In this paper we present a mathematical model and a numerical workflow for the simulation of a thermal single-phase flow with reactive transport in porous media, in the presence of fractures. The latter are thin regions which might behave as high or low permeability channels depending on their physical parameters, and are thus of paramount importance in underground flow problems. Chemical reactions may alter the local properties of the porous media as well as the fracture walls, changing the flow path and possibly occluding some portions of the fractures or zones in the porous media. To solve numerically the coupled problem we propose a temporal splitting scheme so that the equations describing each physical process are solved sequentially. Numerical tests shows the accuracy of the proposed model and the ability to capture complex phenomena, where one or multiple fractures are present.
Flow in fractured porous media is of high relevance in a variety of geotechnical applications, given the fact that they ubiquitously occur in nature and that they can have a substantial impact on the hydraulic properties of rock. As a response to this, an active field of research has developed over the past decades, focussing on the development of mathematical models and numerical methods to describe and simulate the flow processes in fractured rock. In this work, we present a comparison of different finite volume-based numerical schemes for the simulation of flow in fractured porous media by …
In this work we present a model reduction procedure to derive a hybrid-dimensional framework for the mathematical modeling of reactive transport in fractured porous media. Fractures are essential pathways in the underground which allow fast circulation of the fluids present in the rock matrix, often characterized by low permeability. However, due to infilling processes fractures may change their hydraulic properties and become barriers for the flow creating impervious blocks. The geometrical as well as the physical properties of the fractures require a special treatment to allow the subsequent…
In this work we present the mathematical models for single-phase flow in fractured porous media. An overview of the most common approaches is considered, which includes continuous fracture models and discrete fracture models. For the latter, we discuss strategies that are developed in literature for its numerical solution mainly related to the geometrical relation between the fractures and porous media grids.
In this work, we present an uncertainty quantification analysis to determine the influence and importance of some physical parameters in a reactive transport model in fractured porous media. An accurate description of flow and transport in the fractures is key to obtain reliable simulations, however, fractures geometry and physical characteristics pose several challenges from both the modeling and implementation side. We adopt a mixed-dimensional approximation, where fractures and their intersections are represented as objects of lower dimension. To simplify the presentation, we consider only …
Call for participation: Verification benchmarks for single-phase flow in three-dimensional fractured porous media
University of Bergen（卑尔根大学）；Christian Michelsen Research（克里斯蒂安·米歇尔森研究所）；University of Stuttgart（斯图加特大学）；Politecnico di Milano（米兰理工大学）；University of Göttingen（哥廷根大学）
Flow in fractured porous media occurs in the earth's subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, we present a portfolio of four benchmark cases for single-phase flow in three-dimensional fractured porous media. The cases are specifically designed to test the methods' capabilities in handling various complexities common to the geometrical structures of fracture n…
Modeling flows in fractured porous media is important in applications. One main challenge in numerical simulation is that the flow is strongly influenced by the fractures, so that the solutions typically contain complex features, which require high computational grid resolutions. Instead of using uniformly fine mesh, a more computationally efficient adaptively refined mesh is desirable. In this paper we design and analyze a novel residual-type a posteriori error estimator for staggered DG methods on general polygonal meshes for Darcy flows in fractured porous media. The method can handle fairl…
In this paper, we present and analyze a staggered discontinuous Galerkin method for Darcy flows in fractured porous media on fairly general meshes. A staggered discontinuous Galerkin method and a standard conforming finite element method with appropriate inclusion of interface conditions are exploited for the bulk region and the fracture, respectively. Our current analysis weakens the usual assumption on the polygonal mesh, which can integrate more general meshes such as elements with arbitrarily small edges into our theoretical framework. We prove the optimal convergence estimates in $L^2$ er…
An accurate modeling of reactive flows in fractured porous media is a key ingredient to obtain reliable numerical simulations of several industrial and environmental applications. For some values of the physical parameters we can observe the formation of a narrow region or layer around the fractures where chemical reactions are focused. Here the transported solute may precipitate and form a salt, or vice-versa. This phenomenon has been observed and reported in real outcrops. By changing its physical properties this layer might substantially alter the global flow response of the system and thus…
Gradient discretization of Hybrid Dimensional Darcy Flows in Fractured Porous Media with discontinuous pressures at the matrix fracture interfaces
Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis（尼斯索菲亚安提波利斯大学）；INRIA Sophia Antipolis Méditerranée（法国国家信息与自动化研究所索菲亚安提波利斯地中海中心）；CSTJF, TOTAL S.A.（道达尔公司）
We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model is discretized using the gradient discretization method [26], which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the c…
We present a hybrid-mixed finite element method for a novel hybrid-dimensional model of single-phase Darcy flow in a fractured porous media. In this model, the fracture is treated as an $(d-1)$-dimensional interface within the $d$-dimensional fractured porous domain, for $d=2, 3$. Two classes of fracture are distinguished based on the permeability magnitude ratio between the fracture and its surrounding medium: when the permeability in the fracture is (significantly) larger than in its surrounding medium, it is considered as a {\it conductive} fracture; when the permeability in the fracture is…
We present a novel hybridizable discontinuous Galerkin (HDG) method on unfitted meshes for single-phase Darcy flow in a fractured porous media. In particular we apply the HDG methodology to the recently introduced reinterpreted discrete fracture model (RDFM) [Xu & Yang, 2021 submitted] that use Dirac-$δ$ functions to model both conductive and blocking fractures. Due to the use of Dirac-$δ$ function approach for the fractures, our numerical scheme naturally allows for unfitted meshes with respect to the fractures, which is the major novelty of the proposed scheme. Moreover, the scheme is locall…
In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum that coupled by the specific transfer term. To illustrate the idea of our approach, we consider a dual continua background model with discrete fractures networks that generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media. We present fine grid approximation based on the finite element method and Discrete Fr…
Development of models and dedicated numerical methods for dynamics in fractured rocks is an active research field, with research moving towards increasingly advanced process couplings and complex fracture networks. The inclusion of coupled processes in simulation models is challenged by the high aspect ratio of the fractures, the complex geometry of fracture networks and the crucial impact of processes that completely change characteristics on the fracture-rock interface. This paper provides a general discussion of design principles for introducing fractures in simulators, and defines a framew…
Numerical simulations of flow and transport in porous media usually rely on hybrid-dimensional models, i.e., the fracture is considered as objects of a lower dimension compared to the embedding matrix. Such models are usually combined with non-conforming discretizations as they avoid the inherent difficulties associated with the generation of meshes that explicitly resolve fractures-matrix interfaces. However, non-conforming discretizations demand a more complicated coupling of different sub-models and may require special care to ensure conservative fluxes. We propose a novel approach for the …
Nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media
Institute for Scientific Computation, Texas A&M University（德州农工大学科学计算研究所）；Department of Computational Technologies, North-Eastern Federal University（东北联邦大学计算技术系）；Department of Mathematics, The Chinese University of Hong Kong (CUHK)（香港中文大学数学系）；Department of Mathematic, Texas A&M University（德州农工大学数学系）；Multiscale model reduction laboratory, North-Eastern Federal University（东北联邦大学多尺度模型降维实验室）
Many subsurface engineering applications involve tight-coupling between fluid flow, solid deformation, fracturing, and similar processes. To better understand the complex interplay of different governing equations, and therefore design efficient and safe operations, numerical simulations are widely used. Given the relatively long time-scales of interest, fully-implicit time-stepping schemes are often necessary to avoid time-step stability restrictions. A major computational bottleneck for these methods, however, is the linear solver. These systems are extremely large and ill-conditioned. Becau…
In this paper, we study a model for the transport of an external component, e.g., a surfactant, in variably saturated porous media. We discretize the model in time and space by combining a backward Euler method with the linear Galerkin finite elements. The Newton method and the L-Scheme are employed for the linearization and the performance of these schemes is studied numerically. A special focus is set on the effects of dynamic capillarity on the transport equation.
We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functi…
Flow in fractured porous media is modeled frequently by discrete fracture-matrix approaches where fractures are treated as dimensionally reduced manifolds. Generalizing earlier work we focus on two-phase flow in time-dependent fracture geometries including the fracture's aperture. We present the derivation of a reduced model for immiscible two-phase flow in porous media. For the reduced model we present a fully conforming finite-volume discretization coupled with a moving-mesh method. This method permits arbitrary movement of facets of the triangulation while being fully conservative. In numer…
Flow and multicomponent reactive transport in saturated/unsaturated porous media are modeled by ensembles of computational particles moving on regular lattices according to specific random walk rules. The occupation number of the lattice sites is updated with a global random walk (GRW) procedure which spreads the particles from a lattice site with computational costs comparable to those for a single random walk step in sequential procedures. To cope with the nonlinearity and the degeneracy of the Richards equation the GRW flow solver uses linearization techniques similar to the $L$-scheme deve…
We present a family of multi-stage preconditioners for coupled thermal-compositional-reactive reservoir simulation problems. The most common preconditioner used in industrial practice, the Constrained Pressure Residual (CPR) method, was designed for isothermal models and does not offer a specific strategy for the energy equation. For thermal simulations, inadequate treatment of the temperature unknown can cause severe convergence degradation. When strong thermal diffusion is present, the energy equation exhibits significant elliptic behavior that cannot be accurately corrected by CPR's second …
In this work, we introduce a time memory formalism in poroelasticity model that couples the pressure and displacement. We assume this multiphysics process occurs in multicontinuum media. The mathematical model contains a coupled system of equations for pressures in each continuum and elasticity equations for displacements of the medium. We assume that the temporal dynamics is governed by fractional derivatives following some works in the literature. We derive an implicit finite difference approximation for time discretization based on the Caputo time fractional derivative. A Discrete Fracture …
Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution ha…
Computational identification of unknown adsorption and desorption rates is discussed in conjunction with reactive flow considered at pore scale. The reactive transport is governed by incompressible Stokes equations, coupled with convection-diffusion equation for species transport. The surface reactions, namely adsorption and desorption, are accounted via Robin boundary conditions. Henry and Langmuir isotherms are considered. Measured concentration of the specie at the outlet of the domain has to be provided to carry out the identification procedure. Deterministic and stochastic parameter ident…
When modelling fluid flow in fractured reservoirs, it is common to represent the fracturesas lower-dimensional inclusions embedded in the host medium. Existing discretizationsof flow in porous media with thin inclusions assume that the principal directions of theinclusion permeability tensor are aligned with the inclusion orientation. While this mod-elling assumption works well with tensile fractures, it may fail in the context of faults,where the damage zone surrounding the main slip surface may introduce anisotropy thatis not aligned with the main fault orientation. In this paper, we introdu…
Reactive transport in saturated/unsaturated porous media is numerically upscaled to the space-time scale of a hypothetical measurement through coarse-grained space-time (CGST) averages. The reactive transport is modeled at the fine-grained Darcy scale by the actual number of molecules involved in reactions which undergo advective and diffusive movements described by global random walk (GRW) simulations. The CGST averages verify identities similar to a local balance equation which allow us to derive expressions for the flow velocity and the intrinsic diffusion coefficient in terms of averaged m…
A solution technique is proposed for flows in porous media that guarantees local conservation of mass. We first compute a flux field to balance the mass source and then exploit exact co-chain complexes to generate a solenoidal correction. A reduced basis method based on proper orthogonal decomposition is employed to construct the correction and we show that mass balance is ensured regardless of the quality of the reduced basis approximation. The method is directly applicable to mixed finite and virtual element methods, among other structure-preserving discretization techniques, and we present …
