Depth of Investigation and Depletion in Unconventional Reservoirs With Fast-Marching Methods

Abstract

Summary The concept of depth of investigation is fundamental to well-test analysis. Much of the current well-test analysis relies on solutions based on homogeneous or layered reservoirs. Well-test analysis in spatially heterogeneous reservoirs is complicated by the fact that Green's function for heterogeneous reservoirs is difficult to obtain analytically. In this paper, we introduce a novel approach for computing the depth of investigation and pressure response in spatially heterogeneous and fractured unconventional reservoirs. In our approach, we first present an asymptotic solution of the diffusion equation in heterogeneous reservoirs. Considering terms of highest frequencies in the solution, we obtain two equations: the Eikonal equation that governs the propagation of a pressure “front” and the transport equation that describes the pressure amplitude as a function of space and time. The Eikonal equation generalizes the depth of investigation for heterogeneous reservoirs and provides a convenient way to calculate drainage volume. From drainage-volume calculations, we estimate a generalized pressure solution on the basis of a geometric approximation of the drainage volume. A major advantage of our approach is that one can solve very efficiently the Eikonal equation with a class of front-tracking methods called the fast-marching methods. Thus, one can obtain transient-pressure response in multimillion-cell geologic models in seconds without resorting to reservoir simulators. We first visualize the depth of investigation and pressure solution for a homogeneous unconventional reservoir with multistage transverse fractures, and identify flow regimes from a pressure-diagnostic plot. And then, we apply the technique to a heterogeneous unconventional reservoir to predict the depth of investigation and pressure behavior. The computation is orders-of-magnitude faster than conventional numerical simulation, and provides a foundation for future work in reservoir characterization and field-development optimization.