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

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.

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Finite element modeling of fluid flow in fractured porous media using unified approach

Vietnam Journal of Earth Sciences ◽

10.15625/0866-7187/15572 ◽

2020 ◽

Vol 43 (1) ◽

pp. 13-22

Author(s):

Hai-Bang Ly ◽

Hoang-Long Nguyen ◽

Minh-Ngoc Do

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Porous Media ◽

Fluid Flow ◽

Numerical Solutions ◽

Fractured Porous Media ◽

Flow In Porous Media ◽

Soil Science ◽

The Finite Element Method ◽

Element Method

Understanding fluid flow in fractured porous media is of great importance in the fields of civil engineering in general or in soil science particular. This study is devoted to the development and validation of a numerical tool based on the use of the finite element method. To this aim, the problem of fluid flow in fractured porous media is considered as a problem of coupling free fluid and fluid flow in porous media or coupling of the Stokes and Darcy equations. The strong formulation of the problem is constructed, highlighting the condition at the free surface between the Stokes and Darcy regions, following by the variational formulation and numerical integration using the finite element method. Besides, the analytical solutions of the problem are constructed and compared with the numerical solutions given by the finite element approach. Both local properties and macroscopic responses of the two solutions are in excellent agreement, on condition that the porous media are sufficiently discretized by a certain level of finesse. The developed finite element tool of this study could pave the way to investigate many interesting flow problems in the field of soil science.

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Modeling Flow and Pressure Fields in Porous Media with High Conductivity Flow Channels and Smart Placement of Branch Cuts for Variant and Invariant Complex Potentials

Fluids ◽

10.3390/fluids4030154 ◽

2019 ◽

Vol 4 (3) ◽

pp. 154 ◽

Cited By ~ 5

Author(s):

Khanal ◽

Weijermars

Keyword(s):

Porous Media ◽

Complex Potentials ◽

Flow In Porous Media ◽

The Real ◽

Pressure Fields ◽

Flow Channels ◽

Simple Variant ◽

Simple Flows ◽

Time Invariant ◽

Branch Cut

A long overdue distinction between so-called variant and invariant complex potentials is proposed here for the first time. Invariant complex potentials describe physical flows where a switch of the real and imaginary parts of the function will still describe the same type of physical flow (but only rotated by π/2). Such invariants can be formulated with Euler’s formula to depict the same flow for any arbitrary orientation with respect to the coordinate system used. In contrast, variant complex potentials, when swapping their real and imaginary parts, will result in two fundamentally different physical flows. Next, we show that the contour integrals of the real and imaginary part of simple variant and invariant complex potentials generally do not generate any discernable branch cut problems. However, complex potentials due to the multiple superpositions of simple flows, even when invariant, may involve many options for selecting the branch cut locations. Examples of such branch cut choices are given for so-called areal doublets and areal dipoles, which are powerful tools to describe the streamlines and pressure fields for flow in porous media with enhanced permeability flow channels. After a discussion of the branch cut solutions, applications to a series of synthetic and field examples with enhanced permeability flow channels are given with examples of the streamline and pressure field solutions.

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