Calculating Effective Absolute Permeability in Sandstone/Shale Sequences

Abstract

Summary Two averaging algorithms are proposed for determining block effective absolute permeability. The experimental relationship between the effective permeability, the volume fraction of shale, and the anisotropy of the shales is first observed through repeated flow simulations. A power-averaging model and a percolation model are proposed to fit the experimentally observed relationship. The power-averaging model provides a surprisingly easy and efficient way to calculate block effective absolute permeability. A simple graph is given to determine the averaging power from the geometric anisotropy (aspect ratio) of the shales for both vertical and horizontal steady-state flow. The effective absolute permeability can then be calculated with the averaging power, the volume fraction of shale, and the component sandstone mill shale permeabilities. The effective permeability of a sandstone/shale sequence is affected when the shales are large with respect to the size of the gridblocks. A correction for large shales relative to small gridblocks is also proposed. Introduction One goal of reservoir engineering is to develop reservoir-management plans to achieve optimal recovery under certain economic constraints. Reservoir simulation provides the means to perform this optimization by predicting recovery before production. The simulation programs solve mathematical equations that describe the flow of fluids through a numerical model of the reservoir. This paper considers the problem of building an accurate numerical model of absolute permeability. The problem is difficult because flow-simulation programs implicitly assume that each gridblock is homogeneous. In reality, however, the reservoir unit that each gridblock represents is rarely homogeneous. Therefore, to describe the reservoir accurately, it is necessary to define effective properties that represent the small-scale heterogeneity found within each gridblock. This task is not too difficult if the reservoir is relatively homogeneous. Unfortunately, in the most common elastic reservoir, a sandstone/shale sequence, there are severe discontinuities. The flow transport properties within the constituent shales and sandstone differ drastically. The sandstone matrix contains the movable fluids, while shales provide obstacles for fluid flow. Neither the sandstone nor the shales are homogeneous. The impact of the heterogeneity within the sandstone and shale, however, is not as important as that of the transition between the sandstone and shale. The study presented here will consider the sole heterogeneity introduced by the transition between the two rock types. The impact of the shales on the block effective absolute permeability will depend on the volume and spatial distribution of the shales. As the volume, Vsh, increases, the effective absolute permeability decreases. Other important factors are the shape and orientation of the shales with respect to the direction of fluid flow. The shales have the most impact on vertical flow and the least impact on horizontal flow. When the shales within a reservoir can be correlated between wells, the reservoir is essentially separated into distinct layers that should be handled separately in the simulation program. The shales that must be accounted for by averaging are the ones that are discontinuous between wells. These discontinuous shales can significantly affect reservoir performance. Discontinuous shales can be modeled by stochastic processes. To study the effect of discontinuous or stochastic shales on the block effective absolute permeability, various sandstone/shale sequences were simulated and single-phase steady-state flow simulations were performed to obtain effective permeabilities. The relationship between the resultant effective permeability and the volume fraction of shale was then observed for various shale geometries. Two models for the averaging process will be considered: a power-average and a percolation-theory-based model. The resultant models can be applied to real reservoirs to estimate effective absolute permeability. Calculating Effective Absolute Permeability The sandstone/shale sequence is modeled as a three-dimensional (3D) grid network where each gridblock is either sandstone or shale. A particular sandstone/shale sequence is created by adding shales of a given geometry and orientation until a specified target volume fraction is met. Shales are assumed to be ellipsoids of equal size and are positioned independently of other shales. The shale geometry may look quite complex because of overlapping of the shale units and truncation of the shales on the boundary of the grid network. These effects are illustrated in Fig. 1, which shows two horizontal and two vertical sections from a particular sandstone/shale configuration. After the sandstone/shale configuration is created, the effective permeability is found by observing the steady-state flow rate for an applied pressure gradient. The commercial flow simulator ECLIPSE has been used to determine the steady-state flow rate. The block effective permeability can then be calculated directly from Darcy's law. The outlined procedure can be repeated with different volume fractions of shale. Thus, the relationship between the effective permeability and the volume fraction of shale is empirically observed rather than analytically derived. The relationship between the effective permeability and the proportion of shale for small isotropic shales (i.e., each shale is the size of an elemental cubic grid unit) is shown in Fig. 2. The permeability of each gridblock in the 20 × 20 × 10 network is assigned either a sandstone permeability (1,000 md) or a shale permeability (0.01 md) before flow simulation. The three traditional averaging processes (the arithmetic, geometric, and harmonic averages) are shown for reference. Although none of the three conventionally used averages adequately represents the effective permeability, a clear functional relation appears between the effective permeability and the volume fraction of shale for Vsh less than 0.4. Commonly occurring shales are highly anisotropic and are not spatially uncorrelated as in the previous simulation. Figs. 3 and 4 show the results of models constructed for directions parallel and perpendicular to the major anisotropy. A 10:1 horizontal-to-vertical anisotropy ratio was considered for the shale geometry, with an equal range in all horizontal directions. The horizontal permeability (parallel to the major direction of continuity) is consistently greater than the vertical permeability. The arithmetic average appears to represent the horizontal permeability adequately for shale volumes less than 40%. The vertical permeability also appears clearly related to the volume fraction of shale for Vsh less than 40 % but cannot be modeled with conventional averaging techniques. The numerical results presented in Figs. 2 through 4 suggest that an averaging process can be defined that varies continuously with the degree of shale continuity.