Finite Element Modeling of Diffusion in Fractured Porous Media by Using Hierarchical Material Properties
Abstract
Fractured porous media challenge modeling approaches due to high computational costs and excessive mesh refinement imposed by the extreme scale variability of fractures and the heterogeneity of the surrounding porous rock. To overcome such difficulties, we utilize the hierarchical finite element method (HiFEM) that has been developed previously to simulate the electrical potential distribution in complex geologic environments. The method employs the hierarchical basis functions in classical finite element analysis to enable representation of material properties on each dimensional component of a given 3D unstructured finite element, thereby inherently allowing for interactions at the boundary between fracture and a host rock. In this study, we extend its application to transient fluid flow and heat conduction in the Laplace domain. Timedomain flow solutions are obtained by numerical inverse Laplace transform. We evaluate the accuracy of the method using different flow models and demonstrate its robustness for largescale, rock mass models featuring complex fracture networks. Moreover, for the computation of nodal Darcian velocity fields in fractured porous media where the fractures are represented as 2D features, a new approach that employs the Yeh's Galerkin model for both volume and facet elements is proposed. Results show that HiFEM can produce accurate flow solutions for fractured porous media without any need of coupling or transfer mechanism while still being computationally economical and numerically robust, even for largescale simulations.
Key Points

We extend the hierarchical finite element method to model transient fluid flow and heat conduction in complex fractured porous media

The method allows to explicitly represent material properties on each dimensional component of a 3D tetrahedral finite element mesh

A new approach implementing the Yeh's model to compute the velocity of 3D fractured porous media with 2D embedded fractures is proposed
1 Introduction
Fractured rocks are the host for many engineered structures related to energy, water, waste, and transportation. For this reason, the functioning of such infrastructure, as well as optimization of its engineering efficiency, critically depends on characterizing, modeling, and monitoring fractured rock sites (National Academies of Sciences, 2015). Yet, modeling fluid flow in fractured porous media is a longstanding challenge due to the extensive numerical discretization and intractable computational cost imposed by discrete fractures and the matrix they are embedded within, where length scales span millimeterscale fractures to the kmscale regions of the surrounding rock, all while attempting to achieve high model realism and accuracy for the correct management and reliable prediction of hydraulic and thermal behaviors of fractured rock masses. Not surprisingly, there has been a surge of interest on this research problem and it still remains interesting to many researchers due to its importance in a wide scope of applications such as geologic radioactive waste disposal (e.g., C. F. Tsang et al., 2015), CO_{2} storage (e.g., March et al., 2018), geothermal energy extraction (e.g., Bauer et al., 2017), hydrocarbon recovery (e.g., Spence et al., 2014) and subsurface water resource management (Bear et al., 2012).
Several numerical approaches for modeling fluid flow in complex fractured rocks have been proposed in the hydroscience literature which can be categorized into three major concepts based on their assumptions made on underlying of fluid flow physics: (a) continuum models, (b) discrete fracture network (DFN) and (c) hybrid methods.
The continuum models describe fractures and matrix (host rock) as separate overlapping continua where matrix blocks are isolated by fractures, and the crosscontinuum flow is allowed via transfer functions (Barenblatt et al., 1960; Blaskovich et al., 1983; Hill & Thomas, 1985; Warren & Root, 1963; Wu & Pruess, 1988). The influence of discontinuous fractures is implicitly incorporated into the overall flow by representing them with upscaled continuous hydraulic properties which is why the continuum models are prevailing and computationally economical, especially for high density natural fracture systems at large scales. However, the continuum models spatially average hydraulic properties of fractures and thus are justified only when a representative elementary volume (REV) and its associated permeability tensor exist (e.g., Hitchmough et al., 2007; Long et al., 1982; Ren et al., 2017; Rong et al., 2013). This is a nontrivial problem. Continuum models require users to carefully define a proper REV characteristic size and determine a meaningful transfer function that will not oversimplify the physics of fluid flow. The main drawback is their inability to include discrete fractures with their explicit geometric and hydraulic properties in the flow model.
In contrast, the DFN models conceptualize the flow as occurring only through fractures by assuming the matrix as impervious and explicitly representing the entire architecture of complex discrete fracture networks in the flow model (Cacas et al., 1990; Erhel et al., 2009; Mustapha, 2011; Y. W. Tsang & Tsang, 1987; Zhang, 2015; Zhang & Yin, 2014). DFN models have been used widely to study the connectivity and effects of geometric and hydraulic characteristics of complex fracture networks on the flow and transport dynamics (De Dreuzy et al., 2000; Hyman et al., 2015; Koudina et al., 1998; Reeves et al., 2008; Renshaw, 1999; Sisavath et al., 2004). The lack of surrounding porous rock renders the DFN models computationally convenient particularly for modeling of fractured systems where the matrix is highly impermeable compared to the fractures (which is definitely not true for heat conduction); however, the applications of the DFN models are still constrained to small domains of complex fractures due to the expensive mesh and numerical cost required for field scale applications. While the physical assumption of impermeable rock matrix provides computational efficiency for modeling flow through complex fracture networks, it becomes the main shortcoming when it comes to capture the fluid flow in permeable formations.
The extreme physical assumptions of the continuum and the DFN models and their strict choices between continuous and discontinuous representations of fractures have motivated more flexible modeling approaches that can be applicable to a broader spectrum of fluid flow problems. The main goal is to find a comprise between smallto largescale fractures and the heterogeneity of the porous surrounding media in the flow model by simultaneously allowing implicit and explicit representations. One group of such hybrid methods is the Discrete FractureMatrix models (DFM) where explicitly represented large fractures and conduits are treated as DFNs and the surrounding matrix and statistically homogenized small fractures are evaluated as equivalent porous continua. Considerable interest has been drawn to the DFM models, which has resulted in several variants therein (Hoteit & Firoozabadi, 2008; KarimiFard & Durlofsky, 2016; Lee et al., 2001). Classical DFM methods generally obligate the matrix grid to be aligned with the fractures (KarimiFard et al., 2004; Paluszny et al., 2007). To relax this restriction, nonconforming variations have been proposed such as finitevolume based embedded DFM methods (EDFM; Hajibeygi et al., 2011; Moinfar et al., 2014; Nobeck et al., 2016; White & Fu, 2020), the eXtended finite element method (D’Angelo & Scotti, 2012; Martin et al., 2005) and the Lagrange multiplier finite element method (Kopek et al., 2019). Nonetheless, DFM methods still need a coupling mechanism to describe how the matrix and the fractures interact and a strategy to deal with the intersecting fractures (Ahmed et al., 2015; KarimiFard et al., 2004; Reichenberger et al., 2006).
The advancement made on the concept of merging the discrete and continuum hydrological features and heterogeneities in the flow model is not limited to the DFM models. Other numerical approaches based on finite element method (Durlofsky, 1994; Hoteit & Firoozabadi, 2006; Matringe et al., 2006), finite volume method (Bogdanov, 2003; Caillabet et al., 2000; Grillo et al., 2010), and finite difference method (Huang et al., 2014; Yan et al., 2016) have been also developed to simulate flow in multiscale fractured media. Numerical manifold methods have been proposed to avoid the need for the mesh discretization of complex fracture networks in the flow analysis of the fractured porous medium model (Hu & Rutqvist, 2020; Shi, 1991; Zhang et al., 2016). These methods employ the blockcutting analysis and thus both isolated and deadend fractures are artificially extended to form smaller blocks during the 3D generation to increase the numerical accuracy. The transient flow of fluids in the heterogeneous fractured formations has also been solved by the boundary element methods (BEM) where the fractures embedded within the matrix are treated as internal boundaries and the solution of the entire system is obtained via coupling of those connecting regions (Andersson & Dverstorp, 1987; Elsworth, 1986a, 1986b; Lough et al., 1998). Yet another set of models use the pipe network idea that represents the hydraulic properties of fracture networks and/or porous media as representative systems of weighted flow pipes (Li et al., 2014; Ren et al., 2017; Xu et al., 2014). While the pipe network models can drastically reduce the model size for complex flow problems (a topologically similar expression of a DFN), the BEM models capture the flow at mutually intersecting fractures more precisely. However, both the BEM and the pipe network models become less rigorous while describing the interactive flow between fractures and heterogeneous porous rock.
Recently, Weiss (2017) has proposed the hierarchical finite element method (HiFEM) that aims to achieve optimized computational efficiency and model realism for complex fractured geologic environments. Note that the HiFEM is different than the hierarchical finite element method that uses various sets of higher order polynomials as element interpolation functions (e.g., Bardell, 1991; Zhu, 1982). The HiFEM implements hierarchical basis functions for the material properties of the model into the finite element analysis which enables the representation of material properties not only on volumes but also on the lower dimensional elements of the unstructured tetrahedral finite mesh such as facets and edges. The novelty of the method comes from its strategy to incorporate the explicit geometric and physical properties of the model features with insignificant length scales into the whole geologic model by defining the hierarchical material functions in the first place before any calculations for the solution are made. Therefore, the HiFEM enables the physical process to inherently occur along and across the boundaries of the planar or curvilinear features (e.g., fractures and wellbores) embedded in the surrounding porous medium even if they are isolated. To obtain the numerical solution, it actually avoids the need of coupling and/or transfer mechanisms or specific boundary conditions and their imposed physical assumptions. Moreover, the explicit representations of fractures as connected facets and wellbores as connected edges in the 3D finite element model drastically reduce the excessive computational cost and mesh refinement. While there are other methods that enables to represent fractures as 2D planes in 3D media (e.g., the control volume finite element/Box method), the HiFEM allows to simultaneously represent 2D and 1D features in 3D media if needed and does not require a specific triangulation (e.g., Delaunay) since a dual grid is not needed. The HiFEM has been originally developed for electrical conduction in geologic media governed by Poisson's equation. Beskardes and Weiss (2020) and Weiss (2017) showed the HiFEM's robustness and efficiency for the largescale, heavily infrastructured geoelectrical models consisting of many complex fractures that could not even be simulated by other numerical methods with comparable level of model realism.
In this article, motivated with the previous success of the HiFEM in the geophysical electromagnetic problems, we extend its application to transient flow diffusion and heat conduction. Moreover, we propose a nonstandard, new approach to compute the nodal velocity for 3D fractured porous media with 2D embedded fractures from HiFEM's head fields. This novel scheme for the efficient diffusion and velocity field simulations around high permeability fractures is presented along with a set of validating comparisons and a detailed performance analysis. In Section 2, the derivation of the finite element analysis with the hierarchical hydraulic conductivities is presented. The flow diffusion equation is solved in the Laplace domain and the transient solutions of the fluid flow are obtained by using the numerical inverse Laplace transform. In Section 3, we consider a set of flow models where the accuracy and consistency of each dimensional conductivity representation of the HiFEM (volumetric, facetbased, edgebased) are sequentially tested. While the equivalency of the facetbased representation to the fully volumetric HiFEM's solution is tested with a fracture model, that of the edgebased representation is validated by considering the heat conduction of a vertical wellbore model. Moreover, the HiFEM approach is applied to largescale, complex fractured models to evaluate its computational performance and robustness for flow simulations. Last, to compute the Darcian velocity fields of the facetbased models, a new approach that solves the Yeh's Galerkin model for both 3D and 2D features in the flow problem is presented. The efficiency of the approach is demonstrated by considering a comparison against the velocity field of a volumetric model.
2 Derivation of HiFEM for Diffusion
2.1 Governing Equations
Spatially constant or linearly variable nonzero initial conditions can be also handled as volumetric source terms.
The time independency of the transformed diffusion problem (Equation 3) renders its numerical solution possible without time stepping, and therefore avoids both the necessity to satisfy the stability condition which dictates smaller time steps for finer mesh and the requirement to compute the solutions for each time step regardless of the interest at a single time (e.g., Cai & Costache, 1994; MoralesCasique & Neuman, 2009; Sudicky & McLaren, 1992).
2.2 Finite Element Analysis With Hierarchical Material Properties
According to the work of Weiss (2017), the material properties can be represented at each dimensional geometric simplex such as volumes, facets, and edges of an unstructured finite element mesh by introducing a hierarchical composite function for the material property in the finite element analysis. This approach has been previously applied to the Poisson equation with a focus on geophysical electrostatic problems where the hierarchical material function defines the electrical conductivity. In this article, we apply the HiFEM to the physics of fluid flow and heat diffusion and solve the resulting finite element system in the mixed spaceLaplace domain.
Here, n_{V}, n_{F}, and n_{E} are the number of volumes, facets and edges in the finite element mesh, respectively. Considering a reference frame with the orthogonal unit vectors , and (Figure 1), the local basis functions (Equations 8–10) employ particular combinations of unit vectors to allow the material property to be defined on each dimensional component of a given finite element. Such that, all three unit vectors are included for volume elements whereas the basis function employs only and directions for facets that span the 2D plane of the eth facet and solely the vector for edges that is parallel to the edge e. Such reduction in the dimension of the basis functions of facets and edges requires to incorporate the reduced dimension into the material property. Here, K^{(e)} represents the scalar conductivity of tetrahedron e, f^{(e)} is the conductance of facet e and t^{(e)} is the product of conductivity and crosssectional area of edge e. Since the conductivity function, K composes the contributions from the conductivities represented on facets and edges with the volumetric finite element solution, f^{(e)} and t^{(e)} should be considered in the forms of differences from the volumetric conductivity, such as (K_{facet} − K) ⋅ T and (K_{edge} − K) ⋅ A, where T is thickness and A is area, respectively. In this case, the conductivities on facets or edges are much higher than the tetrahedron's conductivity (K ≪ K_{facet} or K ≪ K_{edge}), K may be neglected in the calculations of f^{(e)} and t^{(e)}.
Here, is assumed to be constant within an element.
The integrations over individual elements are calculated analytically by assuming constant material properties and linear basis functions of elements.
Because the basis functions are specifically tailored for facets and edges, the conductivity composite function K keeps the contributions from the facet and edge conductivities strictly local within the volumetric element. This hierarchical conductivity enables representing thin conductive fractures as connected facets or narrow, curvilinear conductive features as connected edges embedded in rock matrix without the mesh refinement necessary for their small dimensions, and therefore renders possible the responses of such complex models without enormous computational cost. In fluid flow problems, the significance of the conductivity represented on edges may be useful when representing a stream or canal at the landsurface boundary of model domain using edge elements that follow its trajectory. Often, the curvilinear features with small radii compared to their lengths such as well casings are impermeable and do not contribute to the fluid flow; on the other hand, the high thermal and electrical conductivities of such metallic infrastructures have farreaching impacts on the overall temperature and electrical potential distributions, which obligates to incorporate those structures into the models as realistic as possible for heat conduction and electrostatics problems. By using the HiFEM, for those physical problems, geometrically complex well systems and their conductivity distributions can be economically modeled by representing them as connected edges with a corresponding conductivityproduct t, as will soon be demonstrated for a heat conduction problem.
It can be seen from the above equations, the implementation of the hierarchical conductivity function into the finite element analysis inherently accounts for the contributions from volumes, facets, and edges in the global finite element solution. Therefore, there is no internal boundary condition, transfer function, or any other special treatment needed in the finite element analysis, and no physical simplification required regarding the physics of flow along facets and edges. One shortcoming is that the method is unsuitable for simulating a system with fractures acting as semipermeable barriers to the flow since pressure continuity across the lower dimensional entities is assumed. Moreover, the HiFEM approach does not impose any restriction on the location or connectivity of fractures; in other words, the nonintersecting, isolated, and deadend fractures do not have to be eliminated or be constrained in the HiFEM model and all fractures can straightforwardly contribute to the flow.
2.3 Computation of Hydraulic Heads
After imposing the Dirichlet boundary condition at the edges of the model volume and the Neumann boundary condition at airEarth interface in the Laplace domain (Equations 4 and 5), the global complex symmetric (not Hermitian) finite element system of linear equations given in Equation 20 is solved using the quasiminimalresidual (QMR) iterative solver (Weiss & Newman, 2002). Note that it is possible to use more general surface boundary conditions such as a recharge source at the landsurface boundary (e.g., river or lake). Due to the large number of unknowns, iterative solvers are a reasonable choice since they require less storage space and only matrixvector products rather than matrixmatrix products as usually performed in direct solvers. The QMR is similar to the Biconjugate Gradient method but an improved iterative approach due to its efficient strategy that implements a modified Lanczos process to generate vectors for the Krylov subspace (Freund & Nachtigal, 1994). Therefore, it provides faster convergence and better numerical stability. In our study, the solution on QMR is iterated considering a target residual L^{2}norm of 10^{−13} to yield high precision results.
After obtaining the values of the unknown hydraulic head in Laplace space, the time domain finite element solutions h(t) at each node within the mesh are obtained by using the numerical inverse Laplace transform. We use the accelerated Fourier series method for the inversion of the Laplace transform. The accelerated method achieves a nonlinear double acceleration with Padé approximation. Details of the implementation of the method can be found in de Hoog et al. (1982). The Fourier series method is a generally robust quadraturebased approach that requires complexvalued Laplace parameter in contrast to realonly methods like the GaverStehfest. While there is no optimum method for the numerical inverse Laplace transform due to its illposed nature, the Fourier series method is known to be robust for nonoptimal choice of Laplace parameters (p) for diffusion (Kuhlman, 2013). In addition, the method is quite economical since the Laplace parameter p is not a function of time which enables reuse of the same values to invert different h(t). Specifically, de Hoog et al.’s accelerated approach is capable of simultaneously inverting groups of time that span one log_{10} cycle by using single vector. In our experience, we find that a total of 41 evaluations is sufficient for the inversion. In addition, the Laplacedomain finite element approach allows various ways to parallelize the modeling task.
The HiFEM and the iterative solver have been developed in Fortran 90 from scratch. We implement an MPI parallelization scheme across both the Laplace parameters and the nodes of the computational mesh. The processors equally split the task of calculating Laplace domain solutions for each Laplace parameter. Then, the computational domain is evenly decomposed and shared among the processors to calculate the numerical inverse Laplace transform at each node.
The computational precision of the HiFEM heavily depends on the efficiency of the conforming mesh generator. The discretization of fracture networks with large number of finite, arbitrarily oriented fractures embedded 3D host media is still a challenging task. The conforming meshes of such dense, complex fractured media impose tiny angles and complicated intersections between fractures which some widely used meshing methods fail to deal with or end up generating an enormous number of elements such as with Delaunay triangulation/tetrahedralization (e.g., Bogdanov et al., 2003; Mustapha, 2011) and advancingfront techniques (Koudina et al., 1998; Mourzenko et al., 2011). The possible remedies to increase the mesh quality include modifying the challenging model features during mesh generation (e.g., KarimiFard & Durlofsky, 2016; Marys̆ka et al., 2005; Mustapha, 2011) or constraining the topology and/or geometry of fracture network prior to meshing (Hyman et al., 2014). In addition, the recent progress has been done by the Virtual Elements Method (Beirão da Veiga et al., 2013) in solving flow problems in networks of discrete fractures without restrictions on the element shape ratio (Benedetto et al., 2014; Fumagalli & Keilegavlen, 2019). Recognizing the aforementioned challenges on mesh generation, we use the meshing package “Cubit” (Blacker et al., 1994) to generate the tetrahedral meshes in our study by employing the advancingfront technique for discretization, and the fracture geometries for a given set of fracture parameters (e.g., size, orientation, aperture, etc., distributions) are generated by an external script and incorporated into meshing.
3 Validations and Applications
In this section, several models are established to verify the validity of the HiFEM approach for each dimensional conductivity representation (volumetric, facet, and edge) and to demonstrate its effectiveness for transient fluid flow modeling in largescale porous media with embedded complex networks of discrete fractures. First, the accuracy of the HiFEM approach is compared against the analytical solution of the transient flow in a homogeneous medium. It can be easily seen that by defining solely volumetric conductivities, the HiFEM formulation becomes equivalent to the classical finite element analysis. By comparing the numerical solutions of the fully volumetric models with those of their corresponding facet and edgebased representations, the concept of hierarchical conductivities as well as the internal consistency of the HiFEM approach are validated. While a vertical fracture flow model is considered to verify the facetbased representation of fractures in the form of connected facets, we temporarily move our attention to the heat conduction of a vertical wellbore model to test the efficiency of the edgebased conductivities that represent thermally conductive wellbores as connected edges. In addition to the validation cases, we also consider the transient fluid flow in largescale fractured porous media consisted of complex fracture networks and a performance analysis to demonstrate the capabilities of the proposed approach.
3.1 Transient Flow in Homogeneous Medium
The homogenous model is constructed as a cuboid with dimensions of 10 × 10 × 5 km. A continuous source with a volumetric rate of 1.16 × 10^{−3} m^{3}/s (100 m^{3}/day) is located at the coordinates of (−100, 0, −2,500 m). The hydraulic head distribution is obtained by assigning zero fixedhead boundaries on left, right and at the bottom, and noflow boundary at the top. Assuming isotropy of the geologic properties, the hydraulic conductivity of the porous medium is 10^{−4} m/s and the specific storage coefficient is 2 × 10^{−4} m^{−1}. By generating a tetrahedral mesh with gradually increasing element size from the source location that consists of 398,000 tetrahedra, the transient HiFEM solutions are obtained at the times between 10^{−4} and 10^{2} days. Figure 2 compares the hydraulic heads obtained from the HiFEM simulation against the analytical solution (Crank, 1975) along a radial profile from the source location and the corresponding root mean squared errors (RMSE) at different times. The RMSEs at different times are less than 0.026 indicating the accuracy of the HiFEM numerical solution for fully volumetric flow models.
The accuracy of the HiFEM for the Poisson's equation of the geophysical electrostatics has already been benchmarked against the analytical solutions of various models including buried disk, vertical cylinder, and vertical dike (Beskardes & Weiss, 2018; Weiss, 2017), which can be considered as reference for its validation for the steadystate flow models.
3.2 Transient Flow With a Vertical Fracture—The FacetBased Conductivity Representation
A vertical fracture model is considered to validate the equivalency of HiFEM's facetbased representation of hydraulically conductive fractures to a volumetric representation. The model consists of a homogeneous porous host rock with a size of 10 × 10 × 5 km and a large, vertical fracture cutting the entire domain. We compare the transient flow solutions of the facetbased model where the fracture is represented as a 2D plane in the form of connected facets with those of the fully volumetric model where the fracture's aperture is explicitly represented. The host rock possess a continuous source at the center of the computational domain and the fracture is located 100 m from the source. The volumetric flow rate of the source is 1.16 × 10^{−3} m^{3}/s (100 m^{3}/day). To allow an affordable mesh refinement and a feasible computational cost for the volumetric model, the fracture aperture is defined as unrealistically large, 1 m, which still results in ∼3 M tetrahedra for the fracture and a total of ∼20.6 M tetrahedra for the entire computational mesh whereas the facetbased model defines the fracture with a modest 19,150 facets in a mesh of ∼2.98 M tetrahedra. The hydraulic conductivity of the host medium is 10^{−7} m/s and that of the fracture is 10^{−4} m/s which corresponds to a conductance (f) of 10^{−4} m^{2}/s. The specific storage of the host medium and the fracture are 2 × 10^{−3} and 2 × 10^{−4} m^{−1}, respectively. The transient flow solutions of both volumetric and facetbased models are obtained with the boundary conditions that are identical with those used in the homogeneous model and zero initial conditions.
Figures 3a and 3b show the hydraulic head distributions obtained from the volumetric and the facetbased fracture models. The facetbased model produces consistent results with the volumetric model successfully capturing the distortive effect of the highly conductive vertical fracture on the overall head pattern. A comparison of both model solutions at 10, 100, and 1,000 days is also shown in Figures 3c and 3d. The radial profiles across and along the fracture indicate that the solutions of both models agree well at different times, suggesting that the HiFEM's facetbased representation of fractures is capable of producing accurate results without resorting to an enormous number of elements. Also, the reduction in the mesh refinement inherently lessens the computational cost. The QMR algorithm used in this study requires the computation of two inner products and four vector norms per iteration which corresponds to 12n − 2 floating point operations (flops) per iteration. Considering only the cost of the QMR computation, the facetbased model (∼0.5 M nodes) necessitate ∼6 M flops per iteration whereas ∼42 M flops are needed for the volumetric model (∼3.5 M nodes), which indicates a drastic relieve (about a factor of 7) for the computational cost.
In addition, we also examine the effects of the spatial discretization of the fracture on the solution of the facetbased model. The meshes are generated by using different aggregation ratios, and the resulting flow solutions at 300 days are compared. Figure 4a indicates that the mesh where the fracture is highly exaggerated inherently results in the decrease in computational precision. Moreover, the head profiles across the fracture show that, regardless of the coarseness of mesh, the precisions of the solutions do not significantly decrease across the fracture (Figure 4b). Yet, the heads along the fracture indicate that the solutions in the use of dense meshes are sufficiently accurate for far distances, whereas the unreasonably coarse meshes produce accurate results only near the source (Figure 4c). This implies that the excessive sparse discretization of model features should be avoided to maintain the computational precision of HiFEM simulations as is the case for any other meshreliant methods.
3.3 Transient Heat Conduction With a Metallic Wellbore—The EdgeBased Conductivity Representation
The homogenous model consists of a line heater with a length of 10 m and a constant power of 5 kW. The size of the model domain is the same as the previous models. The porous medium is defined as granitic rock with the thermal properties shown in Table 1. The temperature rise in the granitic rock is shown for the radial distances from the source location (Figure 5). The HiFEM's solution and the analytic solution coincide very well for different times and radial distances with the RMSEs less than 0.24.
Granite  Steel casing  

Thermal conductivity (k), W/(m°C)  2.5  75 
Specific heat capacity (c), J/(kg°C)  897  460.5 
Density (ρ), kg/m^{3}  2,600  7,850 
Following the successful benchmarking of the volumetric HiFEM's solution for the transient heat conduction, we now proceed with the verification of the equivalency of the numerical solutions obtained from the edgebased and the volumetric model. Figure 6 illustrates the synthetic vertical wellbore problem employed for this test. The wellbore is located at the origin of the computational mesh with the size of 10 × 10 × 5 km and extends 30 m in depth. A heat source with a constant power of 10 kW is placed 10 m apart from the wellbore. The wellbore is defined as a steel solid cylinder with a radius of 0.2 m and is embedded within a homogeneous granitic rock. Table 1 shows the material properties of both the wellbore and the porous medium used in the thermal simulations. The use of the edgebased conductivity representation for this problem drastically reduced the mesh size by a factor of 10 from ∼3.7 to ∼0.33 M tetrahedra as well as the computational cost required by the QMR from ∼44 M flops to ∼4 M flops per iteration. The volumetric model explicitly defines the conductivity and the area of the wellbore whereas the edgebased model represents the wellbore as a set of 120 connected edges on which a conductivityarea product (k ⋅ A) of ∼9.11 Wm/°C is defined to implicitly incorporated its insignificant area into modeling (Figures 6a and 6b). As indicated on the crosssections (Figures 6c and 6d), the temperature distributions at 100 days obtained from both the volumetric and the edgebased models are quite consistent. Moreover, the temperature rises in both models show a good agreement at different times and radial distances (Figure 6e), which validates the accuracy and efficiency of the HiFEM's edgebased conductivity representation. Overall, the results obtained from each hierarchical conductivity representation confirms the internal consistency of the HiFEM's formulation.
Furthermore, these results clearly indicate that the HiFEM may straightforwardly be used for a wide variety of applications in thermal conduction problems especially that involve even more complex metallic infrastructure designs. In addition, due to the flexibility of the HiFEM's hierarchical conductivity representation, canister models for longterm geologic storage can be economically simulated by considering the facetbased thermal conductivities or the complex well completion designs such as concentric well casings can be modeled with a high level of realism by combining both edge and facetbased conductivity representations.
3.4 Application to Transient Flow in Fractured Porous Media
To illustrate the efficiency of the proposed approach on challenging geologic models, transient flow in a porous medium consisting of manmade hydrofractures and a network of preexisting random fractures is simulated. The problem considers a scenario of a pumping well intersected by complex fracture networks. The simulation domain is discretized using a 10 × 10 × 5 km tetrahedral mesh. We define the fractured zone as a ∼150 m^{3} zone embedded in the largescale homogeneous host rock and located at the center of the finite element mesh at a depth of 2,500 m (Figures 7a and 7b). We consider three fracture network models with 10, 50, and 100 fractures shown in Figure 7d (N10, N50, and N100 models). The fractures are regarded as planar ellipses with uniform apertures. Their apertures are defined as 0.01 m which is in the range of 10^{−3}–10^{−2} m often given by the petroleum industry (e.g., Jia et al., 2016). The fracture centers and depths are stochastically generated following a uniform distribution truncated for a range of [−50, −50 m] and [−2,450, 2,550 m], respectively. The sizes of the randomly oriented fractures are populated from a uniform distribution and vary between 10 and 25 m. The fracture networks are generated exhibiting a progressive network growth with increasing number of fractures. While the fractures are dominantly isolated in the N10 model, a connected fracture network is achieved in the larger network models (N50 and N100). Five additional large fractures with an equal size (ellipse axes: 20 and 50 m) are also defined near the source with 10 m spacing to allow more flow into the network fractures as commonly seen in hydraulic fracturing applications. A line source with a length of 40 m is located at the center of the host medium and its constant strength is set to be 5.78 × 10^{−3} m^{3}/s (500 m^{3}/day) that is equally distributed along the nodes of the line source. The specific storage coefficients of the host rock and the fractures are 2 × 10^{−3} and 2 × 10^{−4} m^{−1}, respectively. The hydraulic conductivity of the host rock is 10^{−9} m/s. All fractures have a conductivity of 10^{−4} m/s and are represented by connected facets with a conductance of 10^{−6} m^{2}/s. The mesh details of the three fractured rock models are presented in Table 2.
Model  n_{fracture}  n_{facet}  n_{tet} 

N10  10  7,038  224,475 
N50  50  20,235  510,303 
N100  100  41,393  822,446 
Figures 7c and 7d show the hydraulic head distributions for the fractured models at 100 days. The head fields exhibit reasonable flow patterns pointing out that the HiFEM can successfully handle different levels of network connectivity; in other words, isolated and deadend fractures are inherently incorporated to the flow diffusion solution without any specific treatment. Such fractures are often strictly discarded in the DFN flow models since they do not contribute to the flow at network scale, and most DFN solutions are so computationally expensive that they need to trim the network as much as possible to make the problem solvable. However, the presence of isolated fractures within host media may significantly change the flow pattern especially when the fluid flows between matrix and isolated fractures. Moreover, the head distributions of the most complex model (N100) at different times (Figure 8) show that the flow solutions are robust along the boundaries of sharp conductivity contrasts and their resulting distortions in the head field can be successfully captured. These points out the efficiency of the HiFEM on accounting for both the interactions of mutually intersecting fractures and those between fractures and surrounding rock.
Simulating an approximate twodimensional transient response of a hydraulically stimulated wellbore with discrete natural fractures has been performed by other researchers (e.g., Clarkson & Pedersen, 2010; Jun et al., 2020), but is typically too computationally intensive to simulate large domains, the effects of natural fractures, or to include flow in three dimensions. Here, we illustrate the power of the HiFEM method, which can simultaneously simulate all these effects on a desktop computer. This application geometry is of direct interest for interpreting well testing results in hydrofractured geothermal or tight hydrocarbon extraction problems.
Figure 8 shows the evolution of change in head around the discrete fractures, indicative of the typical ”flow regimes” identifies in stimulated lowpermeability wells (e.g., early linear fracture flow, early radial flow).
In addition, to further evaluate the HiFEM's efficiency in terms of computational cost, a performance analysis is also considered for the fractured models described above. Figures 9a and 9b compare the calculation times of each fractured model's solution performed by different numbers of processors. To provide representative elapsed times for the two ends of the performance levels, 2 and 41 processors are considered for the comparison referring to laptoplevel and supercomputer, respectively. Here, the processors split both the calculations of 41 Laplace's parameters and the nodes in the computational domain for each of the inverse Laplace transform function evaluations. The results are obtained on a multiprocessor Intel Xeon 2 GHz computer. Note that the elapsed times are constant for the transient solutions calculated at times given in each log_{10} cycle because the de Hoog's inverse Laplace transform algorithm uses the same Laplace parameters and the solutions are obtained simultaneously for given times in each log_{10} cycle. Moreover, the transient solutions at late times require more calculation time than those at early times. Figure 9a shows that the calculation times resulting from the 2 processors are in the range of minutes to a couple of hours for the fractured models increasing with the mesh size. Depending on the complexity of the flow model and the desired repetition of simulation, the performance of the HiFEM is satisfactory for a 3D transient DFN simulation on a desktop computer considering kmscale surrounding rock with tiny fractures. In addition, the calculation times of 41 processors indicate that the solutions for all fractured models can be obtained in a couple of minutes. The speedup of the parallel implementation using different number of processors (Figure 9c) suggests that the parallel scheme is more suitable especially in the case of more complex fractured models or multiple simulations as commonly required in sensitivity analyses.
3.5 Velocity Computation
Computing the Darcian velocity field is an important part of modeling the flow since the resulting velocity field is often used to simulate the transport of solutes in a subsurface flow system (both by Lagrangian and Eularian methods). The conventional way to compute the Darcian velocity is to taking numerical derivatives of the computed hydraulic head field; however, this result in discontinuities in the velocity field at the material interfaces, which makes this method not suitable especially for the HiFEM where the fractures are represented as 2D planes with no thickness. For this reason, we take the nonstandard approach of Yeh's Galerkin method (Yeh, 1981) that solves the Darcy's equation by the Galerkin FEM following the calculation of head distribution and computes the nodal velocity field. Yeh's method has been previously applied by several researchers to produce continuous velocity fields. A detailed review can be found in Xie et al. (2017). Here, considering the Yeh's method for the head distributions obtained by the HiFEM, we propose a new approach (HiFEM Yeh) to calculate the velocity fields of 3D fractured porous media with 2D embedded fractures.
Here, we construct [M] and [B] considering the total number of nodes (not the number of the facet nodes) simply to keep the indexing system the same for both volume and facet solutions and solve Equation 24 by applying zero Dirichlet boundary condition for the nonfacet nodes.
The velocities at the interfaces of different geologic materials in the flow system should obey the refraction law (Bear, 1972, 1979). Since Yeh's method does not incorporate the refraction law into the modeling, it cannot produce accurate results near the material interfaces. As a remedy to this problem, one proposed modification is to decompose the model domain into subdomains by material interfaces and sequentially solve each homogenous subdomain by applying the refraction laws as boundary condition as applied in 2D heterogeneous porous media (Xie et al., 2017). Here, in our fractured models, the interfaces themselves are fractures that exhibit high contrast in hydraulic conductivity compared to the host block. For the HiFEM, to improve the accuracy of the velocity fields in the proximity of fractures, we first solve Equation 24 for the fractures using Equations 27 and 28. Then, the velocity values of the neighboring tetrahedral elements at the fracture planes are calculated from the jump functions obtained by the 3D refraction laws (Zhou et al., 2001) and subsequently used as the Dirichlet boundary conditions while solving Yeh's method for the host rock using Equations 25 and 26. We still use a continuous mesh; in other words, the velocity values at the fracture planes are defined not for both sides but only for the fracture side.
Equation 24 is a realvalued system where [M] is symmetric and positive definite. To solve this equation system, we utilize an inhouse, iterative solver that implements the Jacobipreconditioned conjugate gradient method (Weiss, 2017) while the QMR could possibly be adapted to deal with the realvalue system.
To demonstrate the applicability of the HiFEM Yeh approach for fractured porous media, we consider a model with two intersecting fractures in a homogenous host rock (Figure 11a). The study region is 1 × 1 × 0.5 km. The model consists of a horizontal fracture and a dipping fracture rotated 45° around the yaxis. Both fractures are defined as 1 m square planes centered at the origin of the region. The fractures have an aperture of 10^{−2} m and the specific storage of 2 × 10^{−4} m^{−1}. The hydraulic conductivity of the horizontal and the dipping fractures are 10^{−4} and 5 × 10^{−5} m/s, which corresponds to the conductances of 2 × 10^{−6} and 10^{−6} m^{2}/s, respectively. The hydraulic conductivity and the specific storage of the host rock are 10^{−8} m/s and 2 × 10^{−3} m^{−1}. A vertical well with a continuous flow rate is located 2 m away from the center of the fractures. The well is 2 m long and discretized every 0.025 m. The volumetric flow rate is defined as 1.16 × 10^{−3} m^{3}/s (100 m^{3}/day) and equally distributed among the nodes of the well.
In addition to the facetbased model, we also consider the same model as fully volumetric to evaluate how the representation of fractures as lower dimensional elements impact the computed velocity field. While the volumetric model defines the entire model with ∼9 M tetrahedra, the facetbased model defines the fractures with 11,296 facets in a mesh of ∼0.72 M tetrahedra.
Figures 11b and 11c show the hydraulic head distributions of the facetbased and the volumetric models at 1 day. After the head distributions are obtained, the velocity fields can be computed by using the HiFEM Yeh approach and the classic Yeh method for the facetbased and the computationally expensive volumetric models, respectively. Figures 12a and 12b indicate that the velocity fields of the facetbased model are in a good agreement with those of the volumetric model. The profiles along both fractures (Figure 13) show that the velocity fields obtained from both models are consistent while the consistency at fracture tips could probably be further improved by considering a comparison model with much thinner fractures. The results suggest that the HiFEM Yeh approach can compute the velocity fields of facetbased models as robust as those of the volumetric models. This also implies that the lower dimensional elements in fractured models do not have impact on the computed velocity fields if the HiFEM Yeh approach is used. The computation time of the velocity fields for both facet and volumetric models are quite short (seconds and a few minutes with single processor for the given time step, respectively) in comparison to the time needed for the head solutions.
The capability of computing nodal velocity fields for the facedbased fractured rock models increase the applicability of the HiFEM to simulate more complex physical behaviors while still being computationally economical. Immediate applications include both Lagrangian particle tracking and Eulerian advective transport with the fluid flow through the computed Darcian velocity fields. Many different coupling strategies exist for flow, transport and heat for which the HiFEM can possibly be tailored; on the other hand, we will primarily focus on the coupling of the flow and solute transport in our future work.
4 Conclusions
In this article, HiFEM is extended to the physics of fluid flow and heat conduction. We derive the formulae of the HiFEM for the flow diffusion in the Laplace domain and implement the inverse Laplace transform for the timedomain solutions. The concept of the HiFEM relies on the hierarchical conductivities defined at the each dimensional component of the tetrahedral finite mesh while following the standard steps of the classical finite element analysis which enables to represent fractures as 2D planes and wellbores as 1D lines in the 3D surrounding medium as well as the fluid flow to inherently be accounted for at the boundaries between fractures, wellbores and intact porous rock without any mathematical interference for coupling or transfer. Our validation tests show that the HiFEM produces accurate results compared to analytical solutions. Moreover, the facedbased flow model of fractures achieve nearly the same accuracy as the fully volumetric model suggesting that the fracture networks embedded in the porous rock can be simulated by using the HiFEM's facetbased representation with a major reduction in the mesh size and calculation time. In addition, by replacing the hydraulic properties with the thermal properties in the diffusion equation, the heat conduction solution obtained from the edgebased model of a vertical wellbore is also compared with its corresponding fully volumetric solution which provides satisfactory results. Further, the modeling results of the largescale complex fractured porous rock models indicate the capability of the HiFEM to efficiently capture the flow along and across the complex fractures as well as the interactions with wellbores. Our performance analysis for these complex models also points out the HiFEM's efficiency on the computing times. We also propose a new approach (HiFEM Yeh) to compute the nodal Darcian velocity field of the fractured porous media with 2D fractures by employing the Yeh's Galerkin model for both 3D and 2D features, and demonstrate its applicability with a comparison against the volumetric model. As a result, our study suggests that the HiFEM is a strong candidate for rigorous and robust highfidelity modeling of diffusive flow in fractured porous media with affordable computational cost.
Because the current meshing strategies cannot afford to generate highly dense smallscale fracture systems, we envision the application of the HiFEM for the medium to large discrete fractures embedded in heterogeneous porous media, especially for their simulations at large scales. Even though we have not demonstrated here, the statistically homogenized smallscale fractures can also be incorporated into modeling by defining their continuous conductivities over the volume elements. The current formulation of the HiFEM allows facet and edgerepresentations for only conductive model features but the modification for the resistive structures may be feasible as mentioned in Weiss (2017). Besides, since the method relies on the classical FEM, it is quite flexible to employ higher order polynomials to further improve the precision.
In the context of modeling at large scales, the evaluations and predictions regarding fractured porous media often involve not only the analysis of fluid flow but also those of other physical processes such as heat and electric flow. The initial application of the HiFEM in the geophysical electromagnetics as well as its capability to model the fluid flow and the heat conduction as presented in this study indicate the robustness of the method on precisely accounting of not only the interactions between fracture networks and wellbores for the flow diffusion but also the geometric and material properties of metallic infrastructures for the heat and electric flow. Another possible application may be in neutronics for simulating individual fuel rods in an assembly by using hierarchical material properties, where the diffusion equation is solved as well. This flexibility makes the HiFEM almost independent from the physics of interest, which suggests a great potential to simulate multiple physical processes by using a single mesh that describes the geologic environments consisted of both geometrically complex fractures and manmade infrastructures.
This article presents the primary work that introduces the HiFEM to the fluid flow. As our next research problem, we will further extend its application to other flow processes such as diffusionadvection flow and heat transfer, and focus on its applications to real case studies.
Acknowledgments
The authors thank the Editor (Xavier SanchezVila), the Associate Editor and three anonymous reviewers for their constructive comments. Sandia's Laboratory Directed Research and Development program funded this work. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc, for the U.S. Department of Energy's National Nuclear Security Administration under contract DENA0003525. This article describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
Open Research
Data Availability Statement
Data produced in this article for comparison are available via https://doi.org/10.7910/DVN/ORFDJI. Finite element meshes used for this analysis were generated by Cubit, available at http://cubit.sandia.gov, commercially available through CSimSoft.