Scaling-up in the Vicinity of Wells in Heterogeneous Field

Abstract

SPE Member Abstract It is well known that the equivalent gridblock permeability obtained by the scaling-up procedure from fine gridblocks depends on the boundary conditions. In the literature, most boundary conditions considered represent a "linear" flow pattern, while in reservoir simulation, "radial" flow plays an important role and is related directly to well production forecasts. In this paper, we will show the need for a specific treatment of the scaling-up procedure in the vicinity of wells due to the "radial" flow pattern with a high pressure gradient. On reservoir field scale, two types of flow should be considered:the "linear" flow pattern, which usually represents a low pressure gradient region; andthe "radial" flow pattern, which usually represents a high pressure gradient region. The determination of the equivalent absolute permeabilities by "radial" flow patterns is discussed in this paper. The scaled-up parameters consist of transmissibility and numerical productivity index (PI). Introduction Probabilistic techniques are increasingly being used for modelling reservoir heterogeneities. However, the heterogeneities, such as permeability, generated on fine grids by geological/geostatistical models, can not be directly used for flow simulation, due to computational cost and memory storage capacity. Therefore, efficient techniques are needed to scale the fine grid petrophysical parameters up to the coarse grid. Among the scaled-up parameters, the absolute permeability is one of the most important and the most often considered in the literature. In this paper, we will discuss the determination of the equivalent absolute permeability on coarse grids from known values on a fine grid system. Following Durlofsky's classification, the equivalent (effective) properties of heterogeneous fields can be divided into two kinds of problem. The first consists in determining the effective permeability according to spatial distribution or correlation. It provides the effective properties of the medium and the results do not vary with flow conditions. The second kind of problem consists in providing the equivalent permeability so that the simulations on coarse grids are equivalent to those on fine grids by numerical flow simulators. This problem is purely numerical, and it is recognised that the boundary conditions imposed on the fine grid, which is used to determine the equivalent properties on the coarse grid, can significantly influence the scaling-up results. The problem considered in this paper belongs to the second kind. On reservoir field scale, the (coarse) gridblock cannot be considered as infinitesimal. Therefore, the equivalent permeability always depends on boundary conditions and can not be determined singly. White and Home presented a numerical technique to determine equivalent permeability tensors by using several different sets of boundary conditions. Begg et al. determined the equivalent block permeability by performing fine grid simulation using closed and constant pressure boundary conditions. Durlofsky and Romeu and Noetinger calculated this parameter by using periodic pressure boundary conditions. Different scaling-up procedures give different equivalent results. P. 441