Non-Fickian mass transport in fractured porous media

The paper provides an introduction to fundamental concepts of mathematical modeling of mass transport in fractured porous heterogeneous rocks. Keeping aside many important factors that can affect mass transport in subsurface, our main concern is the multi-scale character of the rock formation, which is constituted by porous domains dissected by the network of fractures. Taking into account the well-documented fact that porous rocks can be considered as a fractal medium and assuming that sizes of pores vary significantly (i.e. have different characteristic scales), the fractional-order differential equations that model the anomalous diffusive mass transport in such type of domains are derived and justified analytically. Analytical solutions of some particular problems of anomalous diffusion in the fractal media of various geometries are obtained. Extending this approach to more complex situation when diffusion is accompanied by advection, solute transport in a fractured porous medium is modeled by the advection-dispersion equation with fractional time derivative. In the case of confined fractured porous aquifer, accounting for anomalous non-Fickian diffusion in the surrounding rock mass, the adopted approach leads to introduction of an additional fractional time derivative in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties can be readily modeled and analyzed.     

Modeling non-equilibrium mass transport in biologically reactive porous media

We develop a one-equation non-equilibrium model to describe the Darcy-scale transport of a solute undergoing biodegradation in porous media. Most of the mathematical models that describe the macroscale transport in such systems have been developed intuitively on the basis of simple conceptual schemes. There are two problems with such a heuristic analysis. First, it is unclear how much information these models are able to capture; that is, it is not clear what the model's domain of validity is. Second, there is no obvious connection between the macroscale effective parameters and the microscopic processes and parameters. As an alternative, a number of upscaling techniques have been developed to derive the appropriate macroscale equations that are used to describe mass transport and reactions in multiphase media. These approaches have been adapted to the problem of biodegradation in porous media with biofilms, but most of the work has focused on systems that are restricted to small concentration gradients at the microscale. This assumption, referred to as the local mass equilibrium approximation, generally has constraints that are overly restrictive. In this article, we devise a model that does not require the assumption of local mass equilibrium to be valid. In this approach, one instead requires only that, at sufficiently long times, anomalous behaviors of the third and higher spatial moments can be neglected; this, in turn, implies that the macroscopic model is well represented by a convection–dispersion–reaction type equation. This strategy is very much in the spirit of the developments for Taylor dispersion presented by Aris (1956). On the basis of our numerical results, we carefully describe the domain of validity of the model and show that the time-asymptotic constraint may be adhered to even for systems that are not at local mass equilibrium.     

Multiscale flow and transport model in three-dimensional fractal porous media

This paper proposes a multiscale flow and transport model which can be used in three-dimensional fractal random fields. The fractal random field effectively describes a field with a high degree of variability to satisfy the one-point statistics of Levy-stable distribution and the two-point statistics of fractional Levy motion (fLm). To overcome the difficulty of using infinite variance of Levy-stable distribution and to provide the physical meaning of a finite domain in real space, truncated power variograms are utilized for the fLm fields. The fLm model is general in the sense that both stationary and commonly used fractional Brownian motion (fBm) models are its special cases. When the upper cutoff of the truncated power variogram is close to the lower cutoff, the stationary model is well approximated. The commonly used fBm model is recovered when the Levy index of fLm is 2. Flow and solute transport were analyzed using the first-order perturbation method. Mean velocity, velocity covariance, and effective hydraulic conductivity in a three-dimensional fractal random field were derived. Analytical results for particle displacement covariance and macrodispersion coefficients are also presented. The results show that the plume in an fLm field moves slower at early time and has more significant long-tailing behavior at late time than in fBm or stationary exponential fields. The proposed fractal transport model has broader applications than those of stationary and fBm models. Flow and solute transport can be simulated for various scenarios by adjusting the Levy index and cutoffs of fLm to yield more accurate modeling results.     

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.     

基于正交分解的非线性结构外荷载识别方法研究

提出一种基于切比雪夫正交分解的非线性结构外荷载识别方法及分解阶数确定办法。在识别过程中建立非线性结构体系状态空间方程,并将切比雪夫正交多项式展开系数扩展于状态量,对状态量进行递推估计。通过结构反应频域分析筛选频率范围并确定正交多项式项数。文中将通过6层隔震结构、波形钢腹板PC组合梁桥的数值仿真和3层隔震框架的振动台试验验证所提基于正交分解的荷载识别方法可行性。研究结果表明,基于切比雪夫正交基分解的外荷载识别方法及正交基项数确定方法,适用于非线性结构的荷载识别。从识别效果上看,即使在噪声及模型误差因素的影响下外荷载仍然能够得到较好的识别。     

Transport in chemically and mechanically heterogeneous porous media IV: large-scale mass equilibrium for solute transport with adsorption

In this article we consider the transport of an adsorbing solute in a two-region model of a chemically and mechanically heterogeneous porous medium when the condition of large-scale mechanical equilibrium is valid. Under these circumstances, a one-equation model can be used to predict the large-scale averaged velocity, but a two-equation model may be required to predict the regional velocities that are needed to accurately describe the solute transport process. If the condition of large-scale mass equilibrium is valid, the solute transport process can be represented in terms of a one-equation model and the analysis is simplified greatly. The constraints associated with the condition of large-scale mass equilibrium are developed, and when these constraints are satisfied the mass transport process can be described in terms of the large-scale average velocity, an average adsorption isotherm, and a single large-scale dispersion tensor. When the condition of large-scale mass equilibrium is not valid, two equations are required to describe the mass transfer process, and these two equations contain two adsorption isotherms, two dispersion tensors, and an exchange coefficient. The extension of the analysis to multi-region models is straight forward but tedious.     

A two-dimensional horizontal wave propagation and mud mass transport model

The present study offers a two-dimensional horizontal wave propagation and morphodynamic model for muddy coasts. The model can be applied on a general three-dimensional bathymetry of a soft muddy coast to calculate wave damping, fluid mud mass transport and resulting bathymetry change under wave actions. The wave propagation model is based on time-dependent mild slope equations including the wave energy dissipation due to the wave-mud interaction of bottom mud layers as well as the combined effects of the wave refraction, diffraction and breaking. The constitutive equations of the visco-elastic–plastic model are adopted for the rheological behavior of fluid mud. The mass transport velocity within the fluid mud layer is calculated combining the Stokes’ drift, the mean Eulerian velocity and the gravity-driven mud flow. The results of the numerical model are compared against a series of conducted wave basin experiments, wave flume experiments and field observations. Comparisons between the computed results with both the field and laboratory data reveal the capability of the proposed model to predict the wave transformation and mud mass transport.     

Finite element techniques for convective-diffusive transport in porous media

Operator-splitting techniques are applied to convective-diffusive transport problems in porous media. The convection is treated by applying a modified method of characteristics to time-step along the characteristics of the convective part of the flow. The nonsymmetry in the spatial operator is addressed via a Petrov-Galerkin method which uses a test function to achieve stability through a balancing of the remaining convection, the diffusion, and any possible reaction terms. The use of time-stepping along characteristics allows the use of large time-steps in a stable but accurate fashion. If local phenomena are important, self-adaptive local grid refinement techniques can be coupled with the operator splitting.     

A master equation for reactive solute transport in porous media

The mean value of a density of a cloud of points described by a generalized Liouville equation associated with a convection dispersion equation governing adsorbing solute transport yields a joint concentration probability density. The general technique can be applied for either linear or nonlinear adsorption; here the application is restricted to linear adsorption in one-dimensional transport. The equation generated for the joint concentration probability density is in the general form of a Fokker-Planck equation, but with a suitable coordinate transformation, it is possible to represent it as a diffusion equation with variable coefficients.     

Total variation diminishing schemes for pollutant transport in porous media

A bidimensional numerical model has been used in order to simulate the contaminant transport in the coastal groundwater area (Atlantic margin of the Rharb basin, Morocco). This groundwater is materialized by means of the salt contamination derived from several factors: evapotranspiration, lithological series formations, marine intrusion, and processes of interaction between water and rocks. In order to reduce the numerical diffusion and limit the numerical dispersion, we use the Superbee flux limiter as a total variation diminishing scheme to discretize the convective operator. This kind of discretization was applied to the coastal groundwater of the Rharb basin (Morocco). The results show that the Superbee flux limiter is efficient at drawing the path of the contaminant front with high accuracy. Consequently, this scheme could constitute an approach in water management and allows one to prevent the risks of pollution and to manage the groundwater resource from a durable development perspective. Copyright © 2007 John Wiley & Sons, Ltd.     

Transport in chemically and mechanically heterogeneous porous media: V. Two-equation model for solute transport with adsorption

In this paper we develop the two-equation model for solute transport and adsorption in a two-region model of a mechanically and chemically heterogeneous porous medium. The closure problem is derived and the coefficients in both the one- and two-equation models are determined on the basis of the Darcy-scale parameters. Numerical experiments are carried out for a stratified system at the aquifer scale, and the results are compared with the one-equation model presented in Part IV and the two-equation model developed in this paper. Good agreement between the two-equation model and the numerical experiments is obtained. In addition, the two-equation model is used, in conjunction with a moment analysis, to derive a one-equation, non-equilibrium model that is valid in the asymptotic regime. Numerical results are used to identify the asymptotic regime for the one-equation, non-equilibrium model.     

Numerical mass balance for soil-moisture transport problems

The Galerkin finite element method coupled with the Crank-Nicolson time advance procedure is often used as a numerical analog for unsaturated soil-moisture transport problems. The Crank-Nicolson procedure leads to numerical mass balance problems which results in instability. A new temporal and spatial integration procedure is proposed that exactly satisfies mass balance for the approximating function used. This is accomplished by fitting polynomials continuously throughout the time and space domain and integrating the governing differential equations. To reduce computational effort, the resulting higher order polynomials are reduced to quadratic and linear piece-wise continuous polynomial approximation functions analogous to the finite element approach. Results indicate a substantial improvement in accuracy over the combined Galerkin and Crank-Nicolson methods when comparing to simplified problems where analytical solutions are available.     

A transport phase diagram for pore-level correlated porous media

Transport in porous media is often characterized by the advection–dispersion equation, with the dispersion coefficient as the most important parameter that links the hydrodynamics to the transport processes. Morphological properties of any porous medium, such as pore size distribution, network topology, and correlation length control transport. In this study we explore the impact of correlation length on transport regime using pore-network modelling. Earlier direct simulation studies of dispersion in carbonate and sandstone rocks showed larger dispersion compared to granular homogenous sandpacks. However, in these studies, isolation of the impact of correlation length on transport regime was not possible due to the fundamentally different pore morphologies and pore-size distributions. Against this limitation, we simulate advection–dispersion transport for a wide range of Péclet numbers in unstructured irregular networks with “different” correlation lengths but “identical” pore size distributions and pore morphologies. Our simulation results show an increase in the magnitudes of the estimated dispersion coefficients in correlated networks compared to uncorrelated ones in the advection-controlled regime. The range of the Péclet numbers which dictate mixed advection–diffusion regime considerably reduces in the correlated networks. The findings emphasize the critical role of correlation length which is depicted in a conceptual transport phase diagram and the importance of accounting for the micro-scale correlation lengths into predictive stochastic pore-scale modelling.     

黏弹双相介质中的松弛骨架模型

本文基于Biot理论,考虑了多孔介质中固体骨架的松弛特征,引入了纵波品质因子、横波品质因子与耗散品质因子三参数来描述黏弹双相介质波动方程.采用虚谱法在地震频段进行了波场模拟,模拟结果表明：松弛骨架机制不仅适用于高频段,也可用于解释地震频段下的弹性波衰减现象,以描述固体微细颗粒的中观松弛特征.结合小生境遗传算法对三层模型24个介质参数进行了反演,反演结果表明：无噪波场的反演结果具有较高精度,对于含噪波场,取值在奇异点附近的介质参数反演精度降低.最后,对中国东部某地区的实际资料进行了浅层参数的反演,得到了该地区的表层固体体积模量、固体密度以及品质因子.     

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.     

An efficient discontinuous Galerkin method for advective transport in porous media

We consider a discontinuous Galerkin scheme for computing transport in heterogeneous media. An efficient solution of the resulting linear system of equations is possible by taking advantage of a priori knowledge of the direction of flow. By arranging the elements in a suitable sequence, one does not need to assemble the full system and may compute the solution in an element-by-element fashion. We demonstrate this procedure on boundary-value problems for tracer transport and time-of-flight.