Geofluids

Volume 2019, Article ID 9736729, 13 pages

https://doi.org/10.1155/2019/9736729

## A Novel Numerical Model for Fluid Flow in 3D Fractured Porous Media Based on an Equivalent Matrix-Fracture Network

School of Civil Engineering and Architecture, Nanchang University, Nanchang 330033, China

Correspondence should be addressed to Jianhua Yang; nc.ude.ucn@68auhnaijgnay

Received 1 July 2018; Accepted 10 October 2018; Published 3 January 2019

Academic Editor: Baojun Bai

Copyright © 2019 Chi Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An original 3D numerical approach for fluid flow in fractured porous media is proposed. The whole research domain is discretized by the Delaunay tetrahedron based on the concept of node saturation. Tetrahedral blocks are impermeable, and fluid only flows through the interconnected interfaces between blocks. Fractures and the porous matrix are replaced by the triangular interface network, which is the so-called equivalent matrix-fracture network (EMFN). In this way, the three-dimensional seepage problem becomes a two-dimensional problem. The finite element method is used to solve the steady-state flow problem. The big finding is that the ratio of the macroconductivity of the whole interface network to the local conductivity of an interface is linearly related to the cubic root of the number of nodes used for mesh generation. A formula is presented to describe this relationship. With this formula, we can make sure that the EMFN produces the same macroscopic hydraulic conductivity as the intact rock. The approach is applied in a series of numerical tests to demonstrate its efficiency. Effects of the hydraulic aperture of fracture and connectivity of the fracture network on the effective hydraulic conductivity of fractured rock masses are systematically investigated.

#### 1. Introduction

The research for seepage flow in the subsurface system is of great significance in many geological and environmental engineering applications, for example, petroleum and gas exploitation, greenhouse gas storage, and nuclear waste treatment. Most rocks in nature are cut by fracture or fracture network, and the fractures have significant influence on the deformation and seepage characteristics of rocks [1–4]. It is well known that fluid tends to flow through the easiest path to the downstream. For fractured rock, the conductivity of fractures is often several orders of magnitude higher than that of the porous matrix, which means that fractures and fracture networks can strongly influence or even dominate the conductivity and flow paths of rock masses. Moreover, fluid exchange between the porous matrix and fractures often results in complex hydraulic behaviors, such as seepage flow in aquifers and reservoirs. It is generally hard to solve seepage problems in such complicated materials with analytical methods. On the other hand, many numerical simulation methods, such as finite element method [5, 6], finite volume method [7, 8], boundary element method [9], and numerical manifold method [10, 11], provide efficient tools for the analysis of fluid flow in fractured porous media and can help us better predict the hydraulic properties of underground engineering.

There are three major kinds of model for fluid flow simulation in fractured porous media which based on equivalent continuum/porous media (EPM), discrete fracture network (DFN), or fractured porous media (FPM). The EPM-based model assumes that fluid flow obeys Darcy’s law, in which fractures and matrix are approximated as a continuum and the classical continuous seepage theory is used for analysis [12–14]. This kind of model provides a phenomenological description of flow in fractured porous media in large dimension. However, it is limited by whether the REV exists and cannot reflect exactly the local conductivity anisotropy induced by fractures and the preferential flow effect within fracture networks. DFN-based models assume that rock blocks are impermeable and flow movement only takes place in fracture networks [15–17]. The geometry and topology of fracture networks are generally described explicitly by this kind of approach. DFN-based models are applicable only when the fracture network is well-connected, and the conductivity of the rock matrix is negligible. Methods based on the FPM concept consider that fluid flows simultaneously within fractures and the rock matrix [7, 11, 18–21]. This kind of approach inherits the advantages of EPM and DFN. It not only describes the geometry of the fracture network but also takes into consideration the contribution from the matrix and the fluid exchange between matrix and fractures. However, this method has difficulties in numerical mesh generation due to geometry complexity of fracture networks, and the computational cost is generally very high. The three kinds of model mentioned previously have their advantages in fluid seepage simulation, respectively, among which the FPM-based models are usually more comprehensive, more consistent with physical reality, and thus could provide more details.

To overcome the drawbacks of three previously mentioned approaches, Yao et al. [22] proposed a novel equivalent matrix-fracture network (EMFN) approach, in which flows in the rock matrix and fractures are represented by the flow in an equivalent fracture network. However, Yao et al. [22] only considered 2D problems where the Voronoi diagram was used for discrete 2D domains and the one-dimensional equivalent fracture network was used to study the conductivity evolution of fractured rock masses. It is well known that the 2D fracture network tends to underestimate the conductivity compared with those based on 3D description since the connectivity of the 2D model is inferior to the 3D model [5, 23, 24]. As a result, 3D models are necessary for the prediction of hydraulic properties of fractured porous media.

In this study, the EMFN approach is further developed to model 3D fluid flows in fractured porous media. The porous matrix and fracture network are replaced by an equivalent matrix-fracture network (EMFN). Delaunay tetrahedrons are used to construct the 3D fracture network. It is shown that Delaunay tetrahedrons are efficient and able to handle irregular boundary geometry and complex fracture distributions. The proposed approach is applied to study the conductivity of complex fractured rock masses. Results show that there is a unique relationship between the macroscopic conductivity of the whole interface network and the microscopic conductivity of interfaces, the ratio of which is linearly related to the cubic root of the number of nodes used for mesh generation. With this relationship, the so-called equivalent matrix-fracture network (EMFN) can produce the same conductivity of intact rock, which lays the foundation for the proposed model. The effectiveness of this approach is then demonstrated by several examples.

#### 2. The Equivalent Matrix-Fracture Network Model

##### 2.1. The Numerical Model

In this approach, rock masses and fractures are discretized into a triangular interface network in the 3D domain. Fluid flow occurs on the network and follows Darcy’s law. Firstly, a 2D model is used to illustrate the concept of the equivalent matrix-fracture network model. A 2D schematic of one streamline (solid red line) in the model is depicted in Figure 1(a), where the triangular interfaces are reduced to line segments. A 3D conceptual diagram of the interface network is also shown in Figure 1(b). The triangular interfaces between tetrahedrons are the paths through which fluid flows. Rock matrix and fractures are represented by interface networks in the same mesh. The fracture thickness is not considered. The only difference between the matrix interface and the fracture interface is that they are assigned different transmissivities while building the equivalent equation. In this way, the pore structure of the porous media and the fracture network is replaced by an equivalent interface network, which is called the 3D equivalent matrix-fracture network (EMFN). The core concept of the EMFN model is to use a sufficiently complex and chaotic discrete and permeable interface network to fill a three-dimensional volume, making the effective permeability of the interface network equivalent to the continuous media. Then, the fractures and matrix are simultaneously represented as lower-dimensional domains (2D surfaces) in the volume, and the Galerkin-based finite element method is used to solve the flow problem.