Fracture distributions (simple or complex fractures), fracture-conductivity heterogeneity (uniform or varying conductivity along the fracture), and flow regimes inside the fracture (Darcy or non-Darcy flow) are the three main issues that have been widely investigated for transient-pressure analysis of vertical fracture systems. In this study, we focus on the latter two issues by proposing a semianalytical solution to discuss the transient-pressure behaviors of a varying-conductivity fracture under non-Darcy-flow condition. First, a general fracture-flow equation is established for the uniform-/varying-conductivity fracture under Darcy/non-Darcy flow. Second, for the case of a varying-conductivity fracture, a dimension transformation and an unequal-length-discretization model are proposed to obtain the pressure solution. Then, the transient-pressure response for the case of non-Darcy flow in the fracture can be also obtained by use of an iterative procedure in each timestep in the Laplace domain. It is shown that results from our solutions agree very well with those reported in the literature (Guppy et al. 1982;Poe et al. 1992). Third, the transient-pressure behaviors of the varying-conductivity fracture under Darcy-and non-Darcy-flow condition are discussed in detail. Results show that non-Darcy flow in the fracture mainly reduces the effective conductivity and the transient-pressure curve follows the curve of an equivalently constant conductivity except for the case of extremely small conductivities. The pressure behaviors of varying-conductivity fractures depend on the value of average conductivity, the distribution of conductivity along the fracture, and the maximum-minimum-conductivity ratio. The presence of the varying conductivity not only affects the effective conductivity in the early and late times, but also changes the shape of the pressure curve, especially for the high-conductivity fracture in the early time. It is very difficult to accurately estimate the fracture parameters by well test for most of the cases of varying conductivities under non-Darcy flow in the fracture.

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The principal focus of this work is on transient-pressure behaviors of multiwing fractures connected to a vertical wellbore. The vertical well is fractured with multiple-fracture wings with varied intersection angle, length, and asymmetry factor (AF). In the case of equally spaced fractures connected to a vertical wellbore, three flow regimes may be observed: bilinear-flow regime, formation linear flow, and pseudoradial-flow regime. With the increase of fracture numbers, the interaction of fractures becomes stronger and a "hump" occurs on the curves of pressure derivative for low and moderate fracture conductivities. For an anisotropic formation, the fracture may grow at a specific azimuth, and a fracture cluster develops. Because of the strong interactions among fracture clusters, the end of bilinear flow occurs earlier, and the formation linear flow will not be observed even for high fracture conductivities. In some extreme case in which a vertical well is intercepted with highly asymmetrically distributed fracture clusters, its transient performances of pressure and pressure-derivative curves may deviate from the conventional type curves totally. In addition, it is found that the complexity of multiple fractures near the wellbore can enhance the recovery of oil and gas.

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In this paper, a semi-analytical solution of the pseudosteady-state (PSS) productivity index under non-Darcy flow condition is proposed. Based on the model, a new method of optimization of the fracture conductivity has been developed for non-Darcy flow in the fracture. Meanwhile, the effects of the Reynolds number, the proppant number and the fracture conductivity on the dimensionless productivity index have been discussed in detail. Results show that: (1) the classic UFD (Unified Fracture Design) curves of the Darcy-flow model underestimate the effect of the proppant number and are unsuitable for the non-Darcy-flow fracture optimization. Our discretized model provides a new tool to obtain the optimal fracture parameters for the case of non-Darcy flow condition. (2) for a given penetration ratio, the non-Darcy flow behavior exerts a strong influence on the productivity index with the dimensionless fracture conductivity C fD = 0.1 -1000 and a considerable productivity index drop can be observed. For the extremely low (C fD <0.1) or high (C fD >1000) dimensionless fracture conductivity, the effect of non-Darcy flow becomes negligible. (3) the effect of the non-Darcy flow on the productivity index becomes pronounced at large value 2 of proppant number, N p .For a given proppant number, the optimal fracture conductivity is slowly increasing as the Reynolds number increases. However, the magnitude of the effect on the productivity index is gradually declining. At the Reynolds number less than 5, the non-Darcy flow has a relatively strong impact on the productivity index and an apparent fall of the maximum productivity index can be noticed, especially for a large proppant number. Beyond the value of 5, the declining trend of the maximum productivity index gradually slows down as the Reynolds number increases.