On the Errors Associated with Two-Dimensional Stochastic Solute Transport Models

Many subsurface solute transport studies employ numerical modeling techniques to estimate solute arrival times. Simplifying assumptions must be made to define the modeling domain within a mathematical framework. One common assumption is that the vertical flow is negligible such that the flow field can be simulated with a two-dimensional model. Reducing the vertical dimension reduces the number of flow paths that a solute can take. In a heterogenous medium, artificially removing the 3rd dimension may lead to erroneous results. We investigate the error in the simulated solute breakthrough associated with a two-dimensional model. We also use a stochastic solution of solute arrival time to derive a transform of a two-dimensional ln (k) field so that solute transport more closely resembles three-dimensional transport behavior. The moment equations for two- and three-dimensional domains were solved simultaneously to calculate this transform. The results indicate that the removal of the vertical variability (3D 2D) introduces a 5–10% error in the predicted solute breakthrough. The error tends to increase with increased hydraulic conductivity variance. Numerical experiments confirm that the transform developed herein decreases the relative error of particle breakthrough curves.     

Analytical Solutions for Macrodispersion in a 3D Heterogeneous Porous Medium with Random Hydraulic Conductivity and Dispersivity

A stochastic analysis of macrodispersion for conservative solute transport in three-dimension (3D) heterogeneous statistically isotropic and anisotropic porous media when both hydraulic conductivity and local dispersivity are random is presented. Analytical expressions of macrodispersivity are derived using Laplace and Fourier transforms. The effects of various parameters such as ratio of transverse to longitudinal local dispersivity, correlation length ratio, correlation coefficient and direction of flow on asymptotic macrodispersion are studied. The behaviour of growth of macrodispersivity in preasymptotic stage is also shown in this paper. The variation in local dispersion coefficient causes change in transverse macrodispersivity. The consideration of random dispersivity along with random hydraulic conductivity indicates that the total dispersion is affected and important in the case when the hydraulic conductivity and dispersivity are correlated. It is observed that the pre-asymptotic behavior of the macrodispersivity is not sensitive to the choice of spectral density functions.     

A Solute Transport in Stratified Media

Two-dimensional and steady solute transport in a stratified porous formation is analysed under assumption that the effect of pore-scale dispersion is negligible. The longitudinal dispersion produced as a result of the vertical variation of hydraulic conductivity is analysed by averaging the variability of a solute flux concentration and conductivity. The evolution of the solute flux concentration is expressed with respect to the correlated variable, that is the travel (arrival) time at a fixed location and the averaging procedure is constructed to satisfy the boundary condition where the inlet concentration is a known function of time. In such a statement, a velocity-averaged solute flux concentration is described by a conventional dispersion model (CDM) with a dispersion coefficient which is a function of the arrival time. It is demonstrated that such CDM satisfies the assumption that hydraulic conductivity of the layers is gamma distributed with the parameter of distribution which is chosen to represent a reasonable value of the field scale solute dispersion. The overall behaviour of the model is illustrated by several examples of two-dimensional mass transport.     

Hydraulic radius and transport in reconstructed model three-dimensional porous media

Methods for reconstructing three-dimensional porous media from two-dimensional cross sections are evaluated in terms of the transport properties of the reconstructed systems. Two-dimensional slices are selected at random from model three-dimensional microstructures, based on penetrable spheres, and processed to create a reconstructed representation of the original system. Permeability, conductivity, and a critial pore diameter are computed for the original and reconstructed microstructures to assess the validity of the reconstruction technique. A surface curvature algorithm is utilized to further modify the reconstructed systems by matching the hydraulic radius of the reconstructed three-dimensional system to that of the two-dimensional slice. While having only minor effects on conductivity, this modification significantly improves the agreement between permeabilities and critical diameters of the original and reconstructed systems for porosities in the range of 25–40%. For lower porosities, critical pore diameter is unaffected by the curvature modification so that little improvement between original and reconstructed permeabilities is obtained by matching hydraulic radii.     

Solute Transport Analysis Through Heterogeneous Media in Nonuniform in the Average Flow by a Stochastic Approach

The stochastic approach has been shown to be an excellent tool for the characterisation and analysis of velocity fields and transport processes through heterogeneous porous formations. The main results (linear theory) have been obtained for problems with simplified flow conditions, usually in the assumption of uniform in the average flow, but a great effort is spent to reach theoretical results for more complex situations.This paper deals with 2D heterogeneous aquifers subject to uniform recharge; the stochastic approach is adopted to characterise, as ensemble behaviour, the velocity field and transport processes of a nonreactive solute. The impact of transmissivity conditioning on solute particles trajectories is analysed and an application is carried out. The analytical formulations, obtained by a first order analysis, are compared to the one resulting from constant in the average hydraulic gradient, and their reliability is investigated with numerical tests performed by a Monte Carlo method.The result of this study is that strong non-stationarities are present in the flow and transport process. A detailed analysis shows that the theoretical results cannot be extended to cases with high heterogeneity level, unlike the uniform in the average flow fields.     

Estimation of Macrodispersion by Different Approximation Methods for Flow and Transport in Randomly Heterogeneous Media

We present two- and three-dimensional calculations for the longitudinal and transverse macrodispersion coefficient for conservative solutes derived by particle tracking in a velocity field which is based on the linearized flow equation. The simulations were performed upto 5000 correlation lengths in order to reach the asymptotic regime. We used a simulation method which does not need any grid and therefore allows simulations of very large transport times and distances.Our findings are compared with results obtained from linearized transport, from Corrsin's Conjecture and from renormalization group methods. All calculations are performed with and without local dispersion. The variance of the logarithm of the hydraulic conductivity field was chosen to be one to investigate realistic model cases.While in two dimensions the linear transport approximation seems to be very good even for this high variance of the logarithmic hydraulic conductivity, in three dimensions renormalization group results are closer to the numerical calculations. Here Dagan's theory and the theory of Gelhar and Axness underestimate the transverse macrodispersion by far. Corrsin's Conjecture always overestimates the transverse dispersion. Local dispersion does not significantly influence the asymptotic behavior of the various approximations examined for two-dimensional and three-dimensional calculations.     

Steady water flow through unsaturated aggregated porous materials

The three-dimensional steady water flow through unsaturated aggregated porous materials composed of simple cubic open packed and tetrahedral close packed assemblies of uniform porous spheres is investigated with electric analogues and numerical computations. Water around the spheres is considered to be discontinuous with flow restricted through isolated annular water lenses held by surface tension forces around the contact points between spheres. It is found that the conductance of individual spheres depends only on the size of the water lenses and is independent of the radius of the sphere. For a simple cubic packing the conductance for small lens radii is given by Weber’s formula for flow from an electrified disc into an infinite medium. It follows that the bulk hydraulic conductivity of these assemblies of porous spheres is also independent of aggregate size over a range of water contents. This independence is also shown in measurements of hydraulic conductivity of aggregates of diatomaceous earth that show a convergence to a single relationship between conductivity and water content when there is no longer continuity of water in the macropore space. The effect of the three-dimensional flow through aggregates on solute leaching is demonstrated by considering the numerical results of the stream-tube pattern in a sphere.     

Laws of viscoplastic fluid flow in anisotropic porous and fractured media

In invariant tensor form, the laws of viscoplastic fluid flow are formulated for capillary and fractured media with a periodic microstructure that has orthotropic and transversely isotropic symmetry in the flow properties. An analysis of the laws of viscoplastic fluid flow in transversely isotropic and orthotropic porous and fractured media shows that in formulating the equations it is necessary to distinguish between the permeability tensor and the limiting gradient tensor, which may differ in the symmetry of the flow characteristics, and that the flow law is multivariant and admits one-, two-, and three-dimensional flows.     

Influence of Microbial Growth on Hydraulic Properties of Pore Networks

From laboratory experiments it is known that bacterial biomass is able to influence the hydraulic properties of saturated porous media, an effect called bioclogging. To interpret the observations of these experiments and to predict possible bioclogging effects on the field scale it is necessary to use transport models, which are able to include bioclogging. For these models it is necessary to know the relation between the amount of biomass and the hydraulic conductivity of the porous medium. Usually these relations were determined using bundles of parallel pore channels and do not account for interconnections between the pores in more than one dimension. The present study uses two-dimensional pore network models to study the effects of bioclogging on the pore scale. Numerical simulations were done for two different scenarios of the growth of biomass in the pores. Scenario 1 assumes microbial growth in discrete colonies clogging particular pores completely. Scenario 2 assumes microbial growth as a biofilm growing on the wall of each pore. In both scenarios the hydraulic conductivity was reduced by at least two orders of magnitude, but for the colony scenario much less biomass was needed to get a maximal clogging effect and a better agreement with previously published experimental data could be found. For both scenarios it was shown that heterogeneous pore networks could be clogged with less biomass than more homogeneous ones.     

Methods of Quantum Field Theory in the Physics of Subsurface Solute Transport

The stochastic theory of subsurface solute transport has received stimulus recently from modeling techniques originating in quantum field theory (QFT), resulting in new calculations of the solute macrodispersion tensor that derive from the solving Dyson equation with a subsequent renormalization group analysis. In this paper, we offer a critical evaluation of these techniques as they relate specifically to the derivation of a field-scale advection–dispersion equation. An approximate Dyson equation satisfied by the ensemble-average solute concentration for tracer movement in a heterogeneous porous medium is derived and shown to be equivalent to a truncated cumulant expansion of the standard stochastic partial differential equation which describes the same phenomenon. The full Dyson equation formalism, although exact, is of no importance to the derivation of an improved field-scale advection–dispersion equation. Similarly, renormalization group analysis of the macrodispersion tensor has not yet provided results that go beyond what is available currently from the cumulant expansion approach.     

Localization of Mean Flow and Apparent Transmissivity Tensor for Bounded Randomly Heterogeneous Aquifers

We explore the concept of apparent transmissivity for bounded randomly heterogeneous media under steady-state flow regime. The novelty of our study consists of investigating a tensorial nature of apparent transmissivity. We demonstrate that apparent transmissivity of bounded domains is anisotropic even though an underlying local transmissivity field is statistically isotropic. For rectangular flow domains, we derive an analytical expression for the apparent transmissivity tensor via localization and perturbation expansion of the nonlocal mean flow equations in the variance of log-transmissivity. In this expression, almost everywhere the off-diagonal terms are several orders of magnitude smaller than the diagonal terms. When the domain size relative to the log-transmissivity correlation scale is large, the longitudinal and transverse components of the apparent transmissivity tensor approach the geometric mean of local transmissivity. While rigorously valid for mean uniform flows only, our expression for the apparent transmissivity tensor leads to mean hydraulic head distributions that compare favorably with those obtained through Monte-Carlo simulations and the nonlocal mean flow equations even in the presence of pumping wells. This agreement deteriorates in the vicinity of wells and as pumping rates increase.     

Reactive Solute Transport in a Nonstationary,Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. Solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.     

裂隙岩体渗透系数确定方法研究

裂隙岩体渗透系数以及渗透主方向的确定对研究岩体渗透性大小及各向异性具有重要意义。高放废物地质处置库介质岩体的渗透性能将直接影响其使用安全性。本文运用离散裂隙网络模拟的方法对我国高放废物处置库甘肃北山预选区3#钻孔附近裂隙岩体进行了渗透性质分析。通过对3#钻孔1715～1780m段压水试验数据的反演,标定了离散裂隙网络渗流模型中的裂隙渗透参数(导水系数T)。利用标定的离散裂隙网络模型对场区裂隙岩体进行了渗流模拟,确定了该区域裂隙岩体的渗流表征单元体(REV)的尺寸大小以及渗透主值和主渗透方向。运用离散裂隙网络模型计算得出的渗透主值的几何均值与现场压水试验计算结果较接近,证明了计算结果的有效性。     

The Effect of varying correlation lengths on Anomalous Transport

Conventional concepts for transport in porous media assume that the heterogeneous distribution of hydraulic conductivities is the source for the contaminant temporal and spatial heavy tail. This tailing, known as anomalous or non-Fickian transport, can be captured by the β parameter in the continuous-time random walk framework. This study shows that with the increase in spatial correlation length between these heterogeneous distributions of hydraulic conductivities, the transport’s anomaly reduces; yet, the β value is unchanged, suggesting a topological component of the conductivity field, captured by the β. This finding is verified by an analysis of the solute transport, showing that the changing conductivity values have a moderate effect on the transport shape.

     

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed.     

A new stochastic method of reconstructing porous media

We present a new stochastic method of reconstructing porous medium from limited morphological information obtained from two-dimensional micro- images of real porous medium. The method is similar to simulated annealing method in the capability of reconstructing both isotropic and anisotropic structures of multi-phase but differs from the latter in that voxels for exchange are not selected completely randomly as their neighborhood will also be checked and this new method is much simpler to implement and program. We applied it to reconstruct real sandstone utilizing morphological information contained in porosity, two-point probability function and linear-path function. Good agreement of those references verifies our developed method’s powerful capability. The existing isolated regions of both pore phase and matrix phase do quite minor harm to their good connectivity. The lattice Boltzmann method (LBM) is used to compute the permeability of the reconstructed system and the results show its good isotropy and conductivity. However, due to the disadvantage of this method that the connectivity of the reconstructed system’s pore space will decrease when porosity becomes small, we suggest the porosity of the system to be reconstructed be no less than 0.2 to ensure its connectivity and conductivity.     

Exact Averaging of Stochastic Equations for Flow in Porous Media

It is well-known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods—for example, the convergence behavior and the accuracy of truncated perturbation series—are not well-studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green’s functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic non-local equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged non-local equations with a unique kernel-tensor. When global symmetry does not exist, the non-local equation with a kernel-tensor involves complications and leads to an ill-posed problem.     

Effective Thermal Conductivity Modelling for Closed-Cell Porous Media with Analytical Shape Factors

This paper presents a fully analytical model for the effective thermal conductivity of two-phase porous media with two-/three-dimensional closed cells, applicable to honeycombs and closed-cell foams. The present model combines an existing analytical expression derived based on the Laplace heat conduction equation with an analytical shape factor which corrects the deviation caused from a non-circular (or non-spherical) pore inclusion. Results demonstrate the validity of the present model capable of analytically estimating the effective thermal conductivity of closed-cell porous media. The simple yet accurate model provides the physical mechanisms of how effective thermal conductivity depends upon the shape of pores.     

多孔材料孔隙尺寸对渗透系数影响的数值模拟

采用有限元方法数值模拟了多孔材料的孔隙尺寸与等效渗透系数之间的非线性关系.有限元模型中的固体骨架和孔隙根据孔隙率的大小随机生成,模型中的材料参数和单元属性用ANSYS中的APDL参数化语言赋值.根据有限元随机模拟断面的流量分布和稳态渗流问题的达西定律,计算在不同孔隙尺寸的等效渗透系数,研究等效渗透系数与孔隙尺寸之间的关系.计算结果表明,在孔隙率不变的情况下,等效渗透系数与孔隙尺寸的平方成正比,该结论与经验公式相一致.而孔隙尺寸不变的条件下,随着孔隙率的增加等效渗透系数近似呈线性增加.     

Stochastic Analysis of Reactive Solute Transport in Heterogeneous,Fractured Porous Media: A Dual-Permeability Approach

A nonlocal, first-order, Eulerian stochastic theory is developed for reactive chemical transport in a heterogeneous, fractured porous medium. A dual-permeability model is adopted to describe the flow and transport in the medium, where the solute convection and dispersion in the matrix are considered. The chemical is under linear nonequilibrium sorption and first-order degradation. The hydraulic conductivities, sorption coefficients, degradation rates in both fracture and matrix regions, and interregional mass transfer coefficient are all assumed to be random variables. The resultant theory for mean concentrations in both regions is nonlocal in space and time. Under spatial Fourier and temporal Laplace transforms, the mean concentrations are explicitly expressed. The transformed results are then numerically inverted to the real space via Fast Fourier Transform method. The theory developed in this study generalizes the stochastic studies for a reactive chemical transport in a one-domain flow field (Hu et al., 1997a) and in a mobile/immobile flow field (Huang and Hu, 2001). In comparison with one-domain transport, the dual-permeability model predicts a larger second moment in the longitudinal direction, but smaller one in the transverse direction. In addition, various simplification assumptions have been made based on the general solution. The validity of these assumptions has been tested via the spatial moments of the mean concentration in both fracture and matrix regions.