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1.
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. 相似文献
2.
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. 相似文献
3.
Due to complex dynamics inherent in the physical models, numerical formulation of subsurface and overland flow coupling can be challenging to solve. ParFlow is a subsurface flow code that utilizes a structured grid discretization in order to benefit from fast and efficient structured solvers. Implicit coupling between subsurface and overland flow modes in ParFlow is obtained by prescribing an overland boundary condition at the top surface of the computational domain. This form of implicit coupling leads to the activation and deactivation of the overland boundary condition during simulations where ponding or drying events occur. This results in a discontinuity in the discrete system that can be challenging to resolve. Furthermore, the coupling relies on unstructured connectivities between the subsurface and surface components of the discrete system, which makes it challenging to use structured solvers to effectively capture the dynamics of the coupled flow. We present a formulation of the discretized algebraic system that enables the use of an analytic form of the Jacobian for the Newton–Krylov solver, while preserving the structured properties of the discretization. An effective multigrid preconditioner is extracted from the analytic Jacobian and used to precondition the Jacobian linear system solver. We compare the performance of the new solver against one that uses a finite difference approximation to the Jacobian within the Newton–Krylov approach, previously used in the literature. Numerical results explores the effectiveness of using the analytic Jacobian for the Newton–Krylov solver, and highlights the performance of the new preconditioner and its cost. The results indicate that the new solver is robust and generally outperforms the solver that is based on the finite difference approximation to the Jacobian, for problems where the overland boundary condition is activated and deactivated during the simulation. A parallel weak scaling study highlights the efficiency of the new solver. 相似文献
4.
《水文科学杂志》2013,58(4):868-882
Abstract Non-Darcian flow in a finite fractured confined aquifer is studied. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly, and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer's equation, which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady-state fracture—block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow in both fractures and blocks. Numerical and analytical solutions give the same results when the Reynolds number is less than 0.1. The effect of nonlinearity on the flow appears when the Reynolds number is greater than 0.1. The higher the rate of flow from the stream to the aquifer, the higher the degree of nonlinearity. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution provide a useful application of nonlinear flow models to fractured aquifers. It is possible to extend the model to different types of aquifer, as well as boundary conditions at the stream side. Time-dependent flow rates in the analysis of recession hydrographs could also be evaluated by this model. 相似文献
5.
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. 相似文献
6.
常规的三维时间域航空电磁模拟通常采用隐式步长方法进行时间离散,需要几次矩阵分解和上百次右端源项回带,计算效率较低.为了提高正演计算效率,本文提出使用有理Krylov方法求解时间域电场扩散方程.首先使用非结构四面体网格进行空间离散,采用Nédélec矢量基函数近似四面体单元内的电场;然后基于有限元离散给出矩阵指数和矢量乘积表示的电场显式解;最后采用有理Arnoldi算法构造Krylov子空间内的正交基函数并进一步求解矩阵指数与矢量的乘积,直接得到任意时刻的电场解向量,避免步长离散过程.此外,本文还提出一种指数加权偏移参数优化方法,使得有理Arnoldi近似在瞬变衰减晚期具备更高的精度,从而降低Krylov子空间阶数并提高计算效率.通过和层状模型解析解的对比验证了有理Krylov方法的精度.针对三维异常体模型使用全局网格和局部网格剖分并和其他数值方法比较,进一步说明了有理Krylov方法的有效性. 相似文献
7.
In this paper, a computational model for the simulation of coupled hydromechanical and electrokinetic flow in fractured porous media is introduced. Particular emphasis is placed on modeling CO2 flow in a deformed, fractured geological formation and the associated electrokinetic flow. The governing field equations are derived based on the averaging theory and the double porosity model. They are solved numerically with a mixed discretization scheme, formulated on the basis of the standard Galerkin finite element method, the extended finite element method, the level-set method and the Petrov–Galerkin method. The standard Galerkin method is utilized to discretize the equilibrium and the diffusive dominant field equations, and the extended finite element method, together with the level-set method and the Petrov–Galerkin method, are utilized to discretize the advective dominant field equations. The level-set method is employed to trace the CO2 plume front, and the extended finite element method is employed to model the high gradient in the saturation field front. The proposed mixed discretization scheme leads to a convergent system, giving a stable and effectively mesh-independent model. The accuracy and computational efficiency of the proposed model is evaluated by verification and numerical examples. Effects of the fracture spacing on the CO2 flow and the streaming potential are discussed. 相似文献
8.
在数值模拟中,隐式有限差分具有较高的精度和稳定性.然而,传统隐式有限差分算法大多由于需要求解大型矩阵方程而存在计算效率偏低的局限性.本文针对一阶速度-应力弹性波方程,构建了一种优化隐式交错网格有限差分格式,然后将改进格式由时间-空间域转换为时间-波数域,利用二范数原理建立目标函数,再利用模拟退火法求取优化系数.通过对均匀模型以及复杂介质模型进行一阶速度-应力弹性波方程数值模拟所得单炮记录、波场快照分析表明:这种优化隐式交错网格差分算法与传统的几种显式和隐式交错网格有限差分算法相比不但降低了计算量,而且能有效的压制网格频散,使弹性波数值模拟的精度得到有效的提高. 相似文献
9.
In our previous study, we developed the Stokes–Darcy (SD) model was developed for flow in a karst aquifer with a conduit bedded in matrix, and the Beavers–Joseph (BJ) condition was used to describe the matrix–conduit interface. We also studied the mathematical well‐posedness of a coupled continuum pipe flow (CCPF) model as well as convergence rates of its finite element approximation. In this study, to compare the SD model with the CCPF model, we used numerical analyses to validate finite element discretisation methods for the two models. Using computational experiments, simulation codes implementing the finite element discretisations are then verified. Further model validation studies are based on the results of laboratory experiments. Comparing the results of computer simulations and experiments, we concluded that the SD model with the Beavers–Joseph interface condition is a valid model for conduit–matrix systems. On the other hand, the CCPF model with the value of the exchange parameter chosen within the range suggested in the literature perhaps does not result in good agreement with experimental observations. We then examined the sensitivity of the CCPF model with respect to the exchange parameter, concluding that, as has previously been noted, the model is highly sensitive for small values of the exchange parameter. However, for larger values, the model becomes less sensitive and, more important, also produces results that are in better agreement with experimental observations. This suggests that the CCPF model may also produce accurate simulation results, if one chooses larger values of the exchange parameter than those suggested in the literature. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
10.
In discrete fracture network (DFN) modeling, fractures are randomly generated and placed in the model domain. The rock matrix is considered impermeable. Small fractures and isolated fractures are often ignored to reduce computational expense. As a result, the rock matrix between fractures could be large and intersections may not be found between a well introduced in the model and the hydraulically connected fracture networks (fracture backbones). To overcome this issue, this study developed a method to conceptualize a well in a three-dimensional (3D) DFN using two orthogonal rectangular fractures oriented along the well's axis. Six parameters were introduced to parameterize the well screen and skin zone, and to control the connectivity between the well and the fracture backbones. The two orthogonal fractures were discretized using a high-resolution mesh to improve the quality of flow and transport simulations around and along the well. The method was successfully implemented within dfnWorks 2.0 (Hyman et al. 2015) to incorporate a well in a 3D DFN and to track particles leaving an injection well and migrating to a pumping well. Verification of the method against MODFLOW/MODPATH found a perfect match in simulated hydraulic head and particle tracking. Using three examples, the study showed that the method ensured the connectivity between wells and fracture backbones, and honored the physical processes of flow and transport along and around wells in DFNs. Recommendations are given for estimating the values of the six introduced well parameters in a real-world case study. 相似文献
11.
A numerical method has been proposed by Ross [Ross PJ. Modeling soil water and solute transport-fast, simplified numerical solutions. Agron J 2003; 95(6): 1352–1361.] to solve one-dimensional soil water movement problems. The Ross method is a noniterative numerical scheme, that can reduce computational time without sacrificing computational accuracy. The main aim of this study is to present a general form of the Ross method for two- and three-dimensional variably saturated flow. The established numerical model (R3D) is widely tested using five problems, in which the numerical solutions of R3D are compared with analytical solutions, laboratory data, and solutions from a traditional iterative numerical model. The comparison shows that R3D accommodates various hydraulic functions and boundary conditions. Results from R3D, which does not require iteration, are as accurate as results from iterative model. With the help of the primary variable switching technique, this model is unconditionally mass conservative, and computes infiltration into dry soil more efficiently. R3D is thus considered as an efficient tool for its high accuracy and efficiency for solving two- and three-dimensional variably saturated flow problems. 相似文献
12.
Two primary concerns in performing watershed overland flow routing are the numerical instability and computational efficiency. The stability of executing an explicit scheme has to be maintained by observing the Courant–Friedrich–Lewy criterion, which is adopted to confirm that the numerical marching speed is larger than the wave celerity. Moreover, there is another criterion of time step devised in previous studies to avoid back‐and‐forth refluxing between adjacent grids. The situation of refluxing usually occurs on flat regions. In light of this, the selection of a small time increment to honor both restrictions simultaneously is believed to decrease the computational efficiency in performing overland flow routing. This study aims at creating a robust algorithm to relax both restrictions. The proposed algorithm was first implemented on a one‐dimensional overland plane to evaluate the accuracy of the numerical result by comparing it with an analytical solution. Then, the algorithm was further applied to a watershed for 2D runoff simulations. The results show that the proposed integrated algorithm can provide an accurate runoff simulation and achieve satisfactory performance in terms of computational speed. 相似文献
13.
We present a fully implicit numerical method to solve the incompressible MHD equations in a strongly rotating Cartesian domain. The equations are solved in a primitive variable formulation using a finite volume discretization. In order to use massively parallel computers, we applied a domain decomposition approach in space. The performance of this model is compared with an earlier model, which treated the convective terms of the equations in an explicit manner. Our results indicate that although the fully implicit method needs about three times the memory of the implicit–explicit method, it is superior in terms of computational efficiency. As an application of this model, we investigated the influence of the Prandtl number in the range of 0.01–1000 on the dynamics of the dynamo. 相似文献
14.
In this study, a numerical manifold method (NMM) model is developed to analyze flow in porous media with discrete fractures in a non-conforming mesh. This new model is based on a two-cover-mesh system with a uniform triangular mathematical mesh and boundary/fracture-divided physical covers, where local independent cover functions are defined. The overlapping parts of the physical covers are elements where the global approximation is defined by the weighted average of the physical cover functions. The mesh is generated by a tree-cutting algorithm. A new model that does not introduce additional degrees of freedom (DOF) for fractures was developed for fluid flow in fractures. The fracture surfaces that belong to different physical covers are used to represent fracture flow in the direction of the fractures. In the direction normal to the fractures, the fracture surfaces are regarded as Dirichlet boundaries to exchange fluxes with the rock matrix. Furthermore, fractures that intersect with Dirichlet or Neumann boundaries are considered. Simulation examples are designed to verify the efficiency of the tree-cutting algorithm, the calculation's independency from the mesh orientation, and accuracy when modeling porous media that contain fractures with multiple intersections and different orientations. The simulation results show good agreement with available analytical solutions. Finally, the model is applied to cases that involve nine intersecting fractures and a complex network of 100 fractures, both of which achieve reasonable results. The new model is very practical for modeling flow in fractured porous media, even for a geometrically complex fracture network with large hydraulic conductivity contrasts between fractures and the matrix. 相似文献
15.
在实际的水力压裂过程中,裂缝总是沿着垂直于最小地应力的方向扩展,地应力的分布形式和多个压裂段之间的互相影响(应力阴影效应)对于形成复杂的裂缝网络具有重要的影响。本文基于扩展有限单元法(XFEM)模拟页岩等多孔介质在水压作用下裂缝的任意扩展,由于在传统有限元法的基础上引入了扩充自由度和可以描述间断的位移阶跃函数,所以裂缝可以独立于网格扩展,而不需要重新剖分网格。通过引入一维流动假设,求解润滑方程,并考虑流体在裂缝内的流动。同时也考虑裂缝向基质中流动的滤失效应。研究实际施工中不同段间距下裂缝的扩展模式和段间距对裂缝形态的影响,结果表明,压裂段间距过小时中间的裂缝会被屏蔽;此外,裂缝会由于应力阴影效应而发生转向。 相似文献
16.
A unified approach to modeling flows of slightly compressible fluids through naturally fractured media is presented. The unified fractional differential model is derived by combining the flow at micro scale for matrix blocks and macro scale for fractures, using the transient interporosity flow behavior at the interface between matrix blocks and fractures. The derived model is able to unify existing transient interporosity flow models formulated for different shapes of matrix blocks in any medium dimensions. The model is formulated in the form of a fractional order partial differential equation that involves Caputo derivative of order 1/2 with respect to time. Explicit solutions for the unified model are derived for different axisymmetrical spatial domains using Hankel or Hankel–Weber finite or infinite transforms. Comparisons between the predictions of the unified model and those obtained from existing transient interporosity flow models for matrix blocks in the form of slabs, spheres and cylinders are presented. It is shown that the unified fractional derivative model leads to solutions that are very close to those of transient interporosity flow models for fracture-dominant and transitional fracture-to-matrix dominant flow regimes. An analysis of the results of the unified model reveals that the pressure varies linearly with the logarithm of time for different flow regimes, with half slope for the transitional fracture-to-matrix dominant flow regime vs. the fracture and matrix dominant flow regimes. In addition, a new re-scaling that involves the characteristic length in the form of matrix block volume to surface area ratio is derived for the transient interporosity flow models for matrix blocks of different shapes. It is shown that the re-scaled transient interporosity flow models are governed by two dimensionless parameters Θ and Λ compared to only one dimensionless parameter Θ for the unified model. It is shown that the solutions of the transient interporosity flow models for different shapes of matrix blocks are almost identical for the re-scaled variables. Furthermore, the driving parameters for solution behavior are identified based on asymptotic approximations for different flow regimes. It is found that the matrix diffusion and the matrix area-to-volume ratio affect the solution behavior only for the transitional fracture-to-matrix dominant flow regime, that the capacitance ratio affects the solution behavior only for transitional and matrix dominant flow regimes and that the fracture diffusion is involved in all three flow regimes. Similar identification of the driving parameters is also presented in the re-scaled case. 相似文献
17.
结合有限差分方法和等效介质理论,模拟了离散分布裂缝介质中地震波的传播. 基于等效介质理论,利用二维有限差分实现封闭裂缝的离散分布;裂缝可以处理成固体岩石中的高度柔性界面,并可以用线性滑动或者位移间断模型进行裂缝的物理描述. 对于含有多组裂隙的破裂固体,其有效柔度可以认为是固体骨架背景柔度和裂缝附加柔度之和. 在一阶近似条件下,固体骨架和裂缝参数可以通过有效各向异性系数联系起来,有效各向异性系数决定了各向异性（裂缝效应）对于地震波传播的影响. 通过与射线理论方法的对比检验,说明本文提出的模拟方法的有效性,并通过几个数值算例说明本方法可有效模拟不同的裂缝分布效应. 结果表明,即使在裂缝密度很小的情况下,具有相同裂缝密度的不同的空间分布可以产生不同的波场特征. 同时,也验证了不同裂缝尺度对波长的不同影响,以及裂缝尺度具有幂率分布（分形）时,尺度对波场的影响. 最后得出结论：在运用建立在等效介质理论基础上的地震各向异性概念来描述裂缝固体的特征时,要倍加小心,等效介质理论中尚未合理处理的裂缝尺度和空间分布对波的传播特征具有重要的影响. 相似文献
18.
Contrast in capillary pressure of heterogeneous permeable media can have a significant effect on the flow path in two-phase immiscible flow. Very little work has appeared on the subject of capillary heterogeneity despite the fact that in certain cases it may be as important as permeability heterogeneity. The discontinuity in saturation as a result of capillary continuity, and in some cases capillary discontinuity may arise from contrast in capillary pressure functions in heterogeneous permeable media leading to complications in numerical modeling. There are also other challenges for accurate numerical modeling due to distorted unstructured grids because of the grid orientation and numerical dispersion effects. Limited attempts have been made in the literature to assess the accuracy of fluid flow modeling in heterogeneous permeable media with capillarity heterogeneity. The basic mixed finite element (MFE) framework is a superior method for accurate flux calculation in heterogeneous media in comparison to the conventional finite difference and finite volume approaches. However, a deficiency in the MFE from the direct use of fractional flow formulation has been recognized lately in application to flow in permeable media with capillary heterogeneity. In this work, we propose a new consistent formulation in 3D in which the total velocity is expressed in terms of the wetting-phase potential gradient and the capillary potential gradient. In our formulation, the coefficient of the wetting potential gradient is in terms of the total mobility which is smoother than the wetting mobility. We combine the MFE and discontinuous Galerkin (DG) methods to solve the pressure equation and the saturation equation, respectively. Our numerical model is verified with 1D analytical solutions in homogeneous and heterogeneous media. We also present 2D examples to demonstrate the significance of capillary heterogeneity in flow, and a 3D example to demonstrate the negligible effect of distorted meshes on the numerical solution in our proposed algorithm. 相似文献
19.
《Advances in water resources》2004,27(9):875-887
A physically based numerical approach is presented for modeling fracture–matrix interaction, which is a key issue for fractured reservoir simulation. Commonly used mathematical models for dealing with such interactions employ a dual- or multiple-continuum concept, in which fractures and matrix are represented as overlapping, different, but interconnected continua, described by parallel sets of conservation equations. The conventional single-point upstream weighting scheme, in which the fracture relative permeability is used to represent the counterpart at the fracture–matrix interface, is the most common scheme by which to estimate flow mobility for fracture–matrix flow terms. However, such a scheme has a serious flaw, which may lead to unphysical solutions or significant numerical errors. To overcome the limitation of the conventional upstream weighting scheme, this paper presents a physically based modeling approach for estimating physically correct relative permeability in calculating multiphase flow between fractures and the matrix, using continuity of capillary pressure at the fracture–matrix interface. The proposed approach has been implemented into two multiphase reservoir simulators and verified using analytical solutions and laboratory experimental data. The new method is demonstrated to be accurate, numerically efficient, and easy to implement in dual- or multiple-continuum models. 相似文献
20.
Due to their high aspect ratio fractures are often conceptualized as lower-dimensional structures embedded into the surrounding host matrix. This simplification is typically made within the context of numerical simulation, for the inverse estimation of the matrix-diffusion coefficient from break-through curves or for the derivation of analytical solutions describing flow and transport in a fracture–matrix system. It is generally justified by the so called Lauwerier assumption stating that the transversal dispersion inside the fracture is infinitely fast therefore hampering the formation of gradients across the width of the fracture. In this study we want to verify the applicability of such lower-dimensional modeling. To that end we investigate the occurrence of fracture-scale gradients in a simplified fracture–matrix model by virtue of analytical as well as numerical investigations. The relevant processes modeled are advection, dispersion, matrix diffusion and linear decay. In addition, we also investigate the impact on the inverse estimation of matrix-diffusion coefficients through analytical solutions, which assume a lower-dimensional fracture. Results show that a lower-dimensional modeling of fractures will only lead to errors for early periods of the time-dependent solution. Such errors may however, extent to the steady state if fast radioactive decay is considered. The estimation of the matrix-diffusion coefficient too is affected by the assumption of a lower-dimensional fracture. We see errors as big as 20% for the estimation procedure, the value of which depends on the ratio of the matrix-diffusion vs. the transversal dispersion coefficient. Our analysis suggest that a lower-dimensional representation of fractures is justified for many typical conditions and that special attention must only be paid in a confined number of cases. 相似文献