The concept of radius of investigation is fundamental to well test analysis and is routinely used to design well tests and to understand the reservoir volume investigated. The radius of investigation can also be useful in identifying new well locations and planning, designing and optimizing hydraulic fractures in unconventional reservoirs. It has additional implications in estimating reserves and understanding stimulated reservoir volumes. There are many definitions of radius of investigation in the literature and Kuchuk (2009) summarized them recently. Although these definitions vary in detail, they all relate to the propagation of a pressure disturbance or impose thresholds on detectable pressure or rate changes. In this article we will focus on the definition proposed by Lee (1982). Lee defines the radius of investigation as the propagation distance of the “peak” pressure disturbance for an impulse source or sink. For simplified flow geometries and homogeneous reservoir conditions, the radius of investigation can be calculated analytically. However, such analytic solutions are severely limited for heterogeneous and fractured reservoirs, particularly for unconventional reservoirs with multistage hydraulic fractures. Generalization of the Concept How can we generalize the concept of radius of investigation to heterogeneous reservoir conditions including unconventional reservoirs with horizontal wells and multistage hydraulic fractures? For such general situations, it will be more appropriately called “the depth of investigation” rather than the radius of investigation. The simplest, not necessarily the most desirable, approach will be to use a numerical reservoir simulator. For example, we can simulate a constant rate drawdown test and observe the pressure response at every grid block in the simulation model. It is as if, we have distributed sensors throughout the reservoir. We can now compute the time derivative of the pressure at each grid block and note the time when the derivative reaches a maximum. We can then simply contour this “peak” arrival time at every grid block. Note that because the constant rate test corresponds to a step function (from 0 to Q), its derivative is an impulse function. Thus, by contouring the arrival time of the maximum of the pressure derivative, we are actually looking at the arrival time of the maximum of an impulse response as defined by Lee (1982). How well does the approach work? Fig. 1a shows the evolution of the radius of investigation for homogeneous radial flow using Lee’s analytic solution. Fig. 1b shows the radius of investigation obtained from numerical simulation. We have superimposed the analytic solution (black lines) on the results from the numerical simulation. We do see a close correspondence, although the numerical results have difficulties resolving the pressure transients away from the well. In spite of its limitations, the numerical approach is very general and can be applied to arbitrary reservoir and well conditions. The computation time and expenses, however, make the numerical simulation approach unfeasible for routine applications.
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 (Deng and Horne 1993). In this paper, we introduce a novel approach for computing the depth of investigation and pressure response in spatially heterogeneous and fractured 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 based on a geometric approximation of the drainage volume. A major advantage of our approach is that the Eikonal equation can be solved very efficiently using a class of front tracking methods called the Fast Marching Methods (FMM). Thus, transient pressure response can be obtained in multimillion cell geologic models in seconds without resorting to reservoir simulators. We first visualize depth of investigation and pressure solution for a homogeneous reservoir with multi-stage transverse fractures and identify flow regimes from pressure diagnostic plot. And then, we apply the technique to a heterogeneous reservoir to predict 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.
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.
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