Flow in Fractured Porous Media Modeled in Closed-Form: Augmentation of Prior Solution and Side-Stepping Inconvenient Branch Cut Locations

Abstract

Carefully chosen complex variable formulations can solve flow in fractured porous media. Such a calculus approach is attractive, because the gridless method allows for fast, high-resolution model results. Previously developed complex potentials to describe flow in porous media with discrete heterogeneities such as natural fractures can be modified to expand the accuracy of the solution range. The prior solution became increasingly inaccurate for flows with fractures oriented at larger angles with respect to the far-field flow. The modified solution, presented here, based on complex analysis methods (CAM), removes the limitation of the earlier solution. Benefits of the CAM model are (1) infinite resolution, and (2) speed of use, as no gridding is required. Being gridless and meshless, the CAM model is computationally faster than integration methods based on solutions across discrete volumes. However, branch cut effects may occur in impractical locations due to mathematical singularities. This paper demonstrates how the augmented formulation corrects physically unfeasible refraction of streamlines across high-permeability bands (natural fractures) oriented at high angles with respect to a far-field flow. The current solution is an important repair. An application shows how a drained rock volume in hydraulically fractured hydrocarbon wells will be affected by the presence of natural fractures.

Comparison of Flow Solutions for Naturally Fractured Reservoirs Using Complex Analysis Methods (CAM) and Embedded Discrete Fracture Models (EDFM): Fundamental Design Differences and Improved Scaling Method

“…Models based on CAM have been previously used to model fluid flow in hydraulically fractured reservoirs [39][40][41]. This paper applies various complex potentials based on areal doublets to model the flow of fluids in natural fractures [7,17].…”

“…The key algorithm used to model natural fractures in CAM was first derived by superposing areal doublet solutions [17], which are accurate for flow through fractures aligned with a far-field flow and can model multiple fractures with different flux strengths. The natural fracture element [17] was recently augmented [7] to accommodate the particle paths for fractures oriented at a large angle to the far-field flow. As the algorithms based on CAM are multivalued in certain branch-cut locations [44], the augmented solution [7] also needed to circumvent the branch cuts to avoid discontinuity in pressure (potential function) plots.…”

“…The natural fracture element [17] was recently augmented [7] to accommodate the particle paths for fractures oriented at a large angle to the far-field flow. As the algorithms based on CAM are multivalued in certain branch-cut locations [44], the augmented solution [7] also needed to circumvent the branch cuts to avoid discontinuity in pressure (potential function) plots. The solution was augmented by gradually superposing two areal doublets with transformed coordinates, based on the angle with the far-field flow.…”

“…The rotated fracture element results in the correct particle paths, even when the fractures occur perpendicular to the principal flow direction. For the intermediate cases, when the angles between the direction of flow and the areal doublet range between 0°and 90°, the particle paths are solved by the superposition of the original element and the rotated element [7]. The key algorithms used in this study are summarized in Section 3 and Appendix A.…”

“…In the present study, we compare the results of CAM with EDFM for natural fractures oriented at different directions with respect to the principal direction of fluid flow. The three-term CAM algorithm [7] developed to trace the particle path deflection by natural fractures oriented at any angle with respect to the far-field flow is used. The outline of this paper is as follows.…”