Coupled solution of heat and moisture flow in unsaturated clay barriers in a repository geometry

An analytical solution in the Laplace transform domain is obtained for the transient heat and moisture transport in an unsaturated clay buffer with a geometry simulating repository conditions. A numerical inversion scheme based on Crump's method is used to obtain the time‐domain solution. The coupled effect of thermally driven moisture transport is especially investigated because of its importance to alter the flow field in low‐permeability buffers. The practical background is based on the case of an engineering bentonite barrier placed in a drift excavated in rock in the context of underground disposal of high‐level radioactive waste. Parametric study has been performed to assess the effects of dimensionless geometry and material parameters on flow field. Despite the simplified assumptions required in order to obtain analytical expressions, the results incorporate the main mechanisms involved in the coupled thermo‐hydraulic (T–H) problem, and they may be eventually used for validation purposes. Copyright © 2006 John Wiley & Sons, Ltd.     

A discretized analytical solution for fully coupled non‐linear simulation of heat and mass transfer in poroelastic unsaturated media

Mathematical simulation of non‐isothermal multiphase flow in deformable unsaturated porous media is a complicated issue because of the need to employ multiple partial differential equations, the need to take into account mass and energy transfer between phases and because of the non‐linear nature of the governing partial differential equations. In this paper, an analytical solution for analyzing a fully coupled problem is presented for the one‐dimensional case where the coefficients of the system of equations are assumed to be constant for the entire domain. A major issue is the non‐linearity of the governing equations, which is not considered in the analytical solution. In order to introduce the non‐linearity of the equations, an iterative discretized procedure is used. The domain of the problem is divided into identical time–space elements that cover the time–space domain. A separate system of equations is defined for each element in the local coordinate system, the initial and boundary conditions for each element are obtained from the adjacent elements and the coefficients of the system of equations are considered to be constant in each step. There are seven governing differential equations that should be solved simultaneously: the equilibrium of the solid skeleton, mass conservation of fluids (water, water vapor and gas) and energy conservation of phases (solid, liquid and gas). The water vapor is not in equilibrium with water and different phases do not have the same temperature. The governing equations that have been solved seem to be the most comprehensive in this field. Three examples are presented for analyzing heat and mass transfer in a semi‐infinite column of unsaturated soil. Copyright © 2009 John Wiley & Sons, Ltd.     

A constitutive model for unsaturated soil–structure interfaces

A constitutive model for simulation of the behavior of unsaturated interfaces is presented here. The model is an extension of an existing critical state compatible interface model for dry and saturated interfaces that was already proposed by one of the authors [Lashkari, A. 2013. Int. J. Numer. Anal. Meth. Geomech. 37 (8): 904–931]. For a proper simulation of the behavior of partially saturated interfaces, the extended model is formulated in terms of two pairs of work conjugate stress–strain‐like variables. The modified model simulations are compared with the existing data of dry, unsaturated, and saturated interfaces. For each interface type, it is shown that the proposed model can capture the essential elements of the behavior using a unique set of parameters. Copyright © 2015 John Wiley & Sons, Ltd.     

Uncoupling of coupled flows in soil—a finite element method

Coupled flow of water, chemicals, heat and electrical potential in soil are of significance in a variety of circumstances. The problem is characterized by the coupling between different flows, i.e. a flow of one type driven by gradients of other types, and by the dual nature of certain flows, i.e. combined convection and conduction. Effective numerical solutions to the problem are challenged due to the coupling and the dual nature. In this paper, we first present a general expression that can be used to represent various types of coupled flows in soil. A finite element method is then proposed to solve the generalized coupled flows of convection-conduction pattern. The unknown vector is first decomposed into two parts, a convective part forming a hyperbolic system and a conductive part forming a parabolic system. At each time step, the hyperbolic system is solved analytically to give an initial solution. To solve the multi-dimensional hyperbolic system, we assume that a common eigenspace exists for the coefficient matrices, so that the system can be uncoupled by transforming the unknown vector to the common eigenspace. The uncoupled system is solved by the method of characteristics. Using the solution of the hyperbolic system as the initial condition, we then solve the parabolic system by a Galerkin finite element method for space discretization and a finite difference scheme for time stepping. The proposed technique can be used for solving multi-dimensional, transient, coupled or simultaneous flows of convection-conduction type. Application to a flow example shows that the technique indeed exhibits optimality in convergence and in stability.     

Three-dimensional heat,moisture and air transfer in unsaturated soils

A new three-dimensional numerical model of coupled heat, moisture and air transfer in unsaturated soil is presented. In particular, the model accommodates moisture transfer in the form of liquid and vapour flow and heat transfer arising from conduction, convection and latent heat of vaporization. The bulk flow of dry air and the movement of air in a dissolved state are also included. The theoretical basis of the model, the finite element solution of the spatial terms and finite difference solution of the temporal terms are briefly presented. Attention is focused on the verification of the new numerical solution. This is achieved via comparisons with independent solutions of heat, moisture and air transfer in an unsaturated soil. The physical problem considered includes the highly non-linear hydraulic properties of sand. Thermal conductivity is also included as a function of soil moisture content. Excellent correlation of results is shown thus providing confidence in the new model. The new model is also applied to a number of test cases which illustrate the need for the development of a model which can fully include three-dimensional behaviour. In particular, three applications are presented each increasing in complexity. The first application illustrates three-dimensional heat transfer. This particular application is verified against existing commercial finite element software. Subsequent applications serve to illustrate how the coupled processes of heat moisture and air transfer combine to yield three-dimensional problems even within a simple geometric domain. Visualization of three-dimensional results is also addressed. © 1998 by John Wiley & Sons, Ltd.     

Subsidence due to pumping from layered soil—a perturbation theory

Persistent pumping of groundwater is known to cause subsidence of the ground surface. When the soil is composed of alternating layers of sand and clay, subsidence is, in general, a consequence of pressure change in and deformation of all layers. Even with various simplifications, the mathematical task for the mechanics of a layered soil is complex and, in the past, many idealizations have been introduced in the literature. Among the existing approximations, a well-known approximation by Hantush¹ for well-hydraulics in leaky aquifers is intuitively attractive. He used the fact known to be strictly true for simple flows in a soil system with horizontal layers, that the flow must be mainly horizontal in a relatively porous aquifer and mainly vertical in a highly impervious aquitard, and invoked the same approximation for transient flows induced by well-pumping. In this paper we shall apply a perturbation analysis to give a mathematical confirmation of the hydrological approximation and to show its degree of accuracy. In particular, we treat a three-layered system, where the middle layer is a relatively soft aquitard, and the well withdraws water from the artesian aquifer at the bottom. The upper aquifer has a phreatic surface. Physically, we examine the often ignored effects of self-weight which is important for thick aquitards. The perturbation theory is worked out for sufficiently weak pumping rates and small soil deformation. Subsidence in the vertical direction due mainly to the deformation of the soft aquitard is studied. Results show that the common assumption of constant total stress with respect to depth is valid only when the storage coefficient is very small.     

Modelling transient heat and moisture transfer in unsaturated soil using a parallel computing approach

A parallel numerical model, employing a finite difference explicit scheme for the analysis of coupled heat and moisture transfer in unsaturated soil, is employed to simulate a laboratory experiment of heating of medium sand. The model, written in a two-dimensional polar co-ordinate formulation, is programmed in the concurrent language Occam and executed on a parallel computing network of transputers. Parallelization is adopted as a means of overcoming computing difficulties, which limited numerical solutions to those at steady state, to enable transient behaviour to be simulated. The parallel algorithm was found to be very efficient, enabling a full solution of transient behaviour to be obtained. An investigation of the ability of the model to accurately simulate the complex, interrelated coupled nature of both two-dimensional transient and steady-state behaviour yielded very good correlation between experimental and numerical results. It can therefore be concluded that overall the results obtained provide confidence in the validity of the approach proposed.     

Sub-sea-floor metamorphism,heat and mass transfer; an additional comment

Attention is drawn to the fact that the warm geothermal brines of Matupi Harbour (New Britain) are of sea water origin, contain amounts of total dissolved solids comparable to those in sea water, but are remarkably enriched in heavy metals (Ferguson and Lambert, 1972). This evidence supports the proposal that metal enriched brines, of sea water origin, may be generated in sub-sea-floor geothermal systems by a high temperature leaching process, and may form a metal enriched sedimentary component on discharge back into sea water (Spooner and Fyfe, 1973).The chloride contents of the Matupi Harbour brines are low, relative to other known hydrothermal solutions which carry comparable amounts of heavy metals in solution. This evidence indicates that high dissolved chloride contents of the order of 160000 ppm, compared with sea water which contains 19800 ppm, are not a prerequisite for a high metal carrying capacity in natural brines.     

Sub-sea-floor metamorphism,heat and mass transfer

The ophiolitic rocks of E. Liguria, Italy contain a spilitic metamorphic assemblage sequence, cross-cut by hydrothermal veins, which developed in the oceanic environment. Metamorphic parageneses indicate that temperatures as high as 400°C were realised at depths as shallow as 300 m below the original rock/water interface. The inferred temperature interval was equivalent to a geothermal gradient of 1300°C/km.It is suggested that metamorphism took place in a sub-sea-floor geothermal system, and that such systems are an integral part of the sea-floor spreading process. Modern evidence is provided to support this hypothesis, and to suggest that heavy metal rich solutions discharged from such systems are responsible for the formation of a metal enriched sedimentary component. A unified model of sub-sea-floor metamorphism and mass transfer is proposed, and possible differences between sub-sea-floor and terrestial geothermal systems are discussed. In the light of the model, the origins of certain aspects of bedded cherts found associated with ophiolitic rocks, of ophiolitic massive sulphide deposits and of certain trace element patterns are considered.     

非饱和介质中热能传输及水分迁移的数值积分解

在给出非饱和介质热能-水分传输的耦合质量控制方程和基于Fourier热传导定律的热能平衡方程的基础上,对热能传输及水分迁移的基本特征和机理进行了分析。其中,考虑了温度势、吸力势和重力势的耦合作用影响。给出有热源时控制方程的简化形式,并对半无限体自由表面作用平面热源条件下介质内非稳态温度场、体积含水率分布场进行数值积分求解。利用这些解答给出常热源强度和变热源强度两种情况下,温度场随时间的变化特征以及水分迁移的演化过程,并分析了重力项对计算结果的影响。     

Modelling two-dimensional heat and moisture transfer in unsaturated soils,including gravity effects

A theoretical model is presented to predict simultaneous transient coupled heat and moisture transfer in partly saturated soils. The formulation is in terms of volumetric moisture content, is two dimensional, includes gravitational flow and takes into account latent heat of vaporization effects. The numerical solution of the problem is accomplished by means of a finite element solution algorithm. Predictions from the numerical model are used to investigate the importance of gravitational flow, for the case of a soil stratum subjected to evaporation losses at the surface. The results achieved show good qualitative agreement with expected behaviour.     

Directional response of a reconstituted fine‐grained soil—Part I: experimental investigation

This paper discusses the results of a large experimental program designed to investigate in a systematic manner the main features of the incremental response of fine‐grained soils. The results are obtained from triaxial stress probing experiments carried out on a French silty clay (Beaucaire Marl). All the tests have been performed on reconstituted specimens, normally consolidated to an initial state which is either isotropic or anisotropic. In the interpretation of the experimental results, extensive use is made of the concept of strain response envelope. The response envelopes obtained for different stress increment magnitudes are remarkably consistent with each other and indicate an inelastic and irreversible material response, i.e. a strong dependence on the stress increment direction, also at relatively small strain levels. A companion paper (Int. J. Numer. Anal. Meth. Geomech., this issue, 2006) assesses the performance of some advanced constitutive models in reproducing the behaviour of reconstituted Beaucaire Marl as observed in this experimental program. Copyright © 2006 John Wiley & Sons, Ltd.     

Fast iterative solution of large undrained soil‐structure interaction problems

In view of rapid developments in iterative solvers, it is timely to re‐examine the merits of using mixed formulation for incompressible problems. This paper presents extensive numerical studies to compare the accuracy of undrained solutions resulting from the standard displacement formulation with a penalty term and the two‐field mixed formulation. The standard displacement and two‐field mixed formulations are solved using both direct and iterative approaches to assess if it is cost‐effective to achieve more accurate solutions. Numerical studies of a simple footing problem show that the mixed formulation is able to solve the incompressible problem ‘exactly’, does not create pressure and stress instabilities, and obviate the need for an ad hoc penalty number. In addition, for large‐scale problems where it is not possible to perform direct solutions entirely within available random access memory, it turns out that the larger system of equations from mixed formulation also can be solved much more efficiently than the smaller system of equations arising from standard formulation by using the symmetric quasi‐minimal residual (SQMR) method with the generalized Jacobi (GJ) preconditioner. Iterative solution by SQMR with GJ preconditioning also is more elegant, faster, and more accurate than the popular Uzawa method. Copyright © 2003 John Wiley & Sons, Ltd.     

An analytical technique for evaluation of coupled heat and mass flow coefficients in unsaturated soil

This study was done to evaluate a phenomenological approach for determining the coupled heat and water diffusion parameters. This technique requires the use of both theory and experimental data. The transport equation was solved analytically using nondimensional analysis. The calculated diffusion parameters of a compacted sand–bentonite-based material are presented. The agreement between the back calculated values and the experimental volumetric water content profiles was good. Practical application of the technique is also discussed.     

Directional response of a reconstituted fine‐grained soil—Part II: performance of different constitutive models

In this paper, the performance of different advanced constitutive models for soils is evaluated with respect to the experimentally observed behaviour of a soft reconstituted clay subject to a wide range of loading directions, see (presented in the companion paper). The models considered include a three‐surface kinematic hardening elastoplastic model; the CLoE hypoplastic model; a recently proposed K‐hypoplastic model for clays, and an enhanced version of the same model incorporating the concept of intergranular strain. A clear qualitative picture of the relative performance of the different models as a function of the loading direction is obtained by means of the incremental strain response envelopes. The definition of suitable error measures allows to obtain further quantitative information in this respect. For the particular initial conditions and loading programme considered in this study, the kinematic hardening and the enhanced K‐hypoplastic models appear to provide the best performance overall. Copyright © 2006 John Wiley & Sons, Ltd.     

Integral equation solution for three-dimensional heat transfer in multiple-fracture enhanced geothermal reservoirs

For the prediction of energy production from multiple-fractured geothermal reservoirs, previous models basically focused on the one-dimensional conduction in the rock containing evenly distributed fractures of equal scale. Here, a novel model is described to numerically investigate the three-dimensional heat transfer in geothermal reservoirs with unevenly spaced disc fractures of various sizes including the aperture and radius. In terms of the water flow through each fracture, an approximate analytical solution is obtained on the assumption that the water pressure disturbances, induced by the fracture margin and extraction (injection) operation, at the injection (extraction) well center and at different locations within the injection (extraction) well range were approximately equal. By the integral equation scheme for two-dimensional planar fractures, the three-dimensional problem of heat exchange is simulated without the reservoir discretization. The singular integral is analytically calculated in polar coordinates whereas the nonsingular integrand is numerically estimated by the Gaussian quadrature method in Cartesian coordinates. Compared with the one-dimensional simplification, the three-dimensional heat conduction remarkably alters the prediction of extraction temperature. In addition, the reservoir temperature field is also significantly influenced by the spacings and dimensions of fractures. The present model may be used for the estimation, design, and optimization of a geothermal reservoir.     

Poroelastic-plastic consolidation — analytical solution

Consolidation of a poroelastic material that yields according to Drucker–Prager or Mohr–Coulomb criteria leads to a Stefan problem for time-dependent pore fluid pressure. The solution to the Stefan problem for a column of infinite depth is known and is adapted to poroelastic/plastic consolidation of a weightless material under a uniform surface load applied instantaneously and subsequently maintained constant. In this approach, the plastic potential and yield criterion need not be the same. If yielding occurs concurrently with application of load, then collapse is instantaneous. Otherwise, yielding may occur during the consolidation period. If so, then the elastic–plastic zone first appears at the surface and subsequently moves down the column. Depth to the elastic–plastic boundary is given by the simple expression Z = 2β√t where β is a constant determined from continuity conditions at the elastic–plastic boundary. Time-dependent surface displacement that occurs during consolidation is directly proportional to Z. There is little difference between elastic–plastic and purely elastic results in a numerical example because there is little difference in the respective consolidation coefficients. Elastic–plastic finite element results obtained from a column of finite depth are in close agreement with analytical results as long as the pore pressure at the bottom of the column does not change significantly from the value induced by application of the surface load. The analytical solution provides for: (1) efficient evaluation of material properties effects on consolidation, including strength and fluid compressibility, and (2) an accurate way of validating poroelastic/plastic computer codes that are based on Drucker–Prager and Mohr–Coulomb criteria. Copyright © 1999 John Wiley & Sons, Ltd.     

Coupled modelling of hydro‐mechanical behaviour of unsaturated compacted expansive soils

The Barcelona basic model cannot predict the mechanical behaviour of unsaturated expansive soils, whereas the Barcelona expansive model (BExM) can only predict the stress–strain behaviour of unsaturated expansive soils without the water‐retention behaviour being incorporated. Moreover, the micro‐parameters and the coupling function between micro‐structural and macro‐structural strains in the BExM are difficult to determine. Experimental data show that the compression curves for non‐expansive soils under constant suctions are shifted towards higher void ratios with increasing suction, whereas the opposite is true for expansive soils. According to the observed water‐retention behaviour of unsaturated expansive soils, the air‐entry value increases with density, and the relationship between the degree of saturation and void ratio is linear at constant suction. According to the above observation, an elastoplastic constitutive model is developed for predicting the hydraulic and mechanical behaviour of unsaturated expansive soils, based on the existing hydro‐mechanical model for non‐expansive unsaturated soil. The model takes into consideration the effect of degree of saturation on the mechanical behaviour and that of void ratio on the water‐retention behaviour. The concept of equivalent void ratio curve is introduced to distinguish the plastic potential curve from the yield curve. The model predictions are compared with the test results of an unsaturated expansive soil, including swelling tests under constant net stress, isotropic compression tests and triaxial shear tests under constant suction. The comparison indicates that the model offers great potential for quantitatively predicting the hydraulic and mechanical behaviour of unsaturated expansive soils. Copyright © 2011 John Wiley & Sons, Ltd.