Permeability of Self-Affine Fractures

The permeability of self-affine fractures with various roughness exponents H is investigated by direct three-dimensional numerical simulations. A scaling behavior with an exponent H is exhibited in the self-affine scale range. Permeability can be related to the fractional open area and to the percolation probability by simple models.     

The Impact of Mineral Dissolution on Drainage Relative Permeability and Residual Trapping in Two Carbonate Rocks

Transport in Porous Media - Carbon dioxide injection into deep saline aquifers is governed by a number of physico-chemical processes including mineral dissolution and precipitation, multiphase...     

Permeability evolution induced by the precipitation of an advected solute: Effect of the microscopic and the macroscopic scales competition on the clogging mechanism

A model is proposed for coupling the one-dimensional transport of solute with surface precipitation kinetics which induces the clogging of an initially homogeneous porous medium. The aim is to focus the non-linear feedback effect between the transport and the chemical reaction through the permeability of the medium. A Lagrangian formulation, used to solve the coupled differential equations, gives semi-analytical expressions of the hydrodynamic quantities. A detailed analysis reveals that the competition between the microscopic and macroscopic scales controls the clogging mechanism, which differs depends on whether short or long times are considered. In order to illustrate this analysis, more quantitative results were obtained in the case of a second and zeroth order kinetic. It was necessary to circumvent the semi-analytic character of the solutions problem by successive approximation. A comparison with results obtained by simulations displays a good agreement during the most part of the clogging time.Nomenclature a(x, t) Capillary tube radius (L) - A _(aq) Chemical species in the aqueous phase - A _n(s) Chemical species of the solid phase - C(x, t) Aqueous concentration in a capillary tube ([mole/L³] in the case of a permanent injection - [mole/L³/L] in the case of an instantaneous injection) - C(x, t) C(x, t)/C ₀ Dimensionless aqueous concentration in a capillary tube - C ₀ Aqueous concentration imposed at the inlet and also initial concentration in an elementary volume of fluid (mole/L³) - C _i(t) Concentration in a fluid element i (mole/L³) - C(_R) (t, C_o) Aqueous concentration in a stirred reactor (mole/L³) - d_ij (t) Length belonging to the volume, inside a fluid element i, which interacts with a precipitate element j (L) - dM _ij(t) Mass exchange between a fluid element i and a precipitate element j (M) - dN ₀ Number of molecules in an elementary volume of fluid injected at the inlet of a capillary tube during dt ₀ - dN(x, t, C₀) Number of molecules in an elementary volume of fluid - dt ₀ Time injection of an elementary volume of fluid (T) - D(x, t) Dispersion coefficient (L²/T) - Da(t, x) Damköhler number - D _m Molecular diffusion coefficient (L²/T) - F(x, t) Advective flux (mole/L²/T) - k ₁ Kinetic constant of dissolution (mole/L³/T) - k ₂ Kinetic constant of precipitation ([mole/L³]^{1 - n} /T) - k ₂ Kinetic constant of precipitation in the case of a zeroth order kinetics (mole/L³/T) - K(x, t) Permeability in a capillary tube (L²) - K(x, t) K (x, t)/K₀ Dimensionless permeability - k ₀ Permeability of a capillary tube at t = 0 (L²) - L Length of a capillary tube (L) - m Molecular weight of the reactive species (M/mole) - n Stochiometry of the chemical reaction and kinetic order of the precipitation reaction - P(x, t) Precipitate concentration in a capillary tube (mole/L³) - P _j(t) Concentration in a precipitate element j (mole/L³) - P(_r) (t, C_o) Precipitate concentration in a stirred reactor (mole/L³) - Pr(x, t) Local pressure in a capillary tube - (M/T²/L³) Pr(x, t) Pr(x, t)/Pr(x, 0) Dimensionless local pressure in a capillary tube - Q(t) Flow rate (L³/T) - Q(t) Q(t)U ₀/S₀ Dimensionless flow rate - R(x, t) Chemical flux between the aqueous and the solid phase in a capillary tube (mole/L³/T) - R _i(t) Chemical flux between an aqueous element i and the solid phase (mole/L³/T) - R _(R)[t, C₀] Chemical flux between the aqueous and the solid phase in a stirred reactor (mole/L³/T) - S(x, t) Cross sectional area of a capillary tube accessible to the aqueous phase (L²) - S(x, t) S(x, t)/S₀ Dimensionless cross-sectional area - S ₀ Cross-sectional area of a capillary tube at t = 0 (L²) - tlim(x) Time at which the precipitation front concentration vanishes in the case of zeroth order kinetics (T) - t _max Time of maximum propagation of the precipitation front in the case of zeroth order kinetics (T) - tmin(x) Time at which the precipitation front arrives at x (T) - t _p L/U ₀. Time necessary for an elementary volume of fluid, moving with the velocity U ₀, to reach the oulet of the medium - t _U max Time of maximum value of the velocity field in the case of zeroth order kinetics (T) - t ₀ Time at which an elementary volume of fluid has left the inlet of a capillary tube (T) - t _0m (x, t) Time at which the last elementary volume of fluid has left the inlet of a capillary tube to reach x at a time lower or equal to t (T) - U(x, t) Fluid velocity (L/T) - U(x, t) U(x, t)/U ₀. Dimensionless fluid velocity - U _j(x, t) Fluid velocity defined from the precipitate element j (L/T) - U _l (t₀, t) Lagrangian fluid velocity (L/T) - U _l (t ₀, t) U _l (x, t)/U ₀. Dimensionless lagrangian fluid velocity - U ₀ Velocity of the fluid at t = 0 (L/T) - V _ij(t) Volume, inside a fluid element i, which interacts with a precipitate element j (L³) - x _i(t) Front position of the fluid element i (L) - x _j Front position of the precipitate element j (L) - X _front(t) Position of the precipitation front (L) - x _lim(t) Position of the precipitation front when the value of its concentration is zero (L) - xmax Position of the maximum propagation of the precipitation front in the case of zeroth order kinetics and for high value of C ₀ (L) - Xmin (t) Position of the precipitation front (L) - x _inf* ^supmax Position of the maximum propagation of the precipitation front in the case of zeroth order kinetics and for small value of C_o (L) Greek Symbols t Time step used during the numerical computation (T) - Pro Imposed pressure drop (M/L/T²) - Injection time of reactive species (T) - Density of the precipitate (M/L³) - Dynamic viscosity (M/L/T) - <Ri(t)> _infi ^supj Mean chemical flux between a precipitate element j and all the fluid elements i susceptible to interact with the precipitate element j (mole/L³/T)     

Achieving Rapid Absorption and Extensive Liquid Uptake Capacity in Porous Structures by Decoupling Capillarity and Permeability: Nanoporous Modified Calcium Carbonate

Porous media with rapid absorption properties are greatly sought after in the fields of super absorbers and catalysts. Natural materials, such as diatomite, or synthetic zeolite feature strongly in industrial reaction processes. Most, or all, of such materials, however, are surface acidic. A novel rapidly absorbing alkaline porous structure, with a high absorption capacity, is presented here. As in the case of diatomite or zeolite, the pigment design incorporates strong capillarity within a highly permeable packed medium. A model is proposed for general use with highly absorbing media that can be proven microscopically to have separate domains of micro- or nano-capillarity embedded within a permeable matrix. The new pigment morphology, based on natural ground calcium carbonate (gcc), exhibits this property using special surface structure modifications. It is contrasted with standard gcc by using consolidated tablet blocks made from a suspension of the pigment and chosen mixtures thereof. The blocks are characterised after drying by mercury porosimetry, and the absorption dynamic of a selected liquid is studied. It is shown that using a self-assembly method of discrete pore structures provides a much faster absorption rate and a liquid capacity for up to 10 times more fluid than a conventional homogeneously distributed pore concept. In such unique discrete network systems, the mercury intrusion curve provides a separable analysis of permeability and capillarity in respect to the inflection point of the cumulative intrusion curve. The discrete decoupled properties each follow the absorption behaviour predicted by previous modelling (Ridgway and Gane, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 206(1–3), 2002). The absorption driving force is shown to be determined by the proportion of fine pores present up to a size equal to a Bosanquet inertially-defined optimum within the timescale of absorption. Combining the wetting force, from the capillarity-controlled fine pore structure, with the experimental flow resistance of the sample, consisting of the assembly of particles, it is possible to predict the trends in absorption dynamic using the pore and throat model Pore-Cor.* Use of this model allows existing materials as well as new synthetic designs to be modelled prior to manufacture. The novel alkaline material is compared with independent absorption data for diatomite and shown to be comparable. *Pore-Cor is a software product name of the Environmental and Fluid Modelling Group, University of Plymouth, Devon PL4 8AA, U.K.     

Pore-Scale Analysis of NAPL Blob Dissolution and Mobilization in Porous Media

A pore-scale analysis of nonaqueous phase liquid (NAPL) blob dissolution and mobilization in porous media was presented. Dissolution kinetics of residual NAPLs in an otherwise water-saturated porous medium was investigated by conducting micromodel experiments. Changes in residual NAPL volume were measured from recorded video images to calculate the mass transfer coefficient, K and the lumped mass transfer rate coefficient, k. The morphological characteristics of the blobs such as specific and intrinsic area were found to be independent of water flow rate except at NAPL saturations below 2%. Dissolution process was also investigated by separating the mass transfer into zones of mobile and immobile water. The fractions of total residual NAPL perimeters in contact with mobile water and immobile water were measured and their relationship to the mass transfer coefficient was discussed. In general, residual NAPLs are removed by dissolution and mobilization. Although these two mechanisms were studied individually by others, their simultaneous occurrence was not considered. Therefore, in this study, mobilization of dissolving NAPL blobs was investigated by an analysis of the forces acting on a trapped NAPL blob. A dimensional analysis was performed to quantify the residual blob mobilization in terms of dimensionless Capillary number (Ca _I). If Ca _I is equal to or greater than the trapping number defined as , then blob mobilization is expected.     

Onset of Surface Damage in Modular Orthopedic Implants: Influence of Normal Contact Loading and Stress-assisted Dissolution

Surface damage at interfaces of modular implants results from repeated fretting contacts between metallic surfaces in a corrosive environment. As a first step in understanding this complex process, multi-asperity contact experiments were conducted to characterize roughness evolution due to action of contact loads and exposure to a reactive environment. Cobalt–chromium specimens with surface roughness similar to modular implant were first subjected to only contact loading and subsequently, to alternating contact loads and exposure to reactive environment. During repeated normal contact loading, amplitude of surface roughness reached a steady value after decreasing during the first few cycles. However during the second phase surface roughness amplitude continuously evolved—decreasing during contact loading and increasing on exposure to corrosive environment. The increase in roughness amplitude during surface reaction depended on the magnitude of applied contact loads. A damage mechanism that incorporates contact-induced residual stress development and stress-assisted dissolution is proposed to elucidate the measured surface roughness evolution.     

塔河油田超埋深碳酸盐岩油藏基质的力学试验研究

碳酸盐岩油气藏在全球范围内分布广泛,其中30%以上为缝洞型碳酸盐岩油气藏。在我国缝洞型油藏占已探明的碳酸盐岩油藏储量的2/3,是今后增储的主要领域。为准确揭示碳酸盐岩油藏基质的力学特性,本文以塔河油田奥陶系油藏地层为研究背景,通过现场深孔钻井取样得到埋深达5300~6200m的碳酸盐岩油藏基质岩样,并通过力学试验首次获得超埋深碳酸盐岩油藏基质的弹模、泊松比、抗压强度、抗拉强度、粘聚力、内摩擦角等力学参数,借助电镜扫描试验揭示出超埋深碳酸盐岩的微细观破裂机制。该项研究成果可为分析碳酸盐岩油藏溶洞的垮塌破坏机理提供有效的试验参数。     

K+ uptake by root systems grown in soil under salinity: II. Sensitivity analysis

A numerical algorithm for the solution of multicomponent transport of Ca²⁺, Mg²⁺, Na⁺, K⁺, Cl^– in soil and their uptake by plant roots has been developed. The model emphasizes adsorptiondesorption due to cation exchange mechanism, dissolution-precipitation of CaCO₃, and pH changes at the root surface controlled by the anion-cation influx balance. A fully implicit finite difference scheme is used for numerical implementation. Sensitivity analysis was conducted to evaluate the effect of each parameter on nutrient uptake. Each parameter (independent of all others) was varied between 0.25 to 4 times its speculated average level. Predicted K⁺ uptake was found to be more sensitive to changes of root radius and the parameter indicating maximal influx of K⁺. Effective diffusion coefficient and soil moisture are less influential. The influence of C_aCO₃ dissolution and different kinds of boundary conditions were also considered.Nomenclature A, B, E matrices of coefficients for finite difference scheme - b _i coefficients of equation for H⁺ concentration - C ^I concentration of theI-component in water - C ₀ ^I initial concentration of theI-component - _I ^r0 concentration of theI-component at the inner side of the root surface - C _I ^r1 concentration of theI-component at the external boundary - C _Na ^cr critical concentration for Na⁺ influx into root - CEC cation exchange capacity - D ^*I effective diffusion coefficient of theI-component in soil - F ^I concentration of theI-component on the exchange complex - G vector of coefficients in finite difference scheme - h Hill's cooperativity index for K⁺ influx - h ⁰ value ofh whenC ^Na=0 - J ^I uptake of theI-component by a root length unit - J _I ^r0 influx at the root surface of theI-component - J _max maximal influx of K⁺ - J _max ⁰ value ofJ ^max whenC ^Na=0 - K _a apparent Michaelis-Menten coefficient for K⁺ influx - K _a ⁰ value ofK _a whenC ^Na=0 - K _i selectivity coefficient of the exchange reaction - P _m ^I permeability of root surface for theI-component - P_CO2 partial pressure of CO₂ - r radial distance - r ₀ root radius - r ₁ half the distance between adjacent root (external boundary) - R ^I retardation factor of the-component in mass balance equation for theI-component - t time - t ₀ initial time - T simulation time - v ₀ water radial velocity at the root surface - x _i coordinate of nodes of finite difference mesh Greek coefficient of linear change of K⁺ influx - coefficient of linear change of Na⁺ influx - _s mass density of soil solid phase - soil porosity - volumetric content of liquid in soil - _i coefficients in formulae for parameters of K⁺ influx - parameter of perturbation in finite difference scheme - gg ^I activity coefficient of theI-component - accuracy of iteration convergence - time step for finite difference scheme - steps of finite difference mesh Special Symbols [...] activity symbol     

Nonaqueous Phase Liquid Dissolution in Porous Media: Current State of Knowledge and Research Needs

Our understanding of nonaqueous phase liquid (NAPL) dissolution in the subsurface environment has been increasing rapidly over the past decade. This knowledge has provided the basis for recent developments in the area of NAPL recovery, including cosolvent and surfactant flushing. Despite these advances toward feasible remediation technologies, there remain a number of unresolved issues to motivate environmental researchers in this area. For example, the lack of an effective NAPLlocation methodology precludes effective deployment of NAPL recovery technologies. The objectives of this paper are to critically review the state of knowledge in the area of stationary NAPL dissolution in porous media and to identify specific research needs. The review first compares NAPL dissolutionbased mass transfer correlations reported for environmental systems with more fundamental results from the literature involving model systems. This comparison suggests that our current understanding of NAPL dissolution in smallscale (on the order of cm) systems is reasonably consistent with fundamental mass transfer theory. The discussion then expands to encompass several issues currently under investigation in NAPL dissolution research, including: characterizing NAPL morphology (i.e. effective size and surface area); multicomponent mixtures; scale-related issues (dispersion, flow by-passing); locating NAPL in the subsurface and enhanced NAPL recovery. Research needs and potential approaches are discussed throughout the paper. This review supports the following conclusions: (1) Our knowledge related to local dissolution and remediation issues is maturing, but should be brought to closure with respect to the link between NAPL emplacement theory (as it impacts NAPL morphology) and NAPL dissolution; (2) The role of nonideal NAPL mixtures, and intra-NAPL mass transfer processes must be clarified; (3) Valid models for quantifying and designing NAPL recovery schemes with chemical additives need to be refined with respect to chemical equilibria, mass transfer and chemical delivery issues; (4) Computational and large-scale experimental studies should begin to address parameter up-scaling issues in support of model application at the field scale; and (5) Inverse modeling efforts aimed at exploiting the previous developments should be expanded to support field-scale characterization of NAPL location and strength as a dissolving source.     

Sorbing and Non-Sorbing Solute Migration in Rough Fractures with a Multi-Species LGA Model: Dispersion Dependence on Retardation and Roughness

A multi-species Lattice-Gas cellular automaton model was applied to the study of the migration of sorbing and non-sorbing tracers in 2D smooth fractures and in a series of increasing roughness fractures. A tenfold increase of the dispersion of the non-sorbing tracer was calculated in the highest roughness case compared to the smooth fracture. Up to a threefold increase of the dispersion of the sorbing tracers was calculated compared to the non-sorbing tracer. These enhanced dispersions were found to be of a Fickian form and were interpreted in terms of the classical Taylor–Aris dispersion. The effects of roughness and retardation over the increase of dispersion were identified and quantified through a semi-empirical relation. These effects were found additive and independent.     

考虑二次梯度项及动边界的双重介质低渗透油藏流动分析

在传统试井模型的非线性偏微分方程中根据弱可压缩流体的假设,忽略了二次梯度项,对于低渗透油藏这种方法是有疑问的.低渗透问题一个显著的特点就是流体的流动边界随着时间不断向外扩展.为了更好地研究双重介质低渗透油藏中流体的流动问题,考虑了二次梯度项及活动边界的影响,同时考虑了低渗透油藏的非达西渗流特征,建立了双重介质低渗透油藏流动模型.采用Douglas-Jones预估-校正差分方法获得了无限大地层定产量生产时模型的数值解,分别讨论了不同参数变化时压力的变化规律及活动边界随时间的传播规律,还分析了考虑和忽略二次梯度项影响时模型数值解之间的差异随时间的变化规律,做出了典型压力曲线图版,这些结果可用于实际试井分析.     

Pore-Level Modeling of Gas and Condensate Flow in Two- and Three-Dimensional Pore Networks: Pore Size Distribution Effects on the Relative Permeability of Gas and Condensate

We present a mechanistic model of retrograde condensation processes in two- and three-dimensional capillary tube networks under gravitational forces. Condensate filling-emptying cycles in pore segments and gas connection–isolation cycles are included. With the pore-level distribution of gas and condensate in hand, we determine their corresponding relative permeabilities. Details of pore space and displacement are subsumed in pore conductances. Solving for the pressure field in each phase, we find a single effective conductance for each phase as a function of condensate saturation. Along with the effective conductance for the saturated network, the relative permeability for each phase is calculated. Our model porous media are two- and three-dimensional regular networks of pore segments with distributed size and square cross-section. With a Monte Carlo sampling we find the optimum network size to avoid size effects and then we investigate the effect of network dimensionality and pore size distribution on the relative permeabilities of gas and condensate.     

Dynamic interaction between a sub-interface crack and the interface in a bi-material: time-domain BEM analysis

Summary Transient response of a sub-interface crack in a bi-material is studied with emphasis on the dynamic interaction between the crack and the interface, by combining the traditional time-domain displacement boundary element method (BEM) and the non-hypersingular traction BEM. Computations are performed for an unbounded bi-material with a crack subjected to impact tensile loading on its faces or incident impact waves and a bounded rectangular bi-material plate under remote impact tensile loading. Numerical results of the dynamic stress intensity factors (DSIFs) and dynamic interface tractions are presented for various material combinations and crack locations. It is shown that pronounced increases in DSIFs and the interface tractions may be caused in some cases because of the dynamic interaction between the crack and the interface.This work was initialized during the second author's stay at Institute of Mechanics, TU Darmstadt, Germany under the support of the Alexander von Humboldt Foundation. Discussion on the BEM formulation with Dr. Seelig is gratefully acknowledged. The first two authors are also grateful for the partial support by the China National Natural Science Foundation under Grant No. 10025211 and the NJTU Scientific Paper Fund (PD195).     

Rearrangement of fine particles in porous media causing reduction of permeability and formation of preferred pathways of flow: Experimental findings and a conceptual model

The rearrangement of matrix configuration due to the redistribution of clay particles was studied by introducing different suspensions into porous media. Clay (kaolinite) with low CEC (cation exchange capacity) and small flocs of high CEC clay (smectite) accumulated in regions of slow flow. In some experiments this was followed by rapid accumulation in regions of high velocity, impairing flow to a considerable degree. Clay with high CEC formed more voluminous structures which could bridge over a passage and impair the flow.In either case, whether small or large units of clay are involved, the final flow occurred mainly through open, preferred pathways. Such processes of redistribution and appearance of flow pathways may occur in subsurface porous media aquifers where forced gradients exist (e.g. in aquifers near the perforation of pumping wells and in places where large quantity of water are naturally or artificially recharged and in producing gas and oil fields).     

Validation of the Slow Compressional Wave in Porous Media: Comparison of Experiments and Numerical Simulations

We perform numerical simulation of ultrasonic experiments on poroelastic samples, in which Biot's slow compressional wave had been observed. The simulation is performed using OASES modeling code, which allows to compute elastic wave fields in layered poroelastic media. Modeled were the experiments of Plona (1980), Rasolofosaon (1988), and our own measurements. In all the three situations, a good agreement between experiment and simulations has been observed. This further confirms the fact that Biot's theory of poroelasticity, on which the simulations were based, adequately describes the behavior of the porous materials under investigations at ultrasonic frequencies.     

Macroscopic and microscopic examination of the relationship between crack velocity and path and Rayleigh surface wave speed in single crystal silicon

The speed of Rayleigh surface waves, denoted C_R, is the accepted upper limit for Mode I crack velocity in monolithic solids. In the current contribution, we discuss several critical issues associated with the velocity of Rayleigh surface waves and crack velocity in single crystal (SC) brittle solids, and the global and local influence of C_R on crack path selection in particular.Recent cleavage experiments in SC silicon showed that crack velocity at certain cleavage planes and crystallographic orientations cannot exceed a small fraction of C_R, and thereafter the crack deflects to other cleavage planes. Indeed, C_R defined by the continuum mechanics ignores atomistic phenomena occurring during rapid crack propagation, and therefore is limited in predicting the crack velocity. Examination of these anomalies shows that this limitation lies in microstructural lattice arrangement and in anisotropic phonon radiation during rapid crack propagation. Globally, C_R has no influence on the crack deflection phenomenon. However, the misfit in C_R between the original plane of propagation and the deflected plane generates local instabilities along the deflection zone.     

The effect of pore-structure on hysteresis in relative permeability and capillary pressure: Pore-level modeling

The effect of pore-structure upon two-phase relative permeability and capillary pressure of strongly-wetting systems at low capillary number is simulated. A pore-level model consisting of a network of pore-bodies interconnected by pore-throats is used to calculate scanning loops of hysteresis between primary drainage, imbibition and secondary drainage. The pore-body to pore-throat aspect ratio strongly influences the pattern of hysteresis. Changes in the patterns of hysteresis often attributed to consolidation can be understood in terms of changes in aspect ratio. Correlation between the sizes of neighboring pore-throats affects the shape of the relative permeability curves, while the width and shape of the pore-size distribution have only a minor influence.     

Effect of gap and flow orientation on two-phase flow in an oil-wet gap: Relative permeability curves and flow structures

Naturally fractured reservoirs contain about 25–30% of the world supply of oil. In these reservoirs, fractures are the dominant flow path. Therefore, a good understanding of transfer parameters such as relative permeability as well as flow regimes occurring in a fracture plays an important role in developing and improving oil production from such complex systems. However, in contrast with gas–liquid flow in a single fracture, the flow of heavy oil and water has received less attention. In this research, a Hele-Shaw apparatus was built to study the flow of water in presence of heavy oil and display different flow patterns under different flow rates and analyze the effect of fracture orientations on relative permeability curves as well as flow regimes. The phase flow rates versus phase saturation results were converted to experimental relative permeability curves. The results of the experiments demonstrate that, depending on fracture and flow orientation, there could be a significant interference between the phases flowing through the fracture. The results also reveal that both phases can flow in both continuous and discontinuous forms. The relative permeability curves show that the oil–water relative permeability not only depends on fluid saturations and flow patterns but also fracture orientation.     

Representation of the glass-transition in mechanical and thermal properties of glass-forming materials: A three-dimensional theory based on thermodynamics with internal state variables

In the vicinity of the glass transition, glass-forming materials exhibit pronounced frequency-dependent changes in the mechanical material properties, the thermal expansion behaviour and the specific heat. The frequency dependence becomes apparent under harmonic stress, strain or temperature excitations. The Prigogine-Defay ratio is a characteristic number which connects the changes in magnitude of these quantities at the glass transition. In order to represent the thermoviscoelastic properties of glass-forming materials in continuum mechanics, a three-dimensional approach which is based on the Gibbs free energy as thermodynamic potential is developed in this article. The Gibbs free energy depends on the stress tensor, the temperature and a set of internal variables which is introduced to take history-dependent phenomena into account. In the vicinity of an equilibrium reference state, the specific Gibbs free energy is approximated up to second order terms. Evaluating the Clausius-Duhem inequality, the constitutive relations for the strain tensor, the entropy and the internal variables are derived. In comparison with other approaches, the entropy, the strain tensor and the internal variables are functionals not only of the stress tensor but also of the temperature. Applying harmonic temperature- or stress-controlled excitations, complex frequency-dependent relations for the specific heat under constant stress, for the thermal expansion coefficients as well as for the dynamic mechanical compliance are obtained. The frequency-dependence of these quantities depicts the experimentally observed behaviour of glass-forming materials as published in literature. Under the assumption of isotropic material behaviour, it is shown that the developed theory is compatible with the Prigogine-Defay inequality for arbitrary values of the material parameters.     

Coupling between shallow water and solute flow equations: analysis and management of source terms in 2D

A two‐dimensional model for the simulation of solute transport by convection and diffusion into shallow water flow over variable bottom is presented. It is based on a finite volume method over triangular unstructured grids. A first order upwind technique is applied to solve the flux terms in both the flow and solute equations and the bed slope source terms and a centred discretization is applied to the diffusion and friction terms. The convenience of considering the fully coupled system of equations is indicated and the methodology is well explained. Three options are suggested and compared in order to deal with the diffusion terms. Some comparisons are carried out in order to show the performance in terms of accuracy and computational effort of the different options. Copyright © 2005 John Wiley & Sons, Ltd.