale to capture the heterogeneity seen in well logs; however, these fine-scale distributions contain too many grids to fit into a reservoir flow simulator. Therefore, we scale up to a coarser grid with arithmetic averaging for porosity and a renormalization algorithm for permeability.7,8 In this paper, we apply the 3D distributions to a typical well-pattern element on 20-acre well spacing. Future work needs to develop techniques for larger well spacings.
The main purpose of our study is to compare all three methods on a mature waterflood. First, we review our field case history, then we describe 2D vertical cross sections. Finally, we describe each technique in more detail and give its flow simulation results.
Waterflood Field Case
The waterflood chosen is in southeastern New Mexico. Most of the development drilling occurred in the early 1940's. Wells produce from Grayburg dolomitic sands and San Andres dolomite pay zones at between 3,600 and 4,300 ft. The primary production mechanism of the reservoir was solution-gas drive.
Waterflood operations on five-spot patterns were phased in during the mid to late 1960's. The original 40-acre well spacing was infill drilled to 20 acres to form inverted nine-spot patterns in the early 1970's. The infill wells generally have sidewall neutron porosity (SNP) logs, which correlate well with available core porosity data. Operations of the inverted nine-spot patterns remained relatively unchanged for almost 20 years. This long period of nearly constant operating conditions allows use of a typical well-pattern element to represent a larger area. We focus on the waterflood performance in a 640-acre area.
2D Porosity and Permeability Distributions
Fractal porosity cross sections are generated in the following way. First, we place corrected and depth-adjusted SNP logs from two adjacent wells on the vertical edges of the cross section. Then, fractal geostatistics techniques3,8 are used to fill in the porosity array.
Fig. 1 shows the resulting cross section of the study area. The distribution is consistent with the fractional Gaussian noise (fGn) structure seen in vertical well logs in our field case. The fGn structure is assumed to exist in the horizontal direction on the basis of data from other reservoirs and sedimentary rocks. These data include porosity logs from horizontal wells9 and outcrop and core photographs.8,10
Horizontal permeabilities are assigned to each grid as a function of porosity by the formula
The values of a and b are set to 0.001 and 33.3, respectively. The value of b corresponds to the slope in a log10(permeability) vs. porosity plot. The value of a is the intercept. As is typical in carbonate reservoirs, log10(permeability) vs. porosity plots show considerable data scatter. The above values of a and b give a line that passes through the scatter but are certainly not unique. At this value of b, the range of permeability covers five log cycles over the usual range of porosity found in a well (2 to 17 porosity units).
Vertical permeability is set to 0.1 times the horizontal permeability. Core data indicate this value is reasonable. The simulated waterflood results in our field case turned out to be relatively insensitive to the vertical/horizontal permeability ratio. This insensitivity is a result of the laterally continuous dense dolomite layers that hinder crossflow and of small gravity effects.
Each grid in Fig. 1 is 2×2 ft, which corresponds to the vertical spatial resolution of the logging tool. Even though this is a 2D distribution, it still contains more grids (512×150=76,800) than usually used in reservoir flow simulations. We obtain a coarser grid (32×150) for porosity by arithmetic averaging (Fig. 2). To obtain a coarser grid for single-phase permeability, we use a renormalization algorithm,7,8 FRACTAM. Laboratory (small-scale) relative permeability curves are applied to all grids regardless of grid size.
We tested this procedure on a single realization by coarsening the grid to various final sizes. Simulated water cuts for successively fewer grids in the horizontal and vertical directions all fall within a narrow band until the grid becomes too coarse. The narrow band is much smaller than the variation among realizations. All grid sizes in this study fall within this narrow band. If the chosen coarse grids are thicker than the high-porosity layers in Fig. 1, individual layers are smeared out and the distribution becomes too homogeneous. Nonreservoir rock is not explicitly identified before flow simulations. Instead, all the fine and coarse grids have nonzero porosity and permeability values. The flow simulations determine the vertical sweep efficiency.
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