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.

Highlights

  • Fast and accurate solutions for flow in fractured porous media are sought after in the petroleum industry, which increasingly extracts oil and gas from hydraulically and naturally fractured shale formations

  • An alternative model approach, advocated here, uses a gridless and meshless analytical modeling technique based on complex analysis methods (CAM), and has been stepwise developed in two key, earlier papers

  • Equation (8) correctly solves the streamlines for a generic areal doublet/dipole element and can be used to represent the streamline deflections occurring near high permeability fractures in fractured porous media

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Summary

Introduction

Fast and accurate solutions for flow in fractured porous media are sought after in the petroleum industry, which increasingly extracts oil and gas from hydraulically and naturally fractured shale formations. We revisit the superposition process used to derive the expressions for an areal doublet and an areal dipole, developed in the two prior studies [5,6] to model flow in fractured porous media. This study augments the prior solutions [5,6] by removing certain limitations when the method is applied to fractures oriented at high angles to the far-field flow. The augmentation proposed in the present study provides a comprehensive solution, valid for any orientation of the natural fractures and does not require (manual) adjustments for high-angle fractures, as was the case in the prior code.

Complex Potential Solutions
Line Doublets and Line Dipoles
Areal Doublets and Areal Dipoles
Branch Cut Effects
Augmented Solution
Shortcomings of the Prior Solution
Areal Doublet Solution
Areal Dipole Solution
Superpositions
Application
Flow in Fractured Reservoirs
Field Application
Augmented Solution for Flow Near Hydraulic Fractures with Natural Fractures
Comparison of Results Augmented and Earlier Solution
Verification
Discussion
Effect of Permeability Contrast
Model Strengths and Weaknesses
Conclusions
Full Text
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