Predicting effective permeabilities of fractured rock masses is a crucial component of reservoir modeling. Its often realized with the discrete fracture network (DFN) method, whereby single-phase incompressible fluid flow is modeled in discrete representations of individual fractures in a network. Depending on the overall number of fractures, this can result in high computational costs. Equivalent continuum models (ECMs) provide an alternative approach by subdividing the fracture network into a grid of continuous medium cells, over which hydraulic properties are averaged for fluid flow simulations. While continuum methods have the advantage of lower computational costs and the possibility of including matrix properties, choosing the right cell size to discretize the fracture network into an ECM is crucial to provide accurate flow results and conserve anisotropic flow properties. Whereas several techniques exist to map a fracture network onto a grid of continuum cells, the complexity related to flow in fracture intersections is often ignored. Here, numerical simulations of Stokes flow in simple fracture intersections are utilized to analyze their effect on permeability. It is demonstrated that intersection lineaments oriented parallel to the principal direction of flow increase permeability in a process we term intersection flow localization (IFL). We propose a new method to generate ECMs that includes this effect with a directional pipe flow parameterization: the fracture-and-pipe model. Our approach is compared against an ECM method that does not take IFL into account by performing ECM-based upscaling with a massively parallelized Darcy flow solver capable of representing permeability anisotropy for individual grid cells. While IFL results in an increase in permeability at the local scale of the ECM cell (fracture scale), its effects on network-scale flow are minor. We investigated the effects of IFL for test cases with orthogonal fracture formations for various scales, fracture lengths, hydraulic apertures, and fracture densities. Only for global fracture porosities above

Discontinuities in rocks provide major pathways for subsurface fluid migration. Thus, fractured reservoirs are frequent targets for oil, gas, or water production, geothermal energy recovery, and

Numerical modeling of fluid flow is most accurately based on the Navier–Stokes equations

A large number of numerical methods to compute effective permeabilities of fractured media have been developed

According to

Several techniques to generate equivalent continuum models (ECMs) of DFNs have been developed in 2D

Next to these issues, this study focuses on an often ignored but potentially important aspect in fracture network modeling given by the complexity of fracture intersection flow. To our knowledge, only a few studies have presented 3D flow simulations within fracture intersections

Panels

2D sketch of the half-hypotenuse assumption in an idealized rectangular fracture crossing (gray regions indicate rock matrix, white regions fracture pore space). The hydraulic apertures (

Fluid flow in porous and fractured media is described by the well-known Navier–Stokes equations

Here, the 3D staggered-grid, finite-difference code LaMEM

As the two main structural features (fractures and intersections) composing a fracture network differ significantly in terms of their hydraulics (Fig.

Workflow for generating an equivalent continuum model of a DFN. Panel

Fracture-intersection-caused changes in permeability tensor characteristics. Panel

Panel

The left panel shows a comparison of directional permeabilities obtained from high-resolution Stokes flow simulations (

The use of the ECM approach instead of the DFN method to predict the effective permeabilities of fractured media crucially depends on the capability to reflect the anisotropic flow properties at the scale of the continuum cells. Therefore, it is essential to integrate the geometry of a DFN into the generation procedure of the ECM instead of generating the grid cell conductivities in a stochastic manner

To our knowledge, there is no approach to generate an ECM of a DFN that takes the effect of IFL (Fig.

Generally, the DFN approach offers a straightforward way to characterize structurally complex fracture networks. Most commonly, every fracture is modeled as a geometric primitive (here a disk) with a prescribed length

To map each individual fracture to its corresponding grid cells, we first
assume a horizontal disk (normal vector

To test the half-hypotenuse assumption (see Fig.

Panel

The figure demonstrates the difference between the vertical component of the permeability tensor

In the previous section, we demonstrated the effects of IFL on the permeability of systems at the scale of a local ECM cell (i.e., system sizes near the fracture aperture) by comparing analytically derived cell permeabilities to the results of direct flow simulations (Stokes equations). If the intersection orientation aligns with the applied pressure gradient and connects inlet and outlet pressure faces, the permeability of the system is increased (i.e., case b in Fig.

the cubic overall system side length

the constant size

the total number of fractures for each set by

the scaling parameter

the isotropic and constant permeability of the rock matrix by

Panels

Panels

Absolute permeability values

The resolution dependency of ECM methods to upscale the permeabilities of fracture networks is a crucial aspect that has to be considered to provide accurate upscaling results. Artificial connectivity is one of the main issues that arises if the resolution of the ECM is insufficient, as demonstrated by

After calculating all fracture intersections with ADFNE's built-in function Intersect (see b and d in Fig.

Including a pipe flow model into the ECM generation process improves the
representation of permeability anisotropy therein and can have impacts on
overall permeabilities as well. For example, at the scale of the intersection
itself, it significantly modifies the shape and absolute values of the
permeability tensor (Fig.

If small DFNs with sizes closer to the mean hydraulic radius of the intersections (e.g., micro-fracture networks of

For flow simulations at reservoir scales (similar to the test cases considered in Sect. 7), the only computationally feasible solution is to use parameterization concepts (e.g., Sect. 3). For that, we were able to demonstrate that the presented fracture-and-pipe ECM method is capable of providing converged effective permeability tensors if the ECM resolution, i.e., the ratio of system size to discretization step size, is sufficiently large. This resolution dependency for 3D ECMs has not been reported at this level of detail so far but was expected based on previous works of

Based on analytical solutions of flow in fracture networks with constant apertures,

This study analyzed the complexity of fracture intersection flow by conducting Stokes flow simulations in simple fracture crossings. Intersections that are aligned with the pressure gradient initiating the flow cause an increase in permeability, as they act similarly to a pipe. This results in intersection flow localization (IFL); i.e., intersections represent preferred pathways for the fluids compared to the connected fractures. We thus extended the state-of-the-art methodology to generate equivalent continuum models (ECMs) for effective permeability computations of discrete fracture networks (DFNs) to incorporate IFL effects. Those are integrated using a directional pipe flow parameterization with a hydraulic radius half the hypotenuse size in a right-angled triangle with side lengths of both intersecting hydraulic apertures. By assessing the permeabilities of fracture intersections numerically, we could demonstrate that for system sizes close to the approximated pipe radius (millimeters to centimeters), the effect of IFL on permeability can be almost 1 order of magnitude. At network scales (meters to kilometers), the impact of IFL on the system's effective permeability is generally minor. Only for fracture systems with high global fracture porosities (above

Next to the effects of IFL, we investigated the resolution dependency of current ECM-based upscaling approaches, as the cell size with which the ECM is discretized represents the most crucial aspect for the accuracy of ECM-based effective permeability predictions. Based on a resolution test with two different DFN scenarios, we suggest that the ECM cell size should be smaller than a third of the minimal fracture size and larger than the maximal hydraulic aperture of the system to conserve constant permeabilities and full anisotropy of flow. Within that range, we conclude that ECM methods equivalently serve as geometric upscaling procedures for fluid flow problems. It is important to note that the accuracy of ECM methods to predict flow is always linked to the quality of the input DFN. Improving the DFN method to better characterize natural fracture systems, especially in terms of fracture termination rules and spatial clustering, is still an ongoing topic of research.

In the following, we will explain our method to obtain the effective permeability tensor of continuum cell representations for fractured porous media. The governing equations for steady-state single-phase flow equations for an incompressible, isothermal, and isoviscous fluid without sources and sinks are given in compact form by the following system of mass (Eq.

Pressure boundary conditions for an applied gradient in the

Benchmark case 1 from

Three principal directions of the applied pressure gradient have to be considered to predict the full tensor of permeability. Thus, the flow simulation procedure has to be repeated three times such that each principal flow direction (

The single-continuum discretization scheme used might appear simplistic compared to more sophisticated mesh representations

All numerical simulation tools used in this paper are past or modified versions of the open-source software package LaMEM (

MOK wrote the initial draft of the paper, performed numerical simulations, analyzed the data, and generated the figures. AAP supervised and helped design the study, helped implement the numerical methods, and edited the paper. SA helped design the study and edited the paper. BJPK edited the paper and discussed the results.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors gratefully acknowledge the computing time granted on the supercomputer Mogon II at Johannes Gutenberg University Mainz (hpc.uni-mainz.de). The authors sincerely thank two anonymous referees for reviewing this paper.

This work has been funded by the Federal Ministry of Education and Research (BMBF; program GEO:N, grant no. 03G0865A) and the M3ODEL consortium at Johannes Gutenberg University Mainz.

This paper was edited by Juliane Dannberg and reviewed by two anonymous referees.