Analytical Solution for Transient Two-Dimensional Filtration nearby a Fracture of Finite Conductivity

The problem of 2D filtration flow in the neighborhood of a producing vertical fracture of limited size and finite conductivity is revisited. New asymptotic solutions specific of 2D inflows to a fracture at early times with pseudo-1D inflows, and at large times with pseudo-radial inflows, are obtained. In general, the problem does not allow for an exact analytical solution to be obtained for any time and any fracture conductivity, even using the Laplace transform. In the limiting cases, the equations are simplified and can be solved either exactly or approximately. The vertex solutions are then interpolated to construct an approximate solution for the profiles of pressure, flowrate and inflow along a fracture at any time and any fracture conductivity. The derived general solution is compared with the numerical solver and analytical solutions, previously obtained for limiting cases of infinite conductivity, steady and pseudo-steady-state flow regimes. The obtained approximation makes it possible to estimate the accurate solution of a fluid filtration problem much faster than numerical methods, while maintaining the accuracy sufficient for field operations. This makes the results of this work to apply to extensive multivariate simulations of fractured well production and inverse problems, such as the problem of optimization and interpretation of field data.