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This paper first describes a mathematical model of a vertical fracture with constant conductivity in three crossflow rectangular layers. Then, three forms of vertical fracture (linear, logarithmic, and exponential variations) with varying conductivity are introduced to this mathematical model. A novel mathematical model and its semi-analytical solution of a vertical fracture with varying conductivity intercepting a three-separate-layered crossflow reservoir is developed and executed. Results show that the transient pressures are divided into three stages: the linear-flow phase, the medium unsteady-flow stage, and the later pseudo-steady-flow phase. The parameters of the fracture, reservoir, and the multi-permeability medium directly influence the direction, transition, and shape of the transient pressure. Meanwhile, the fracture conductivity is higher near the well bottom and is smaller at the tip of the fracture for the varying conductivity. Therefore, there are many more differences between varying conductivity and constant conductivity. Varying conductivity can correctly reflect the flow characteristics of a vertical fractured well during well-test analysis.

energy exploration and exploitation

A mathematical model and semi-analytical solution for transient pressure of vertical fracture with varying conductivity in three crossflow rectangular layers