Thermal and Hydraulic Matrix-Fracture Interaction in Dual-Permeability Simulation

Abstract

Summary The shape factor concept, originally introduced by Barenblatt in 1960, provides an elegant and powerful upscaling method for fractured reservoir simulation. The shape factor determines the fluid and heat transfer between matrix and fractures when there is a difference in pressure or temperature between matrix blocks and the surrounding fractures. An appropriate specification of the shape factor is therefore critical for accurate modeling. Since its introduction, many different values for the shape factor have been proposed in the literature, among which the well-known Warren-Root and Kazemi shape factors. The aim of this paper is to show that the selection of the appropriate shape factor should not only depend on the “shape” and dimensions of matrix blocks, but should also take into consideration the character of the dominant underlying physical recovery mechanisms. We will show that by taking into account the dominant physical recovery mechanism, the apparent discrepancies in the shape factor values reported in the literature can be overcome. We derive a general expression for the shape factor that not only captures existing shape factor expressions, but also allows extensions to recovery mechanisms requiring a dual permeability approach. The paper is organized as follows. First, we briefly review the shape factors presented in the literature. We then derive the general expression for the (single-phase) matrix-fracture shape factor. Subsequently, we analytically derive a new shape factor that captures the transient in pressure/temperature diffusion processes. To compare and contrast the impact of the various shape factors, we consider three cases of increasing complexity. First, we consider pressure/temperature diffusion in a single 1D matrix block following a step change in the boundary conditions. Next, we consider isothermal gas/oil gravity drainage from a homogeneous stack. We compare fine-grid single-porosity simulations (in which the matrix is finely gridded and in which the fractures are explicitly represented) with coarse-grid dual-permeability simulations (in which the matrix-fracture interaction is modeled by shape factors). In the third step, we consider gas-oil gravity drainage of the same stack model, but now under steam injection. In this case, steam is injected at the top, and oil recovered from the base of the fracture system. Again, we compare fine-grid single-porosity simulations with coarse-grid dual-permeability simulations. We show that in this case, the constant (asymptotic) shape factor provides a good approximation to the heating of the stack. We will show, however, that with a constant (time-independent) shape factor, the initial fast heating of the matrix blocks cannot be captured. We show that the new transient shape factor, however, enables coarse-grid dual-permeability modeling of thermal recovery processes such that they reproduce fine-grid results.