In the oil industry, many reservoirs produce from partially penetrated wells, either to postpone the arrival of undesirable fluids or to avoid problems during drilling operations. The majority of these reservoirs are heterogeneous and anisotropic, such as naturally fractured reservoirs. The analysis of pressure-transient tests is a very useful method to dynamically characterize both the heterogeneity and anisotropy existing in the reservoir. In this paper, a new analytical solution for a partially penetrated well based on a fractal approach to capture the distribution and connectivity of the fracture network is presented. This solution represents the complexity of the flow lines better than the traditional Euclidean flow models for single-porosity fractured reservoirs, i.e., for a tight matrix. The proposed solution takes into consideration the variations in fracture density throughout the reservoir, which have a direct influence on the porosity, permeability, and the size distribution of the matrix blocks as a result of the fracturing process. This solution generalizes previous solutions to model the pressure-transient behavior of partially penetrated wells as proposed in the technical literature for the classical Euclidean formulation, which considers a uniform distribution of fractures that are fully connected. Several synthetic cases obtained with the proposed solution are shown to illustrate the influence of different variables, including fractal parameters.
This paper reviews some concepts related to the evaluation of the wellbore pressure behavior in a fractal reservoir. The purpose of this work is to show the impact of the mechanical skin on the pressure drop and its derivative behavior, and therefore on the interpretation of the well test data to determine parameters of a reservoir with fractal geometry. We use the solution for an infinite fractal reservoir without and with matrix participation, with a well producing at constant rate, including wellbore storage and skin effects. This solution is in the Laplace domain; applying numerical inversion with the Stehfest algorithm we obtain the pressure solution in time. It is shown that the characteristic power law behavior of both pressure drop and semilogarithmic derivative not only depends on the parameters of fractal dimension (dmf) and connectivity index of fractures (θ), but also on the skin factor. The parallel straight lines behavior may be valid at long times during the transient period, but at early times, the skin and wellbore storage effects may inhibit this behavior. This effect, at early times, is greater for fractal dimension values close to the Euclidian dimension (i.e. dmf~2). Thus, if the fractal dimension (dmf) and/or the connectivity of fractures decrease (θvalue increases), the pressure drop in the reservoir may be larger than the pressure drop due to the skin effect, consequently the parallel behavior of pressure and its derivative would be even valid at early times. But if the skin is big enough, the parallel behavior would only be valid at long times during the transient period. The direct determination of the skin is not possible for these systems because the skin and fractal parameters are affecting the pressure response at the same time. A new definition of dimensionless time, which includes a new equivalent wellbore radius concept to take into account the skin effect in a general way, is proposed. With this definition of dimensioless time, it is possible to get the parallel straight lines behavior even at early times. It is presented a sensibility analysis to fractal dimension, connectivity of the fracture network, skin factor and wellbore storage constant, for a well on a radial fractal reservoir. The proposed methodology is important because the direct determination of the skin is not possible for fractal systems as it can be done for classical Euclidean reservoirs, with a logarithmic behavior, mainly because the skin and fractal parameters are affecting the pressure response at the same time.
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