This paper investigates the problem of a radial (or a penny-shaped) hydraulic fracture propagating in a permeable reservoir. In particular, we consider the fluid exchange between the crack and ambient porous media as a pressure-dependent process. In most of the existing models, the fluid exchange process is represented as one-dimensional pressure-independent leak-off described by Carter’s law. We modify this mechanism by including the dependence of the fluid-exchange rate on the fluid pressure inside the fracture. The proposed approach allows the liquid not only to flow out of the fracture but also to leak-in along the region adjacent to the fracture front. A complete model for hydraulic fracturing involves several physical processes that determine the crack propagation, namely, brittle rock failure, elastic equilibrium of rock, viscous fluid flow inside the fracture channel, and fluid exchange. In order to resolve these phenomena near the fracture front in the numerical scheme, we utilise a special asymptotic multi-scale model for the near-tip region, which is used as a propagation condition for the finite fracture. The main aspect of the analysis is a comparison of fracture characteristics such as aperture profile, radius and others calculated via the proposed model with the results of the radial fracture model that assumes standard pressure-independent fluid exchange mechanism governed by Carter’s law. Based on the comparison, we determine parameter ranges, for which the effect of the pressure-dependent fluid exchange mechanism is essential, and, on the other hand, we outline the zones for which Carter’s leak-off model provides accurate results. Finally, by using an approximate solution for the radial fracture with Carter’s leak-off, we obtain estimates of the effect in the entire parametric space, which allows one to evaluate the influence of the pressure-dependent leak-off for any problem parameters.

本文研究了在可渗透储层中传播的径向（或便士形）水力压裂问题。尤其是，我们将裂缝与周围多孔介质之间的流体交换视为与压力相关的过程。在大多数现有模型中，流体交换过程表示为卡特定律所描述的一维独立于压力的泄漏。我们通过包括流体交换速率对裂缝内部流体压力的依赖性来修改这种机制。所提出的方法不仅使液体从裂缝中流出，而且使液体沿着与裂缝前沿相邻的区域渗入。完整的水力压裂模型涉及确定裂缝扩展的几个物理过程，即脆性岩石破坏，岩石弹性平衡，粘性流体在裂缝通道内部流动，并进行流体交换。为了在数值方案中解决裂缝前沿附近的这些现象，我们对近端区域采用了一种特殊的渐近多尺度模型，该模型被用作有限裂缝的传播条件。分析的主要方面是比较裂缝特征，如通过建议的模型计算出的孔径特征，半径和其他特征，以及径向裂缝模型的结果，该模型假定标准压力无关的流体交换机制受卡特定律控制。在比较的基础上，我们确定了参数范围，对于这些参数范围，依赖于压力的流体交换机制的作用至关重要，另一方面，我们概述了Carter泄漏模型可提供准确结果的区域。最后，