Unified fractional differential approach for transient interporosity flow in naturally fractured media

Abstract

A unified approach to modeling flows of slightly compressible fluids through naturally fractured media is presented. The unified fractional differential model is derived by combining the flow at micro scale for matrix blocks and macro scale for fractures, using the transient interporosity flow behavior at the interface between matrix blocks and fractures. The derived model is able to unify existing transient interporosity flow models formulated for different shapes of matrix blocks in any medium dimensions. The model is formulated in the form of a fractional order partial differential equation that involves Caputo derivative of order 1/2 with respect to time. Explicit solutions for the unified model are derived for different axisymmetrical spatial domains using Hankel or Hankel–Weber finite or infinite transforms. Comparisons between the predictions of the unified model and those obtained from existing transient interporosity flow models for matrix blocks in the form of slabs, spheres and cylinders are presented. It is shown that the unified fractional derivative model leads to solutions that are very close to those of transient interporosity flow models for fracture-dominant and transitional fracture-to-matrix dominant flow regimes. An analysis of the results of the unified model reveals that the pressure varies linearly with the logarithm of time for different flow regimes, with half slope for the transitional fracture-to-matrix dominant flow regime vs. the fracture and matrix dominant flow regimes. In addition, a new re-scaling that involves the characteristic length in the form of matrix block volume to surface area ratio is derived for the transient interporosity flow models for matrix blocks of different shapes. It is shown that the re-scaled transient interporosity flow models are governed by two dimensionless parameters Θ and Λ compared to only one dimensionless parameter Θ for the unified model. It is shown that the solutions of the transient interporosity flow models for different shapes of matrix blocks are almost identical for the re-scaled variables. Furthermore, the driving parameters for solution behavior are identified based on asymptotic approximations for different flow regimes. It is found that the matrix diffusion and the matrix area-to-volume ratio affect the solution behavior only for the transitional fracture-to-matrix dominant flow regime, that the capacitance ratio affects the solution behavior only for transitional and matrix dominant flow regimes and that the fracture diffusion is involved in all three flow regimes. Similar identification of the driving parameters is also presented in the re-scaled case.