Couplex Benchmark Computations Obtained with the Software Toolbox UG

This paper describes the numerical results for the COUPLEX benchmark obtained with the simulation software UG using vertex centered finite volume and higher order discontinuous Galerkin schemes. Multigrid solvers on unstructured grids, local mesh refinement and parallel computation are employed to yield very accurate solutions. Since the full range of results required in the benchmarks is too large to be displayed in this paper we focus on the comparison of discretization schemes, assessment of numerical errors and the presentation of parallel computations.     

On the failure of a discrete axial chain using a continualized nonlocal Continuum Damage Mechanics approach

The failure of a discrete elastic‐damage axial system is investigated using both a discrete and an equivalent continuum approach. The Discrete Damage Mechanics approach is based on a microstructured model composed of a series of periodic elastic‐damage springs (axial Discrete Damage Mechanics lattice system). Such a discrete damage system can be associated with the finite difference formulation of a Continuum Damage Mechanics evolution problem. Several analytical and numerical results are presented for the tensile failure of this axial damage chain under its own weight. The nonlocal Continuum Damage Mechanics models examined in this paper are mainly built from a continualization procedure applied to centered or uncentered finite difference schemes. The asymptotic expansion of the first‐order upward difference equations leads to a first‐order nonlocal model, whereas the asymptotic expansion of the centered finite difference equations leads to a second‐order nonlocal Eringen's approach. To complete this study, a phenomenological nonlocal gradient approach is also examined and compared with the first continualization methods. A comparison of the discrete and the continuous problems for the chains shows the effectiveness of the new micromechanics‐based nonlocal Continuum Damage modeling, especially for capturing scale effects. For both continualized approaches, the length scale of the nonlocal models depends only on the cell size, while for the so‐called phenomenological approach, the length scale may depend on the loading parameter. This apparent load‐dependent length scale, already discussed in the literature with numerical arguments, is found to be sensitive to the postulated structure of the nonlocal model calibrated according to a lattice approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Guidelines for using the finite element method to predict one-dimensional consolidation behaviour

Several finite element schemes, based both on the diffusion and coupled approaches, have been implemented in computer programs and a comparative study carried out to investigate the numerical performance of each scheme. Factors such as stability, convergency, accuracy, computational time and the effects of wide variations in soil parameters (eg laminated soils) have been examined. The study indicates that the numerical performance of each scheme is controlled by a non-dimensional parameter and guidelines have been suggested which allow accurate and economic solutions to be obtained.     

Finite volume approach for finite strain consolidation

The paper presents the finite volume formulation and numerical solution of finite strain one‐dimensional consolidation equation. The equation used in this study utilises a nonlinear continuum representation of consolidation with varying compressibility and hydraulic conductivity and thus inherits the material and geometric nonlinearity. Time‐marching explicit scheme has been used to achieve transient solutions. The nonlinear terms have been evaluated with the known previous time step value of the independent variable, that is, void ratio. Three‐point quadratic interpolation function of Lagrangian family has been used to evaluate the face values at discrete control volumes. It has been shown that the numerical solution is stable and convergent for the general practical cases of consolidation. Performance of the numerical scheme has been evaluated by comparing the results with an analytical solution and with the piecewise piecewise‐linear finite difference numerical model. The approach seems to work well and offers excellent potential for simulating finite strain consolidation. Further, the parametric study has been performed on soft organic clays, and the influence of various parameters on the time ate consolidation characteristics of the soil is shown. Copyright © 2015 John Wiley & Sons, Ltd.     

A time‐discontinuous Galerkin method for the dynamical analysis of porous media

We present a time‐discontinuous Galerkin method (DGT) for the dynamic analysis of fully saturated porous media. The numerical method consists of a finite element discretization in space and time. The discrete basis functions are continuous in space and discontinuous in time. The continuity across the time interval is weakly enforced by a flux function. Two applications and several numerical investigations confirm the quality of the proposed space–time finite element scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

An efficient finite–discrete element method for quasi‐static nonlinear soil–structure interaction problems

An efficient finite–discrete element method applicable for the analysis of quasi‐static nonlinear soil–structure interaction problems involving large deformations in three‐dimensional space was presented in this paper. The present method differs from previous approaches in that the use of very fine mesh and small time steps was not needed to stabilize the calculation. The domain involving the large displacement was modeled using discrete elements, whereas the rest of the domain was modeled using finite elements. Forces acting on the discrete and finite elements were related by introducing interface elements at the boundary of the two domains. To improve the stability of the developed method, we used explicit time integration with different damping schemes applied to each domain to relax the system and to reach stability condition. With appropriate damping schemes, a relatively coarse finite element mesh can be used, resulting in significant savings in the computation time. The proposed algorithm was validated using three different benchmark problems, and the numerical results were compared with existing analytical and numerical solutions. The algorithm performance in solving practical soil–structure interaction problems was also investigated by simulating a large‐scale soft ground tunneling problem involving soil loss near an existing lining. Copyright © 2011 John Wiley & Sons, Ltd.     

Sequential implicit vertex approximate gradient discretization of incompressible two-phase Darcy flows with discontinuous capillary pressure

This work deals with sequential implicit schemes for incompressible and immiscible two-phase Darcy flows which are commonly used and well understood in the case of spatially homogeneous capillary pressure functions. To our knowledge, the stability of this type of splitting schemes solving sequentially a pressure equation followed by the saturation equation has not been investigated so far in the case of discontinuous capillary pressure curves at different rock type interfaces. It will be shown here to raise severe stability issues for which stabilization strategies are investigated in this work. To fix ideas, the spatial discretization is based on the Vertex Approximate Gradient (VAG) scheme accounting for unstructured polyhedral meshes combined with an Hybrid Upwinding (HU) of the transport term and an upwind positive approximation of the capillary and gravity fluxes. The sequential implicit schemes are built from the total velocity formulation of the two-phase flow model and only differ in the way the conservative VAG total velocity fluxes are approximated. The stability, accuracy and computational cost of the sequential implicit schemes studied in this work are tested on oil migration test cases in 1D, 2D and 3D basins with a large range of capillary pressure parameters for the drain and barrier rock types. It will be shown that usual splitting strategies fail to capture the right solutions for highly contrasted rock types and that it can be fixed by maintaining locally the pressure saturation coupling at different rock type interfaces in the definition of the conservative total velocity fluxes. The numerical investigation of the sequential schemes is also extended to the widely used finite volume Two-Point Flux Approximation spatial discretization.

     

TP or not TP,that is the question

We give here a comparative study on the mathematical analysis of two (classes of) discretization schemes for the computation of approximate solutions to incompressible two-phase flow problems in homogeneous porous media. The first scheme is the well-known finite volume scheme with a two-point flux approximation, classically used in industry. The second class contains the so-called approximate gradient schemes, which include finite elements with mass lumping, mixed finite elements, and mimetic finite differences. Both (classes of) schemes are nonconforming and can be expressed using discrete function and gradient reconstructions within a variational formulation. Each class has its specific advantages and drawbacks: monotony properties are natural with the two-point finite volume scheme, but meshes are restricted due to consistency issues; on the contrary, gradient schemes can be used on general meshes, but monotony properties are difficult to obtain.     

Folder: A numerical tool to simulate the development of structures in layered media

We present Folder, a numerical toolbox for modelling deformation in layered media subject to layer parallel shortening or extension in two dimensions. The toolbox includes a range of features that ensure maximum flexibility to configure model geometry, define material parameters, specify numerical parameters, and choose the plotting options. Folder builds on an efficient finite element method model and implements state of the art iterative and time integration schemes. We describe the basic Folder features and present several case studies of single and multilayer stacks subject to layer parallel shortening and extension. Folder additionally comprises an application that illustrates various analytical solutions of growth rates calculated for the cases of layer parallel shortening and extension of a single layer with interfaces perturbed with a single sinusoidal waveform. We further derive two novel analytical expressions for the growth rate in the cases of layer parallel shortening and extension of a linear viscous layer embedded in a linear viscous medium of a finite thickness. These solutions help understand mechanical instabilities in layered rocks and provide a unique opportunity for benchmarking of numerical codes. We demonstrate how Folder can be used for benchmarking of numerical codes. We test the accuracy of single-layer folding simulations using various 1) spatial and temporal resolutions, 2) iterative algorithms for non-linear materials, and 3) time integration schemes. The accuracy of the numerical results is quantified by: 1) comparing them to analytical solutions, if available, or 2) running convergence tests. As a result, we provide a map of the most optimal choice of grid size, time step, and number of iterations to keep the results of the numerical simulations below a given error for a given time integration scheme. Folder is an open source MATLAB application and comes with a user-friendly graphical interface. Folder is suitable for both educational and research purposes.     

Limitations of Lattice Boltzmann Modeling of Micro‐Flows in Complex Nanopores

The multiscale transport mechanism of methane in unconventional reservoirs is dominated by slip and transition flows resulting from the ultra-low permeability of micro/nano-scale pores, which requires consideration of the microscale and rarefaction effects. Traditional continuum-based computational fluid dynamics (CFD) becomes problematic when modeling micro-gaseous flow in these multiscale pore networks because of its disadvantages in the treatment of cases with a complicated boundary. As an alternative, the lattice Boltzmann method (LBM), a special discrete form of the Boltzmann equation, has been widely applied to model the multi-scale and multi-mechanism flows in unconventional reservoirs, considering its mesoscopic nature and advantages in simulating gas flows in complex porous media. Consequently, numerous LBM models and slip boundary schemes have been proposed and reported in the literature. This study investigates the predominately reported LBM models and kinetic boundary schemes. The results of these LBM models systematically compare to existing experimental results, analytical solutions of Navier-Stokes, solutions of the Boltzmann equation, direct simulation of Monte Carlo (DSMC) and information-preservation DSMC (IP_DSMC) results, as well as the numerical results of the linearized Boltzmann equation by the discrete velocity method (DVM). The results point out the challenges and limitations of existing multiple-relaxation-times LBM models in predicting micro-gaseous flow in unconventional reservoirs.     

Numerical modeling of the flow and transport of radionuclides in heterogeneous porous media

This paper is concerned with numerical methods for the modeling of flow and transport of contaminant in porous media. The numerical methods feature the mixed finite element method over triangles as a solver to the Darcy flow equation and a conservative finite volume scheme for the concentration equation. The convective term is approximated with a Godunov scheme over the dual finite volume mesh, whereas the diffusion–dispersion term is discretized by piecewise linear conforming triangular finite elements. It is shown that the scheme satisfies a discrete maximum principle. Numerical examples demonstrate the effectiveness of the methodology for a coupled system that includes an elliptic equation and a diffusion–convection–reaction equation arising when modeling flow and transport in heterogeneous porous media. The proposed scheme is robust, conservative, efficient, and stable, as confirmed by numerical simulations.      

Using a shifted Grünwald-Letnikov scheme for the Caputo derivative to study anomalous solute transport in porous medium

We propose an extension of the shifted Grünwald-Letnikov method to solve fractional partial differential equations in the Caputo sense with arbitrary fractional order derivative α and with an advective term. The method uses the relation between Caputo and Riemann-Liouville definitions, the shifted Grünwald-Letnikov, and the traditional backward and forward finite difference method. The stability of the method is investigated for the implicit and explicit scheme with homogeneous boundary conditions, and a stability criterion is found for the advective-dispersive equation. An application of the method is used to solve contaminant diffusion and advective-dispersive problems. The numerical solution for the fractional diffusion and fractional advection-dispersion is compared with their respective analytical solutions for different time and space grid refinements. The diffusion simulation exhibited a good fit between the analytical and numerical solutions, with the explicit scheme going from stable to unstable as the time and space refinement changes. The fractional advection-dispersion application produced small deviations from the analytical solution. These deviations, however, are analogous to the numerical dispersions encountered in conventional finite difference solutions of the advection-dispersion equation. The new method is also compared with the traditional L2 method. Notably, an example that involves asymmetrical fractional conditions, a fractional diffusivity that depends on time, and a source term show how the methods compare. Overall, this study assesses the quality and easiness of use of the numerical method.     

Finite volume coupling strategies for the solution of a Biot consolidation model

In this paper a finite volume (FV) numerical method is implemented to solve a Biot consolidation model with discontinuous coefficients. Our studies show that the FV scheme leads to a locally mass conservative approach which removes pressure oscillations especially along the interface between materials with different properties and yields higher accuracy for the flow and mechanics parameters. Then this numerical discretization is utilized to investigate different sequential strategies with various degrees of coupling including: iteratively, explicitly and loosely coupled methods. A comprehensive study is performed on the stability, accuracy and rate of convergence of all of these sequential methods. In the iterative and explicit solutions four splits of drained, undrained, fixed-stress and fixed-strain are studied. In loosely coupled methods three techniques of the local error method, the pore pressure method, and constant step size are considered and results are compared with other types of coupling methods. It is shown that the fixed-stress method is the best operator split in comparison with other sequential methods because of its unconditional stability, accuracy and the rate of convergence. Among loosely coupled schemes, the pore pressure and local error methods which are, respectively, based on variation of pressure and displacement, show consistency with the physics of the problem. In these methods with low number of total mechanical iterations, errors within acceptance range can be achieved. As in the pore pressure method mechanics time step increases more uniformly, this method would be less costly in comparison with the local error method. These results are likely to be useful in decision making regarding choice of solution schemes. Moreover, the stability of the FV method in multilayered media is verified using a numerical example.     

模拟多孔介质中生物化学输运的有限颗粒法的一个修正算法

给出了有限颗粒法(FCM) 的一个修正算法, 用来模拟二维多孔介质中复杂的物理、生物化学输运现象.该算法不仅具有与早先的FCM一样的优点, 而且可以在更微观的水平上保证质量守恒, 获得更为准确的颗粒位置, 从而有利于质量交换的高精度计算.计算结果与精确解和早先的FCM的结果做了比较.      

An adaptive time integration scheme for blast loading on a saturated soil mass

This paper presents a time integration scheme capable of simulating blast loading of relatively high frequency on porous media, using coarse meshes. The scheme is based on the partition of unity finite element method. The discontinuity is imposed on the velocity field, while the displacement field is kept continuous. The velocity discontinuity is postulated to occur in the time domain. The developed time integration scheme is unconditionally stable and has controllable numerical dissipation in the high frequency range. An important feature of the time scheme is that it allows for controlling the numerical damping in a consistent way. The time scheme has been implemented in combination with Biot’s theory of wave propagation in saturated porous media. Numerical examples have demonstrated that the proposed time scheme is, in addition to being accurate and stable, highly effective for coarse meshes. This makes the developed scheme suitable for large scale finite element analysis.     

Numerical modeling of stress-permeability coupling in rough fractures

A numerical model is described for coupled flow and mechanical deformation in fractured rock. The mechanical response of rock joints to changes in hydraulic pressure is strongly influenced by the geometric characteristics of the joint surfaces. The concept of this work is to combine straightforward finite element solutions with complex and realistic fracture surface geometry in order to reproduce the non-linear stress-deformation-permeability coupling that is commonly observed in fractures. Building on the numerous studies that have expanded the understanding of the key parameters needed to describe natural rough-walled fractures, new methods have been developed to generate a finite element mesh representing discrete fractures with realistic rough surface geometries embedded in a rock matrix. The finite element code GeoSys/Rockflow was then used to simulate the coupled effects of hydraulic stress, mechanical stress, and surface geometry on the evolving permeability of a single discrete fracture. The modeling concept was experimentally verified against examples from the literature. Modeling results were also compared to a simple interpenetration model.     

三维梁-颗粒模型在岩石材料破坏模拟中的应用

用三维梁-颗粒模型BPM3D(beam-particlemodelinthreedimensions)对岩石类非均质脆性材料的力学性质和破坏过程进行了数值模拟。梁-颗粒模型是在离散单元法基础上,结合有限单元法中的网格模型提出的用于模拟岩石类材料损伤破坏过程的数值模型。在模型中,材料在细观层次上被离散为颗粒单元集合体,相邻颗粒单元由有限单元法中的弹脆性梁单元联结。梁单元的力学性质均按韦伯(Weibull)分布随机赋值,以模拟岩石类材料力学参数的空间变异性。材料内部裂纹通过断开梁单元来模拟。通过自动生成的非均质材料模型对岩石类材料的破坏机理进行研究。岩石类非均质脆性材料在单轴压缩状态下破坏过程细观数值模拟结果显示,岩石材料宏观破坏是由于其内部细观裂纹产生、扩展、贯通的结果。通过数值模拟结果之间的对比分析,揭示出岩石试样宏观破坏模式随细观层次上韦伯分布参数的变化而不同。与实际矿柱破坏形态的对比分析表明了模型的适用性。根据数值模拟结果对岩石类非均质材料的破坏机理进行了探讨。     

SEAWAT 2000: modelling unstable flow and sensitivity to discretization levels and numerical schemes

A systematic analysis shows how results from the finite difference code SEAWAT are sensitive to choice of grid dimension, time step, and numerical scheme for unstable flow problems. Guidelines to assist in selecting appropriate combinations of these factors are suggested. While the SEAWAT code has been tested for a wide range of problems, the sensitivity of results to spatial and temporal discretization levels and numerical schemes has not been studied in detail for unstable flow problems. Here, the Elder-Voss-Souza benchmark problem has been used to systematically explore the sensitivity of SEAWAT output to spatio-temporal resolution and numerical solver choice. A grid size of 0.38 and 0.60% of the total domain length and depth respectively is found to be fine enough to deliver results with acceptable accuracy for most of the numerical schemes when Courant number (Cr) is 0.1. All numerical solvers produced similar results for extremely fine meshes; however, some schemes converged faster than others. For instance, the 3rd-order total variation-diminishing method (TVD3) scheme converged at a much coarser mesh than the standard finite difference methods (SFDM) upstream weighting (UW) scheme. The sensitivity of the results to Cr number depends on the numerical scheme as expected.     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments. The author is grateful to the DFG (German Science Foundation—Deutsche Forschungsgemeinschaft) for the financial support under the grant number Di 430/4-2.