A mixed-dimensional finite volume method for two-phase flow in fractured porous media

We present a vertex-centered finite volume method for the fully coupled, fully implicit discretization of two-phase flow in fractured porous media. Fractures are discretely modeled as lower dimensional elements. The method works on unstructured, locally refined grids and on parallel computers with distributed memory. An implicit time discretization is employed and the nonlinear systems of equations are solved with a parallel Newton-multigrid method. Results from two-dimensional and three-dimensional simulations are presented.     

A numerical method for two-phase flow in fractured porous media with non-matching grids

We propose a novel computational method for the efficient simulation of two-phase flow in fractured porous media. Instead of refining the grid to capture the flow along the faults or fractures, we represent the latter as immersed interfaces, using a reduced model for the flow and suitable coupling conditions. We allow for non matching grids between the porous matrix and the fractures to increase the flexibility of the method in realistic cases. We employ the extended finite element method for the Darcy problem and a finite volume method that is able to handle cut cells and matrix-fracture interactions for the saturation equation. Moreover, we address through numerical experiments the problem of the choice of a suitable numerical flux in the case of a discontinuous flux function at the interface between the fracture and the porous matrix. A wrong approximate solution of the Riemann problem can yield unphysical solutions even in simple cases.     

A meshless method to simulate solute transport in heterogeneous porous media

We derive a meshless numerical method based on smoothed particle hydrodynamics (SPH) for the simulation of conservative solute transport in heterogeneous geological formations. We demonstrate that the new proposed scheme is stable, accurate, and conserves global mass. We evaluate the performance of the proposed method versus other popular numerical methods for the simulation of one- and two-dimensional dispersion and two-dimensional advective–dispersive solute transport in heterogeneous porous media under different Pèclet numbers. The results of those benchmarks demonstrate that the proposed scheme has important advantages over other standard methods because of its natural ability to control numerical dispersion and other numerical artifacts. More importantly, while the numerical dispersion affecting traditional numerical methods creates artificial mixing and dilution, the new scheme provides numerical solutions that are “physically correct”, greatly reducing these artifacts.     

A new upscaling method for fractured porous media

We present a method to determine equivalent permeability of fractured porous media. Inspired by the previous flow-based upscaling methods, we use a multi-boundary integration approach to compute flow rates within fractures. We apply a recently developed multi-point flux approximation Finite Volume method for discrete fracture model simulation. The method is verified by upscaling an arbitrarily oriented fracture which is crossing a Cartesian grid. We demonstrate the method by applying it to a long fracture, a fracture network and the fracture network with different matrix permeabilities. The equivalent permeability tensors of a long fracture crossing Cartesian grids are symmetric, and have identical values. The application to the fracture network case with increasing matrix permeabilities shows that the matrix permeability influences more the diagonal terms of the equivalent permeability tensor than the off-diagonal terms, but the off-diagonal terms remain important to correctly assess the flow field.     

Discontinuous Galerkin methods for advective transport in single-continuum models of fractured media

Accurate simulation of flow and transport processes in fractured rocks requires that flow in fractures and shear zones to be coupled with flow in the porous rock matrix. To this end, we will herein consider a single-continuum approach in which both fractures and the porous rock are represented as volumetric objects, i.e., as cells in an unstructured triangular grid with a permeability and a porosity value associated with each cell. Hence, from a numerical point of view, there is no distinction between flow in the fractures and the rock matrix. This enables modelling of realistic cases with very complex structures. To compute single-phase advective transport in such a model, we propose to use a family of higher-order discontinuous Galerkin methods. Single-phase transport equations are hyperbolic and have an inherent causality in the sense that information propagates along streamlines. This causality is preserved in our discontinuous Galerkin discretization. We can therefore use a simple topological sort of the graph of discrete fluxes to reorder the degrees-of-freedom such that the discretized linear system gets a lower block-triangular form, from which the solution can be computed very efficiently using a single-pass forward block substitution. The accuracy and utility of the resulting transport solver is illustrated through several numerical experiments.     

An efficient numerical model for multicomponent compressible flow in fractured porous media

An efficient and accurate numerical model for multicomponent compressible single-phase flow in fractured media is presented. The discrete-fracture approach is used to model the fractures where the fracture entities are described explicitly in the computational domain. We use the concept of cross flow equilibrium in the fractures. This will allow large matrix elements in the neighborhood of the fractures and considerable speed up of the algorithm. We use an implicit finite volume (FV) scheme to solve the species mass balance equation in the fractures. This step avoids the use of Courant–Freidricks–Levy (CFL) condition and contributes to significant speed up of the code. The hybrid mixed finite element method (MFE) is used to solve for the velocity in both the matrix and the fractures coupled with the discontinuous Galerkin (DG) method to solve the species transport equations in the matrix. Four numerical examples are presented to demonstrate the robustness and efficiency of the proposed model. We show that the combination of the fracture cross-flow equilibrium and the implicit composition calculation in the fractures increase the computational speed 20–130 times in 2D. In 3D, one may expect even a higher computational efficiency.     

Variable-density flow and transport in porous media: approaches and challenges

We review the state of the art in modeling of variable-density flow and transport in porous media, including conceptual models for convection systems, governing balance equations, phenomenological laws, constitutive relations for fluid density and viscosity, and numerical methods for solving the resulting nonlinear multifield problems. The discussion of numerical methods addresses strategies for solving the coupled spatio-temporal convection process, consistent velocity approximation, and error-based mesh adaptation techniques. As numerical models for those nonlinear systems must be carefully verified in appropriate tests, we discuss weaknesses and inconsistencies of current model-verification methods as well as benchmark solutions. We give examples of field-related applications to illustrate specific challenges of further research, where heterogeneities and large scales are important.     

Finite volume model for reactive transport in fractured porous media with distance- and time-dependent dispersion

Abstract

There are very few studies of fractured porous media that use distance- and time-dependent dispersion models, and, to the best of our knowledge, none which compare these with constant dispersion models. Therefore, in this study, the behaviour of temporal and spatial concentration profiles with distance- and time-dependent dispersion models is investigated. A hybrid finite volume method is used to solve the governing equations for these dispersion models. The developed numerical model is used to study the effects of matrix diffusion coefficient, groundwater velocity and matrix and fracture retardation factor on concentration profiles in the application of constant, distance-dependent and time-dependent dispersion models. In addition, an attempt is made to evaluate the applicability of these dispersion models by using the models to simulate experimental data. It was found that a better fit to the observed data is obtained in the case of distance- and time-dependent dispersion models as compared to the constant dispersion model. Thus, these numerical experiments indicate that distance- and time-dependent dispersion models have better simulation potential than the constant dispersion model.     

Variable-density groundwater flow and solute transport in porous media containing nonuniform discrete fractures

Variations in fluid density can greatly affect fluid flow and solute transport in the subsurface. Heterogeneities such as fractures play a major role for the migration of variable-density fluids. Earlier modeling studies of density effects in fractured media were restricted to orthogonal fracture networks, consisting of only vertical and horizontal fractures. The present study addresses the phenomenon of 3D variable-density flow and transport in fractured porous media, where fractures of an arbitrary incline can occur. A general formulation of the body force vector is derived, which accounts for variable-density flow and transport in fractures of any orientation. Simulation results are presented that show the verification of the new model formulation, for the porous matrix and for inclined fractures. Simulations of variable-density flow and solute transport are then conducted for a single fracture, embedded in a porous matrix. The simulations show that density-driven flow in the fracture causes convective flow within the porous matrix and that the high-permeability fracture acts as a barrier for convection. Other simulations were run to investigate the influence of fracture incline on plume migration. Finally, tabular data of the tracer breakthrough curve in the inclined fracture is given to facilitate the verification of other codes.     

A multidimensional streamline-based method to simulate reactive solute transport in heterogeneous porous media

We present a new streamline-based numerical method for simulating reactive solute transport in porous media. The key innovation of the method is that both longitudinal and transverse dispersion are incorporated accurately without numerical dispersion. Dispersion is approximated in a flow-oriented grid using a combination of a one-dimensional finite difference scheme and a meshless approximation. In contrast to previous hybrid alternatives to incorporate dispersion in streamline-based simulations, the proposed scheme does not require a grid and, hence, it does not introduce numerical dispersion. In addition, the proposed scheme eliminates numerical oscillations and negative concentration values even when the dispersion tensor includes the off-diagonal coefficients and the flow field is non-uniform. We demonstrate that for a set of two- and three-dimensional benchmark problems, the new proposed streamline-based formulation compares favorably to two state of the art finite volume and hybrid Eulerian–Lagrangian solvers.     

An efficient numerical model for incompressible two-phase flow in fractured media

Various numerical methods have been used in the literature to simulate single and multiphase flow in fractured media. A promising approach is the use of the discrete-fracture model where the fracture entities in the permeable media are described explicitly in the computational grid. In this work, we present a critical review of the main conventional methods for multiphase flow in fractured media including the finite difference (FD), finite volume (FV), and finite element (FE) methods, that are coupled with the discrete-fracture model. All the conventional methods have inherent limitations in accuracy and applications. The FD method, for example, is restricted to horizontal and vertical fractures. The accuracy of the vertex-centered FV method depends on the size of the matrix gridcells next to the fractures; for an acceptable accuracy the matrix gridcells next to the fractures should be small. The FE method cannot describe properly the saturation discontinuity at the matrix–fracture interface. In this work, we introduce a new approach that is free from the limitations of the conventional methods. Our proposed approach is applicable in 2D and 3D unstructured griddings with low mesh orientation effect; it captures the saturation discontinuity from the contrast in capillary pressure between the rock matrix and fractures. The matrix–fracture and fracture–fracture fluxes are calculated based on powerful features of the mixed finite element (MFE) method which provides, in addition to the gridcell pressures, the pressures at the gridcell interfaces and can readily model the pressure discontinuities at impermeable faults in a simple way. To reduce the numerical dispersion, we use the discontinuous Galerkin (DG) method to approximate the saturation equation. We take advantage of a hybrid time scheme to alleviate the restrictions on the size of the time step in the fracture network. Several numerical examples in 2D and 3D demonstrate the robustness of the proposed model. Results show the significance of capillary pressure and orders of magnitude increase in computational speed compared to previous works.     

An inverse problem methodology to identify flow channels in fractured media using synthetic steady-state head and geometrical data

We present a methodology for identifying highly-localized flow channels embedded in a significantly less permeable medium using steady-state head and geometrical data. This situation is typical of fractured media where flows are often strongly channeled at the scales of interest (10 m–1 km). The objective is to identify both geometrical and hydraulic characteristics of the conducting structures. Channels are identified in decreasing order of importance by successive optimizations of an objective function. The identification strategy takes advantage of the hierarchical flow organization to restrict the dimension of the solution space of each individual optimization step. The characteristics of the secondary channels are strongly determined by the main flow channels. The latter are slightly modified by the secondary channels through the addition of a regularization term to the main channel characteristics in the objective function. As the objective function is strongly non-convex with numerous local minima, inversion is performed using a stochastic algorithm (simulated annealing). We assess the possibilities of the hierarchical identification strategy on simple synthetic steady-state flow configurations where hydraulic data are made up of 25 regularly spaced heads and of the boundary conditions. Those flow structures that are dominated by at most two simple channels can be identified with these head data only. Configurations comprising up to three complex and interconnected channels can still be identified with additional geometrical information including the distances of piezometers to their closest channel. The capabilities of the hierarchical identification strategy are limited to flow structures dominated by at most three equivalent flow channels. We finally discuss the perspectives of application of the method to transient-state data obtained on a more restricted number of piezometers.     

A new scheme for numerical modelling of flow and transport processes in 3D fractured porous media

The objective of this work is to develop a new numerical approach for the three-dimensional modelling of flow and transient solute transport in fractured porous media which would provide an accurate and efficient treatment of 3D complex geometries and inhomogeneities. For this reason, and in order to eliminate as much as possible the number of degrees of freedom, the fracture network, fractures and their intersections, are solved with a coupled 2D–1D model while the porous matrix is solved independently with a 3D model. The interaction between both models is accounted for by a coupling iterative technique. In this way it is possible to improve efficiency and reduce CPU usage by avoiding 3D mesh refinements of the fractures. The approach is based on the discrete-fracture model in which the exact geometry and location of each fracture in the network must be provided as an input. The formulation is based on a multidimensional coupling of the boundary element method-multidomain (BEM-MD) scheme for the flow and boundary element dual reciprocity method-multidomain (BE-DRM-MD) scheme for the transport. Accurate results and high efficiency have been obtained and are reported in this paper.     

A new multiscale scheme for computing statistical moments in single phase flow in heterogeneous porous media

In this work we develop a new multiscale procedure to compute numerically the statistical moments of the stochastic variables which govern single phase flow in heterogeneous porous media. The technique explores the properties of the log-normally distributed hydraulic conductivity, characterized by power-law or exponential covariances, which shows invariance in its statistical structure upon a simultaneous change of the scale of observation and strength of heterogeneity. We construct a family of equivalent stochastic hydrodynamic variables satisfying the same flow equations at different scales and strengths of heterogeneity or correlation lengths. Within the new procedure the governing equations are solved in a scaled geology and the numerical results are mapped onto the original medium at coarser scales by a straightforward rescaling. The new procedure is implemented numerically within the Monte Carlo algorithm and also in conjunction with the discretization of the low-order effective equations derived from perturbation analysis. Numerical results obtained by the finite element method show the accuracy of the new procedure to approximated the two first moments of the pressure and velocity along with its potential in reducing drastically the computational cost involved in the numerical modeling of both power-law and exponential covariance functions.     

流体饱和多孔隙介质弹性波方程边界元解法研究

基于流体饱和多孔隙各向同性介质模型,本文首先推导了流体饱和多孔隙介质中弹性波传播的频率域系统动力方程及边界积分方程,然后给出了流体饱和多孔隙介质弹性波方程的基本解,最后,利用本文给出的边界元方法对流体饱和多孔隙各向同性介质中的弹性波传播进行了数值模拟.结果表明：不论是从固相位移,还是液相位移的地震合成记录都能看到明显的慢速P波,本文提出的流体饱和多孔隙介质弹性波边界元法是有效可行的.     

Numerical simulations of non-ergodic solute transport in three-dimensional heterogeneous porous media

Numerical simulations of non-ergodic transport of a non-reactive solute plume by steady-state groundwater flow under a uniform mean velocity, , were conducted in a three-dimensional heterogeneous and statistically isotropic aquifer. The hydraulic conductivity, K(x), is modeled as a random field which is assumed to be log-normally distributed with an exponential covariance. Significant efforts are made to reduce the simulation uncertainties. Ensemble averages of the second spatial moments of the plume and the plume centroid variances were simulated with 1600 Monte Carlo (MC) runs for three variances of log K, _Y²=0.09, 0.23, and 0.46, and a square source normal to of three dimensionless lengths. It is showed that 1600 MC runs are needed to obtain stabilized results in mildly heterogeneous aquifers of _Y²0.5 and that large uncertainty may exist in the simulated results if less MC runs are used, especially for the transverse second spatial moments and the plume centroid variance in transverse directions. The simulated longitudinal second spatial moment and the plume centroid variance in longitudinal direction fit well to the first-order theoretical results while the simulated transverse moments are generally larger than the first-order values. The ergodic condition for the second spatial moments is far from reaching in all cases simulated and transport in transverse directions may reach ergodic condition much slower than that in longitudinal direction.     

Statistical characteristics of flow as indicators of channeling in heterogeneous porous and fractured media

We introduce two new channeling indicators D_ic and D_cc based on the Lagrangian distribution of flow rates. On the basis of the participation ratio, these indicators characterize the extremes of both the flow-tube width distribution and the flow rate variation along flow lines. The participation ratio is an indicator biased toward the larger values of a distribution and is equal to the normalized ratio of the square of the first-order moment to the second-order moment. Compared with other existing indicators, they advantageously provide additional information on the flow channel geometry, are consistently applicable to both porous and fractured media, and are generally less variable for media generated using the same parameters than other indicators. Based on their computation for a broad range of porous and fracture permeability fields, we show that they consistently characterize two different geometric properties of channels. D_ic gives a characteristic scale of low-flow zones in porous media and a characteristic distance between effectively flowing structures in fractured cases. D_cc gives a characteristic scale of the extension of high-flow zones in porous media and a characteristic channel length in fractured media. D_ic is mostly determined by channel density and permeability variability. D_cc is, however, more affected by the nature of the correlation structure like the presence of permeability channels or fractures in porous media and the length distribution in fracture networks.     

Exact transverse macro dispersion coefficients for transport in heterogeneous porous media

We study transport through heterogeneous media. We derive the exact large scale transport equation. The macro dispersion coefficients are determined by additional partial differential equations. In the case of infinite Peclet numbers, we present explicit results for the transverse macro dispersion coefficients. In two spatial dimensions, we demonstrate that the transverse macro dispersion coefficient is zero. The result is not limited on lowest order perturbation theory approximations but is an exact result. However, the situation in three spatial dimensions is very different: The transverse macro dispersion coefficients are finite – a result which is confirmed by numerical simulations we performed.