Abstract. The flow of fluids through porous media such as groundwater flow or magma migration is a key process in geological sciences. Flow is controlled by the permeability of the rock; thus, an accurate determination and prediction of its value is of crucial importance. For this reason, permeability has been measured across different scales. As laboratory measurements exhibit a range of limitations, the numerical prediction of permeability at conditions where laboratory experiments struggle has become an important method to complement laboratory approaches. At high resolutions, this prediction becomes computationally very expensive, which makes it crucial to develop methods that maximize accuracy. In recent years, the flow of non-Newtonian fluids through porous media has gained additional importance due to, e.g., the use of nanofluids for enhanced oil recovery. Numerical methods to predict fluid flow in these cases are therefore required. Here, we employ the open-source finite difference solver LaMEM (Lithosphere and Mantle Evolution Model) to numerically predict the permeability of porous media at low Reynolds numbers for both Newtonian and non-Newtonian fluids. We employ a stencil rescaling method to better describe the solid–fluid interface. The accuracy of the code is verified by comparing numerical solutions to analytical ones for a set of simplified model setups. Results show that stencil rescaling significantly increases the accuracy at no additional computational cost. Finally, we use our modeling framework to predict the permeability of a Fontainebleau sandstone and demonstrate numerical convergence. Results show very good agreement with experimental estimates as well as with previous studies. We also demonstrate the ability of the code to simulate the flow of power-law fluids through porous media. As in the Newtonian case, results show good agreement with analytical solutions.

Abstract. Quantifying the hydraulic properties of single fractures is a fundamental requirement to understand fluid flow in fractured reservoirs. For an ideal planar fracture, the effective flow is proportional to the cube of the fracture aperture. In contrast, real fractures are rarely planar, and correcting the cubic law in terms of fracture roughness has therefore been a subject of numerous studies in the past. Several empirical relationships between hydraulic and mechanical aperture have been proposed based on statistical variations of the aperture field. However, often, they exhibit non-unique solutions, attributed to the geometrical variety of naturally occurring fractures. In this study, a non-dimensional fracture roughness quantification scheme is acquired, opposing effective surface area against relative fracture closure. This is used to capture deviations from the cubic law as a function of quantified fracture roughness, here termed hydraulic efficiencies. For that, we combine existing methods to generate synthetic 3-D fracture voxel models. Each fracture consists of two random, 25 cm2 wide self-affine surfaces with prescribed roughness amplitude, scaling exponent, and correlation length, which are separated by varying distances to form fracture configurations that are broadly spread in the newly formed two-parameter space (mean apertures in submillimeter range). First, we performed a percolation analysis on 600 000 synthetic fractures to narrow down the parameter space on which to conduct fluid flow simulations. This revealed that the fractional amount of contact and the percolation probability solely depend on the relative fracture closure. Next, Stokes flow calculations are performed, using a 3-D finite differences code on 6400 fracture models to compute directional permeabilities. The deviations from the cubic law prediction and their statistical variability for equal roughness configurations were quantified. The resulting 2-D solution fields reveal decreasing cubic law accordance down to 1 % for extreme roughness configurations. We show that the non-uniqueness of the results significantly reduces if the correlation length of the aperture field is much smaller than the spatial extent of the fracture. An equation was provided that predicts the average behavior of hydraulic efficiencies and respective fracture permeabilities as a function of their statistical properties. A model to capture fluctuations around that average behavior with respect to their correlation lengths has been proposed. Numerical inaccuracies were quantified with a resolution test, revealing an error of 7 %. By this, we propose a revised parameterization for the permeability of rough single fractures, which takes numerical inaccuracies of the flow calculations into account. We show that this approach is more accurate compared to existing formulations. It can be employed to estimate the permeability of fractures if a measure of fracture roughness is available, and it can readily be incorporated in discrete fracture network modeling approaches.

Abstract. 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 30 % does IFL start to increase the systems permeability. For lower fracture densities, the effects of IFL are smeared out in the upscaling process. However, we noticed a strong dependency of ECM-based upscaling on its grid resolution. Resolution tests suggests that, as long as the cell size is smaller than the minimal fracture length and larger than the maximal hydraulic aperture of the considered fracture network, the resulting effective permeabilities and anisotropies are resolution-independent. Within that range, ECMs are applicable to upscale flow in fracture networks.

Abstract. Fluid flow on different scales is of interest for several Earth science disciplines like petrophysics, hydrogeology and volcanology. To parameterize fluid flow in large-scale numerical simulations (e.g. groundwater and volcanic systems), flow properties on the microscale need to be considered. For this purpose experimental and numerical investigations of flow through porous media over a wide range of porosities are necessary. In the present study we sinter glass bead media with various porosities and measure the permeability experimentally. The microstructure, namely effective porosity and effective specific surface, is investigated using image processing. We determine flow properties like tortuosity and permeability using numerical simulations. We test different parameterizations for isotropic low-porosity media on their potential to predict permeability by comparing their estimations to computed and experimentally measured values.

Abstract. The flow of fluids through porous media such as groundwater flow or magma migration are key processes in geological sciences. Flow is controlled by the permeability of the rock, thus an accurate determination and prediction of its value is of crucial importance. For this reason, permeability has been measured across different scales. As laboratory measurements exhibit a range of limitations, the numerical prediction of permeability at conditions where laboratory experiments struggle has become an important method to complement laboratory approaches. At high resolutions, this prediction becomes computationally very expensive, which makes it crucial to develop methods that maximize accuracy. In recent years, the flow of non-Newtonian fluids through porous media has gained additional importance due to e.g., the use of nanofluids for enhanced oil recovery. Numerical methods to predict fluid flow in these cases are therefore required. Here, we employ the open-source finite difference solver LaMEM to numerically predict the permeability of porous media at low Reynolds numbers for both Newtonian as well as non-Newtonian fluids. We employ a stencil rescaling method to better describe the solid-fluid interface. The accuracy of the code is verified by comparing numerical solutions to analytical ones for a set of simplified model setups. Results show that stencil rescaling significantly increases the accuracy at no additional computational cost. Finally, we use our modeling framework to predict the permeability of a Fontainebleau sandstone, and demonstrate numerical convergence. Results show very good agreement with experimental estimates as well as with previous studies. We also demonstrate the ability of the code to simulate the flow of power law fluids through porous media. As in the Newtonian case, results show good agreement with analytical solutions.