Direct Use Geothermal Systems (DUGS) are increasing their installed capacity worldwide and denser developments with multiple doublets are becoming more common. Interference between doublets therefore becomes an additional concern to subsurface uncertainties. Faults can be either barriers or conduits to flow and can affect the fluid pathways inside the reservoir. The interference between two doublets that are separated by a fault has not been previously studied for DUGS. In this work considering subsurface uncertainty in a full factorial design using 5184 3D reservoir simulations we show that a fault can reduce the system lifetime of a two-doublet system by more than 40 % if one doublet is at close proximity to it. Further, we identify that the fault can also improve both the system lifetime and generated Net Present Value (NPV) with appropriate development decisions. Contrary to previous results that did not consider reservoir architecture, a tramline well configuration is preferable when the doublets have the fault in the centre, while a checkerboard configuration is preferable as the distance to the fault decreases. The Heat In Place (HIP) recovery shows a linear relationship with flow rate and well spacing that is not affected by the fault distance or flow properties. The dimensions of the Influence Area (IA) previously considered are insufficient to capture the temperature drop at the producer wells and the fault position can increase this discrepancy. Our results show the importance of fault characterisation and well positioning with respect to a fault considering subsurface uncertainty and how this can affect denser field development of DUGS. Our findings suggest to integrate faults and the relative positioning of well doublets with respect to a fault more strongly in field development plans. Such considerations should also be included in future optimization plans of multi-well geothermal systems. Moreover, the regulatory framework should be revised to achieve a better match between the IA boundary and the production well temperature drop to enable better planning for denser development of DUGS.@en

This paper analyzes the relation between well spacing and Net Present Value of a Hot Sedimentary Aquifer geothermal doublet. First, a sensitivity analysis is carried out to evaluate the effect of uncertainty of geological and production parameters on the Net present Value. Second a finite-element approach is utilized to study the effect of fluvial facies architecture on geothermal energy production. For this purpose detailed fluvial facies architecture models are created utilizing a process-based facies modelling approach. These models and reservoir properties are based on a geological dataset of the Lower Cretaceous Nieuwerkerk Formation in the West Netherlands Basin (WNB). Results of the sensitivity analysis show that a 10% variation in well spacing from a 1000m base case scenario could vary the NPV by 10%. The minimal required well spacing is dependent on the reservoir thickness, flow rate and the allowed production temperature drop. The simulations results show that the theoretical advantage of a reduction in well spacing could be balanced by a poor well connectivity between the wells because of the characteristic of fluvial reservoir architecture. @en

The Netherlands experienced the fastest European expansion of geothermal energy exploitation in the past decade. The first Dutch geothermal sites proved that Hot Sedimentary Aquifers exploitation can play an important role in a future low-carbon energy mix. In this study, we estimate that with the expansion rate of the past four years, geothermal heat production from Lower Cretaceous Hot Sedimentary Aquifers could cover up to 20% of the heat demand in the province of Zuid-Holland by 2050. Although this is a significant amount, we show in this study that only 1% of the potentially recoverable heat will be recovered by 2050. This is because of inefficient doublet deployment on a ‘first-come, first served’ basis with operational parameters that focus on objectives of small decentralised heat grid demands. Instead, similar to the common-practise approach in the hydrocarbon industry, a regional coordinated ‘masterplan’ approach could be used to increase heat recovery. Utilising numerical simulations for flow and heat transfer in the subsurface, we showed that the heat recovery efficiency could be increased by tens of percentages with such coordinated doublet deployment. Based on calculations of the Levelized Costs Of Heat for both deployment strategies, we also show that current financial support schemes do not favour heat recovery optimisation. This study emphasises that although Hot Sedimentary Aquifer resources have the potential to cover a significant part of our energy demand, a radical change in financial support schemes and legislation are required to unlock their true potential.@en

Discrete-fracture and rock matrix (DFM) modelling necessitates a physically realistic discretisation of the large aspect ratio fractures and the dissected material domains. Using unstructured spatially adaptively refined finite-element meshes, we find that the fastest flow often occurs in the smallest elements. Flow velocity and element size vary over many orders of magnitude, disqualifying global Courant number (CFL)-dependent transport schemes because too many time steps would be necessary to investigate displacements of interest. Here, we present a higher-order accurate implicit pressure-(semi)-implicit transport scheme for the advection-diffusion equation that overcomes this CFL limitation for DFM models. Using operator splitting, we solve the pressure and the transport equations on finite-element, node-centred finite-volume meshes, respectively, using algebraic multigrid methods. We apply this approach to field data-based DFM models where the fracture flow velocity and mesh refinement is 2-4 orders of magnitude greater than that of the matrix. For a global CFL of ≤10,000, this implies sub-CFL, second-order accurate behaviour in the matrix, and super-CFL, at least first-order accurate, transports in fast-flowing fractures. Their greater refinement, however, largely offsets this numerical dispersion, promoting a highly accurate overall solution. Numerical and fracture-related mechanical dispersions are compared in the realistic DFM models using second-order accurate runs as reference cases. With a CFL histogram, we establish target error criteria for CFL overstepping. This analysis indicates that for extreme fracture heterogeneity, only a few transport steps can be sufficient to analyse macro-dispersion. This makes our implicit method attractive for quick analysis of transport properties on multiple realisations of DFM models.@en

Discrete-fracture and rock matrix (DFM) modelling necessitates a physically realistic discretisation of the large aspect ratio fractures and the dissected material domains. Using unstructured spatially adaptively refined finite-element meshes, we find that the fastest flow often occurs in the smallest elements. Flow velocity and element size vary over many orders of magnitude, disqualifying global Courant number (CFL)-dependent transport schemes because too many time steps would be necessary to investigate displacements of interest. Here, we present a higher-order accurate implicit pressure-(semi)-implicit transport scheme for the advection-diffusion equation that overcomes this CFL limitation for DFM models. Using operator splitting, we solve the pressure and the transport equations on finite-element, node-centred finite-volume meshes, respectively, using algebraic multigrid methods. We apply this approach to field data-based DFM models where the fracture flow velocity and mesh refinement is 2-4 orders of magnitude greater than that of the matrix. For a global CFL of ≤10,000, this implies sub-CFL, second-order accurate behaviour in the matrix, and super-CFL, at least first-order accurate, transports in fast-flowing fractures. Their greater refinement, however, largely offsets this numerical dispersion, promoting a highly accurate overall solution. Numerical and fracture-related mechanical dispersions are compared in the realistic DFM models using second-order accurate runs as reference cases. With a CFL histogram, we establish target error criteria for CFL overstepping. This analysis indicates that for extreme fracture heterogeneity, only a few transport steps can be sufficient to analyse macro-dispersion. This makes our implicit method attractive for quick analysis of transport properties on multiple realisations of DFM models.@en

A fracture network arrangement is quantified across an isolated carbonate platform from outcrop and aerial imagery to address its impact on fluid flow. The network is described in terms of fracture density, orientation, and length distribution parameters. Of particular interest is the role of fracture cross connections and abutments on the effective permeability. Hence, the flow simulations explicitly account for network topology by adopting Discrete-Fracture-and-Matrix description. The interior of the Latemar carbonate platform (Dolomites, Italy) is taken as outcrop analogue for subsurface reservoirs of isolated carbonate build-ups that exhibit a fracture-dominated permeability. New is our dual strategy to describe the fracture network both as deterministic- and stochastic-based inputs for flow simulations. The fracture geometries are captured explicitly and form a multiscale data set by integration of interpretations from outcrops, airborne imagery, and lidar. The deterministic network descriptions form the basis for descriptive rules that are diagnostic of the complex natural fracture arrangement. The fracture networks exhibit a variable degree of multitier hierarchies with smaller-sized fractures abutting against larger fractures under both right and oblique angles. The influence of network topology on connectivity is quantified using Discrete-Fracture-Single phase fluid flow simulations. The simulation results show that the effective permeability for the fracture and matrix ensemble can be 50 to 400 times higher than the matrix permeability of 1.0 · 10-14 m2. The permeability enhancement is strongly controlled by the connectivity of the fracture network. Therefore, the degree of intersecting and abutting fractures should be captured from outcrops with accuracy to be of value as analogue.@en

A fracture network arrangement is quantified across an isolated carbonate platform from outcrop and aerial imagery to address its impact on fluid flow. The network is described in terms of fracture density, orientation, and length distribution parameters. Of particular interest is the role of fracture cross connections and abutments on the effective permeability. Hence, the flow simulations explicitly account for network topology by adopting Discrete-Fracture-and-Matrix description. The interior of the Latemar carbonate platform (Dolomites, Italy) is taken as outcrop analogue for subsurface reservoirs of isolated carbonate build-ups that exhibit a fracture-dominated permeability. New is our dual strategy to describe the fracture network both as deterministic- and stochastic-based inputs for flow simulations. The fracture geometries are captured explicitly and form a multiscale data set by integration of interpretations from outcrops, airborne imagery, and lidar. The deterministic network descriptions form the basis for descriptive rules that are diagnostic of the complex natural fracture arrangement. The fracture networks exhibit a variable degree of multitier hierarchies with smaller-sized fractures abutting against larger fractures under both right and oblique angles. The influence of network topology on connectivity is quantified using Discrete-Fracture-Single phase fluid flow simulations. The simulation results show that the effective permeability for the fracture and matrix ensemble can be 50 to 400 times higher than the matrix permeability of 1.0 · 10-14 m2. The permeability enhancement is strongly controlled by the connectivity of the fracture network. Therefore, the degree of intersecting and abutting fractures should be captured from outcrops with accuracy to be of value as analogue.@en

A fracture network arrangement is quantified across an isolated carbonate platform from outcrop and aerial imagery to address its impact on fluid flow. The network is described in terms of fracture density, orientation, and length distribution parameters. Of particular interest is the role of fracture cross connections and abutments on the effective permeability. Hence, the flow simulations explicitly account for network topology by adopting Discrete-Fracture-and-Matrix description. The interior of the Latemar carbonate platform (Dolomites, Italy) is taken as outcrop analogue for subsurface reservoirs of isolated carbonate build-ups that exhibit a fracture-dominated permeability. New is our dual strategy to describe the fracture network both as deterministic- and stochastic-based inputs for flow simulations. The fracture geometries are captured explicitly and form a multiscale data set by integration of interpretations from outcrops, airborne imagery, and lidar. The deterministic network descriptions form the basis for descriptive rules that are diagnostic of the complex natural fracture arrangement. The fracture networks exhibit a variable degree of multitier hierarchies with smaller-sized fractures abutting against larger fractures under both right and oblique angles. The influence of network topology on connectivity is quantified using Discrete-Fracture-Single phase fluid flow simulations. The simulation results show that the effective permeability for the fracture and matrix ensemble can be 50 to 400 times higher than the matrix permeability of 1.0 · 10-14 m2. The permeability enhancement is strongly controlled by the connectivity of the fracture network. Therefore, the degree of intersecting and abutting fractures should be captured from outcrops with accuracy to be of value as analogue.@en

A fracture network arrangement is quantified across an isolated carbonate platform from outcrop and aerial imagery to address its impact on fluid flow. The network is described in terms of fracture density, orientation, and length distribution parameters. Of particular interest is the role of fracture cross connections and abutments on the effective permeability. Hence, the flow simulations explicitly account for network topology by adopting Discrete-Fracture-and-Matrix description. The interior of the Latemar carbonate platform (Dolomites, Italy) is taken as outcrop analogue for subsurface reservoirs of isolated carbonate build-ups that exhibit a fracture-dominated permeability. New is our dual strategy to describe the fracture network both as deterministic- and stochastic-based inputs for flow simulations. The fracture geometries are captured explicitly and form a multiscale data set by integration of interpretations from outcrops, airborne imagery, and lidar. The deterministic network descriptions form the basis for descriptive rules that are diagnostic of the complex natural fracture arrangement. The fracture networks exhibit a variable degree of multitier hierarchies with smaller-sized fractures abutting against larger fractures under both right and oblique angles. The influence of network topology on connectivity is quantified using Discrete-Fracture-Single phase fluid flow simulations. The simulation results show that the effective permeability for the fracture and matrix ensemble can be 50 to 400 times higher than the matrix permeability of 1.0 · 10-14 m2. The permeability enhancement is strongly controlled by the connectivity of the fracture network. Therefore, the degree of intersecting and abutting fractures should be captured from outcrops with accuracy to be of value as analogue.@en

A fracture network arrangement is quantified across an isolated carbonate platform from outcrop and aerial imagery to address its impact on fluid flow. The network is described in terms of fracture density, orientation, and length distribution parameters. Of particular interest is the role of fracture cross connections and abutments on the effective permeability. Hence, the flow simulations explicitly account for network topology by adopting Discrete-Fracture-and-Matrix description. The interior of the Latemar carbonate platform (Dolomites, Italy) is taken as outcrop analogue for subsurface reservoirs of isolated carbonate build-ups that exhibit a fracture-dominated permeability. New is our dual strategy to describe the fracture network both as deterministic- and stochastic-based inputs for flow simulations. The fracture geometries are captured explicitly and form a multiscale data set by integration of interpretations from outcrops, airborne imagery, and lidar. The deterministic network descriptions form the basis for descriptive rules that are diagnostic of the complex natural fracture arrangement. The fracture networks exhibit a variable degree of multitier hierarchies with smaller-sized fractures abutting against larger fractures under both right and oblique angles. The influence of network topology on connectivity is quantified using Discrete-Fracture-Single phase fluid flow simulations. The simulation results show that the effective permeability for the fracture and matrix ensemble can be 50 to 400 times higher than the matrix permeability of 1.0 · 10-14 m2. The permeability enhancement is strongly controlled by the connectivity of the fracture network. Therefore, the degree of intersecting and abutting fractures should be captured from outcrops with accuracy to be of value as analogue.@en

A fracture network arrangement is quantified across an isolated carbonate platform from outcrop and aerial imagery to address its impact on fluid flow. The network is described in terms of fracture density, orientation, and length distribution parameters. Of particular interest is the role of fracture cross connections and abutments on the effective permeability. Hence, the flow simulations explicitly account for network topology by adopting Discrete-Fracture-and-Matrix description. The interior of the Latemar carbonate platform (Dolomites, Italy) is taken as outcrop analogue for subsurface reservoirs of isolated carbonate build-ups that exhibit a fracture-dominated permeability. New is our dual strategy to describe the fracture network both as deterministic- and stochastic-based inputs for flow simulations. The fracture geometries are captured explicitly and form a multiscale data set by integration of interpretations from outcrops, airborne imagery, and lidar. The deterministic network descriptions form the basis for descriptive rules that are diagnostic of the complex natural fracture arrangement. The fracture networks exhibit a variable degree of multitier hierarchies with smaller-sized fractures abutting against larger fractures under both right and oblique angles. The influence of network topology on connectivity is quantified using Discrete-Fracture-Single phase fluid flow simulations. The simulation results show that the effective permeability for the fracture and matrix ensemble can be 50 to 400 times higher than the matrix permeability of 1.0 · 10-14 m2. The permeability enhancement is strongly controlled by the connectivity of the fracture network. Therefore, the degree of intersecting and abutting fractures should be captured from outcrops with accuracy to be of value as analogue.@en

This study investigates the effect of thermoelastic interactions between multiple parallel fractures on energy production from a multiple-fracture enhanced geothermal system. A coupled thermo-hydro-mechanical finite element model has been developed that accounts for non-isothermal fluid flow within the fractures, conductive heat transfer in the rock matrix, and the mechanical deformation of the matrix. The model results show that the matrix deformation significantly increases the interactions between the two adjacent fractures. Matrix contraction due to the cooling of the matrix increases the fracture aperture in the adjacent fracture, and facilitates the creation of favourable flow pathways between the injection and production wells. These flow paths reduce the energy production from the system. The effects of fracture spacing, reservoir temperature gradient and mechanical properties of the rock matrix on the production temperature and the net production energy are investigated. It is shown that the spacing calculated based on the assumption of rigid matrix (constant uniform aperture) are too small, and in order to account for the thermoelastic interactions, the spacing between fractures should be further increased to maximise the net energy production from the system. Otherwise, the multiple-fracture system fails to improve the energy recovery from the geothermal reservoir, as initially intended.@en

The low-enthalpy geothermal systems are commonly deployed in sedimentary geological settings that feature significant levels of deposition-induced heterogeneity. In this paper, realistic levels of heterogeneity in the form of varying porosity variance and spatial correlation lengths are considered for a 3D geothermal system. Using 2600 computationally intensive numerical simulations of two doublets placed in a checkboard pattern, the influence of well and doublet spacings on performance metrics of low-enthalpy geothermal systems are investigated. The simulations strongly support that in varyingly heterogeneous systems, the lifetimes of operation are shorter, and depending on isotropicity or anisotropicity of correlated heterogeneity, the lifetimes vary. Most notably the anisotropically correlated heterogeneity can lead to either positive impact (by diverting the cold water plume) or negative impact (by facilitating an early breakthrough of cold water plume) on the lifetime of the operation compared to isotropically correlated heterogeneity. We also calculate the boundary of the region around the wells designated as the “license area” (where the cold water front reaches to or where a threshold temperature drop of 1 °C occurs). By doing so, it is found that the operator can assume larger extents (of up to 50%) for the license areas of the aquifer than the ones conventionally assumed. To minimize the impact of heterogeneity on operation, the best practice was found to place the doublets in the same spacings as of the wells. Moreover, it is found that the well distance can be significantly shorter than what is commonly realised for heterogeneous geothermal aquifers.@en

The reactive mixing between seawater and terrestrial water in coastal aquifers influences the water quality of submarine groundwater discharge. While these waters come into contact at the seawater groundwater interface by density driven flow, their chemical components dilute and react through dispersion. A larger interface and wider mixing zone may provide favorable conditions for the natural attenuation of contaminant plumes. It has been claimed that the extent of this mixing is controlled by both, porous media properties and flow conditions. In this study, the interplay between dispersion and reactive processes in coastal aquifers is investigated by means of numerical experiments. Particularly, the impact of dispersion coefficients, the velocity field induced by density driven flow and chemical component reactivities on reactive transport in such aquifers is studied. To do this, a hybrid finite-element finite-volume method and a reactive simulator are coupled, and model accuracy and applicability are assessed. A simple redox reaction is considered to describe the degradation of a contaminant which requires mixing of the contaminated groundwater and the seawater containing the terminal electron acceptor. The resulting degradation is observed for different scenarios considering different magnitudes of dispersion and chemical reactivity. Three reactive transport regimes are found: reaction controlled, reaction-dispersion controlled and dispersion controlled. Computational results suggest that the chemical components' reactivity as well as dispersion coefficients play a significant role on controlling reactive mixing zones and extent of contaminant removal in coastal aquifers. Further, our results confirm that the dilution index is a better alternative to the second central spatial moment of a plume to describe the mixing of reactive solutes in coastal aquifers.@en

The reactive mixing between seawater and terrestrial water in coastal aquifers influences the water quality of submarine groundwater discharge. While these waters come into contact at the seawater groundwater interface by density driven flow, their chemical components dilute and react through dispersion. A larger interface and wider mixing zone may provide favorable conditions for the natural attenuation of contaminant plumes. It has been claimed that the extent of this mixing is controlled by both, porous media properties and flow conditions. In this study, the interplay between dispersion and reactive processes in coastal aquifers is investigated by means of numerical experiments. Particularly, the impact of dispersion coefficients, the velocity field induced by density driven flow and chemical component reactivities on reactive transport in such aquifers is studied. To do this, a hybrid finite-element finite-volume method and a reactive simulator are coupled, and model accuracy and applicability are assessed. A simple redox reaction is considered to describe the degradation of a contaminant which requires mixing of the contaminated groundwater and the seawater containing the terminal electron acceptor. The resulting degradation is observed for different scenarios considering different magnitudes of dispersion and chemical reactivity. Three reactive transport regimes are found: reaction controlled, reaction-dispersion controlled and dispersion controlled. Computational results suggest that the chemical components' reactivity as well as dispersion coefficients play a significant role on controlling reactive mixing zones and extent of contaminant removal in coastal aquifers. Further, our results confirm that the dilution index is a better alternative to the second central spatial moment of a plume to describe the mixing of reactive solutes in coastal aquifers.@en

The reactive mixing between seawater and terrestrial water in coastal aquifers influences the water quality of submarine groundwater discharge. While these waters come into contact at the seawater groundwater interface by density driven flow, their chemical components dilute and react through dispersion. A larger interface and wider mixing zone may provide favorable conditions for the natural attenuation of contaminant plumes. It has been claimed that the extent of this mixing is controlled by both, porous media properties and flow conditions. In this study, the interplay between dispersion and reactive processes in coastal aquifers is investigated by means of numerical experiments. Particularly, the impact of dispersion coefficients, the velocity field induced by density driven flow and chemical component reactivities on reactive transport in such aquifers is studied. To do this, a hybrid finite-element finite-volume method and a reactive simulator are coupled, and model accuracy and applicability are assessed. A simple redox reaction is considered to describe the degradation of a contaminant which requires mixing of the contaminated groundwater and the seawater containing the terminal electron acceptor. The resulting degradation is observed for different scenarios considering different magnitudes of dispersion and chemical reactivity. Three reactive transport regimes are found: reaction controlled, reaction-dispersion controlled and dispersion controlled. Computational results suggest that the chemical components' reactivity as well as dispersion coefficients play a significant role on controlling reactive mixing zones and extent of contaminant removal in coastal aquifers. Further, our results confirm that the dilution index is a better alternative to the second central spatial moment of a plume to describe the mixing of reactive solutes in coastal aquifers.@en

The reactive mixing between seawater and terrestrial water in coastal aquifers influences the water quality of submarine groundwater discharge. While these waters come into contact at the seawater groundwater interface by density driven flow, their chemical components dilute and react through dispersion. A larger interface and wider mixing zone may provide favorable conditions for the natural attenuation of contaminant plumes. It has been claimed that the extent of this mixing is controlled by both, porous media properties and flow conditions. In this study, the interplay between dispersion and reactive processes in coastal aquifers is investigated by means of numerical experiments. Particularly, the impact of dispersion coefficients, the velocity field induced by density driven flow and chemical component reactivities on reactive transport in such aquifers is studied. To do this, a hybrid finite-element finite-volume method and a reactive simulator are coupled, and model accuracy and applicability are assessed. A simple redox reaction is considered to describe the degradation of a contaminant which requires mixing of the contaminated groundwater and the seawater containing the terminal electron acceptor. The resulting degradation is observed for different scenarios considering different magnitudes of dispersion and chemical reactivity. Three reactive transport regimes are found: reaction controlled, reaction-dispersion controlled and dispersion controlled. Computational results suggest that the chemical components' reactivity as well as dispersion coefficients play a significant role on controlling reactive mixing zones and extent of contaminant removal in coastal aquifers. Further, our results confirm that the dilution index is a better alternative to the second central spatial moment of a plume to describe the mixing of reactive solutes in coastal aquifers.@en

Predicting equivalent permeability in fractured reservoirs requires an understanding of the fracture geometry and aperture, which cannot be fully sampled from subsurface data. We quantify for a range of aperture-size scaling models and the Barton-Bandis conductive shear model the fractures that are critically stressed at in-situ stress conditions as defined by either Coulomb friction or Barton-Bandis peak shearing. These models are applied to realistic large-scale fracture networks. (Sub-)linear length scaling predicts the largest average aperture, and subsequent equivalent permeability exceeds 1 Darcy in a 10mD matrix. In contrast, Barton-Bandis aperture predicts an equivalent permeability of 60mD. Application of critical stress criteria results in a decrease in the fraction of open fractures. For the applied stress conditions, Coulomb predicts that 50% of the network is critically stressed, compared to 80% for Barton-Bandis peak shear. The impact of the reactivated fracture network on equivalent permeability depends on the matrix properties, as in a low-permeable matrix, fracture connectivity controls equivalent permeability, while for a more permeable matrix, large absolute apertures are more important than network connectivity. Quantification of fracture flow regimes using only the ratio of fracture versus matrix permeability is insufficient, as these regimes also depend on fracture connectivity, controlled by aperture. @en

Fracture networks are heterogeneous systems creating anisotropic flow patterns, though field-scale flow models of fractured reservoirs use up-scaling of the fracture network to effective properties to model flow. We study the impact of upscaling aperture from a realistic aperture model, where aperture is defined by a stress-aperture relation that considers aperture variations within single fractures, to an arithmetically averaged distribution. This analysis is done through calculation of the equivalent permeability tensor of a discrete fracture and matrix flow model, taking into account different contrasts between fracture and matrix permeability. Results show that a strong equivalent permeability anisotropy emerges for a low matrix permeability, and that accurate representation of fracture aperture is as important as the fracture topology, particularly when the fracture matrix permeability ratio is large. The orientation of the permeability tensor varies with varying rock matrix permeability, reflecting anisotropy of the permeability that is impossible to extract from the fracture patterns itself. The results also highlight that utilizing a constant aperture for the fracture network may not reproduce the true anisotropy of the permeability. Utilizing an equivalent aperture in flow simulations introduces errors in both the direction and magnitude of principal permeabilities and the equivalent permeability anisotropy.@en