A model for solute transport following an abrupt pressure impact in saturated porous media

A mathematical model is developed of an abrupt pressure impact applied to a compressible fluid with solute, flowing through saturated porous media. Nondimensional forms of the macroscopic balance equations of the solute mass and of the fluid mass and momentum lead to dominant forms of these equations. Following the onset of the pressure change, we focus on a sequence of the first two time intervals at which we obtain reduced forms of the balance equations. At the very first time period, pressure is proven to be distributed uniformly within the affected domain, while solute remains unaffected. During the second time period, the momentum balance equation for the fluid conforms to a wave form, while the solute mass balance equation conforms to an equation of advective transport. Fluid's nonlinear wave equation together with its mass balance equation, are separately solved for pressure and velocity. These are then used for the solution of solute's advective transport equation. The 1-D case, conforms to a pressure wave equation, for the solution of fluid's pressure and velocity. A 1-D analytical solution of the transport problem, associates these pressure and velocity with an exponential power which governs solute's motion along its path line.     

Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature

A mathematical model is developed for saturated flow of a Newtonian fluid in a thermoelastic, homogeneous, isotropic porous medium domain under nonisothermal conditions. The model contains mass, momentum and energy balance equations. Both the momentum and energy balance equations have been developed to include a Forchheimer term which represents the interaction at the solid-fluid interface at high Reynolds numbers. The evolution of these equations, following an abrupt change in both fluid pressure and temperature, is presented. Using a dimensional analysis, four evolution periods are distinguished. At the very first instant, pressure, effective stress, and matrix temperature are found to be disturbed with no attenuation. During this stage, the temporal rate of pressure change is linearly proportional to that of the fluid temperature. In the second time period, nonlinear waves are formed in terms of solid deformation, fluid density, and velocities of phases. The equation describing heat transfer becomes parabolic. During the third evolution stage, the inertial and the dissipative terms are of equal order of magnitude. However, during the fourth time period, the fluid's inertial terms subside, reducing the fluid's momentum balance equation to the form of Darcy's law. During this period, we note that the body and surface forces on the solid phase are balanced, while mechanical work and heat conduction of the phases are reduced.     

Contributions to Theoretical/Experimental Developments in Shock Waves Propagation in Porous Media

Macroscopic balance equations of mass, momentum and energy for compressible Newtonian fluids within a thermoelastic solid matrix are developed as the theoretical basis for wave motion in multiphase deformable porous media. This leads to the rigorous development of the extended Forchheimer terms accounting for the momentum exchange between the phases through the solid-fluid interfaces. An additional relation presenting the deviation (assumed of a lower order of magnitude) from the macroscopic momentum balance equation, is also presented. Nondimensional investigation of the phases' macroscopic balance equations, yield four evolution periods associated with different dominant balance equations which are obtained following an abrupt change in fluid's pressure and temperature. During the second evolution period, the inertial terms are dominant. As a result the momentum balance equations reduce to nonlinear wave equations. Various analytical solutions of these equations are described for the 1-D case. Comparison with literature and verification with shock tube experiments, serve as validation of the developed theory and the computer code.A 1-D TVD-based numerical study of shock wave propagation in saturated porous media, is presented. A parametric investigation using the developed computer code is also given.     

Scale-dependent Macroscopic Balance Equations Governing Transport Through Porous Media: Theory and Observations

A theory is developed providing a rational framework for spatial scale- dependent fluid’s flow and heat transfer, and mass of a component migrating with it through porous media. Introducing the assumption of a non-Brownian type motion and referring to asymptotic expansion in powers of a small defined parameter, we develop a novel approach associated with macroscopic balance equations obtained by averaging over a Representative Elementary Volume (REV). We prove that these equations can be decomposed into a primary part that refers to the REV length scale and a secondary part valid at a length scale smaller than that of the corresponding REV length. Further to our previous development, we obtain two general forms of the primary and secondary macroscopic balance equations. One is based on the assumption that the advective flux of the extensive quantity is dominant over that of the dispersive flux, whereas the other disregards this assumption. Moreover we also introduce the primary and secondary macroscopic forms for the fluid heat- transfer equation. Considering a Newtonian fluid, the resulting primary Navier–Stokes equation can vary from a nonlinear wave equation to a drag-dominant equation at the fluid–solid interface (Darcy’s law). The secondary momentum balance equation describes a wave equation governing the concurrent propagation of the intensive momentum and the dispersive momentum flux, deviating from their corresponding average terms. The primary macroscopic fluid heat-transfer equation accounts for advective and dispersive heat fluxes and the secondary macroscopic heat-transfer equation involves the simultaneous advection of heat deviating from its corresponding intensive average quantity. The primary macroscopic solute mass balance equation accounts for advection and hydrodynamic dispersion. The secondary macroscopic component mass balance equation is in the form of pure advection governing migration of the deviation from the average component concentration. At this stage, we focus on establishing the viability of the developed theory. We do this by arguing that field observations of motion at small spatial scales are coherent with the hyperbolic characteristics of the secondary balance equations. Field observations under natural gradient flow conditions show excessive high concentration (average of 50 mg/L) of colloids under land irrigated by sewage effluents. We argue that this displacement of condensed colloidal parcels manifests the theoretical findings for the smaller spatial scale. Further evidence show the accumulation of particles moving behind the front of an emitted shockwave. We consider this as an experimental proof reinforcing the argument that colloidal migration is subject to the action of a shockwave in the fluid and pure advection transport, governed by the respective suggested hyperbolic macroscopic balance equations of fluid momentum and component mass at the smaller spatial scale.     

考虑膨胀力的非饱和介质热-水-应力耦合二维有限元分析

从建立应力平衡方程、水连续性方程、能量守恒方程和弹塑性矩阵入手,使用Galerk in方法,将各控制方程分别在空间域和时间域进行离散,开发出了一个可考虑膨胀力的用于分析非饱和介质中热-水-应力耦合弹塑性问题的二维有限元程序.通过对一个假定的核废料地下处置库的热-水-应力耦合问题的数值计算,比较了无、有膨胀力时的情况,在定性上验证了该程序的正确性.     

A linear dynamic model for a saturated porous medium

A linear isothermal dynamic model for a porous medium saturated by a Newtonian fluid is developed in the paper. In contrast to the mixture theory, the assumption of phase separation is avoided by introducing a single constitutive energy function for the porous medium. An important advantage of the proposed model is it can account for the couplings between the solid skeleton and the pore fluid. The mass and momentum balance equations are obtained according to the generalized mixture theory. Constitutive relations for the stress, the pore pressure are derived from the total free energy accounting for inter-phase interaction. In order to describe the momentum interaction between the fluid and the solid, a frequency independent Biot-type drag force model is introduced. A temporal variable porosity model with relaxation accounting for additional attenuation is introduced for the first time. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be estimated from directly measurable phenomenological parameters. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for the two P waves and one S wave are calculated. The influences of the porosity relaxation coefficient on the velocities and attenuation coefficients of the three waves of the porous medium are discussed in a numerical example.     

Solution of the incompressible mass and momentum equations by application of a coupled equation line solver

This paper describes an iterative technique for solving the coupled algebraic equations for mass and momentum conservation for an incompressible fluid flow. The technique is based on the simultaneous solution for pressure and velocity along lines. In a manner similar to ADI methods for a single variable, the solution domain is entirely swept line-by-line in each co-ordinate direction successively until a converged solution is obtained. The tight coupling between the equations that is guaranteed by the method results in an economical solution of the equation set.     

On pressure boundary conditions for the incompressible Navier-Stokes equations

The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence-free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time-varying divergence-free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well-defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.     

Shock waves in saturated thermoelastic porous media

The objective of this paper is to develop and present the macroscopic motion equations for the fluid and the solid matrix, in the case of a saturated porous medium, in the form of coupled, nonlinear wave equations for the fluid and solid velocities. The nonlinearity in the equations enables the generation of shock waves. The complete set of equations required for determining phase velocities in the case of a thermoelastic solid matrix, includes also the energy balance equation for the porous medium as a whole, as well as mass balance equations for the two phase. In the special case of a rigid solid matrix, the wave after an abrupt change in pressure propagates only through the fluid.     

Neglected transport equations: extended Rankine–Hugoniot conditions and <Emphasis Type="Italic">J</Emphasis> -integrals for fracture

Transport equations in integral form are well established for analysis in continuum fluid dynamics but less so for solid mechanics. Four classical continuum mechanics transport equations exist, which describe the transport of mass, momentum, energy and entropy and thus describe the behaviour of density, velocity, temperature and disorder, respectively. However, one transport equation absent from the list is particularly pertinent to solid mechanics and that is a transport equation for movement, from which displacement is described. This paper introduces the fifth transport equation along with a transport equation for mechanical energy and explores some of the corollaries resulting from the existence of these equations. The general applicability of transport equations to discontinuous physics is discussed with particular focus on fracture mechanics. It is well established that bulk properties can be determined from transport equations by application of a control volume methodology. A control volume can be selected to be moving, stationary, mass tracking, part of, or enclosing the whole system domain. The flexibility of transport equations arises from their ability to tolerate discontinuities. It is insightful thus to explore the benefits derived from the displacement and mechanical energy transport equations, which are shown to be beneficial for capturing the physics of fracture arising from a displacement discontinuity. Extended forms of the Rankine–Hugoniot conditions for fracture are established along with extended forms of J -integrals.     

Numerical solution of the two‐phase expansion of a metastable flashing liquid jet using the dispersion‐controlled dissipative scheme

This paper presents a study of the stationary phenomenon of superheated or metastable liquid jets, flashing into a two‐dimensional axisymmetric domain, while in the two‐phase region. In general, the phenomenon starts off when a high‐pressure, high‐temperature liquid jet emerges from a small nozzle or orifice expanding into a low‐pressure chamber, below its saturation pressure taken at the injection temperature. As the process evolves, crossing the saturation curve, one observes that the fluid remains in the liquid phase reaching a superheated condition. Then, the liquid undergoes an abrupt phase change by means of an oblique evaporation wave. Across this phase change the superheated liquid becomes a two‐phase high‐speed mixture in various directions, expanding to supersonic velocities. In order to reach the downstream pressure, the supersonic fluid continues to expand, crossing a complex bow shock wave. The balance equations that govern the phenomenon are mass conservation, momentum conservation, and energy conservation, plus an equation‐of‐state for the substance. A false‐transient model is implemented using the shock capturing scheme: dispersion‐controlled dissipative (DCD), which was used to calculate the flow conditions as the steady‐state condition is reached. Numerical results with computational code DCD‐2D v1 have been analyzed. Copyright © 2009 John Wiley & Sons, Ltd.     

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研究了埋置于弹性地基内充液压力管道中非线性波的传播. 假设管壁是线弹 性的,地基反力采用Winkler线性地基模型,管中流体为不可压缩理想流体. 假定系统初始 处于内压为$P_0$的静力平衡状态,动态的位移场及内压和流速的变化是叠加在静 力平衡状态上的扰动. 基于质量守恒和动量定理,建立了管壁和流体耦合作用的非 线性运动方程组; 进而用约化摄动法, 在长波近似情况下得到了KdV方程,表征 着系统有孤立波解.     

Macroscopic modelling of transport phenomena in porous media. 2: Applications to mass,momentum and energy transport

In this second paper, the averaging rules presented in Part 1 are employed in order to develop a general macroscopic balance equation and particular equations for mass, mass of a component, momentum and energy, all of a phase in a porous medium domain. These balance equations involve averaged fluxes. Then macroscopic equations are developed for advective, dispersive and diffusive fluxes, all in terms of averaged state variables of the system. These are combined with the macroscopic balance equations to yield field equations that serve as the core of the mathematical models that describe the transport of extensive quantities in a porous medium domain. It is shown that the methodology of averaging leads to a better understanding of the effective stress concept employed in dealing with transport phenomena in deformable porous media.     

Modified incompressible SPH method for simulating free surface problems

An incompressible smoothed particle hydrodynamics (I-SPH) formulation is presented to simulate free surface incompressible fluid problems. The governing equations are mass and momentum conservation that are solved in a Lagrangian form using a two-step fractional method. In the first step, velocity field is computed without enforcing incompressibility. In the second step, a Poisson equation of pressure is used to satisfy incompressibility condition. The source term in the Poisson equation for the pressure is approximated, based on the SPH continuity equation, by an interpolation summation involving the relative velocities between a reference particle and its neighboring particles. A new form of source term for the Poisson equation is proposed and also a modified Poisson equation of pressure is used to satisfy incompressibility condition of free surface particles. By employing these corrections, the stability and accuracy of SPH method are improved. In order to show the ability of SPH method to simulate fluid mechanical problems, this method is used to simulate four test problems such as 2-D dam-break and wave propagation.     

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

We have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered‐mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix‐free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical simulation of viscous flow around unrestrained cylinders

A 2-D analysis is made for the dynamic interactions between viscous flow and one or more circular cylinders. The cylinder is free to respond to the fluid excitation and its motions are part of the solution. The numerical procedure is based on the finite volume discretization of the Navier–Stokes equations on adaptive tri-tree grids which are unstructured and nonorthogonal. Both a fully implicit scheme and a semi-implicit scheme in the time domain have been used for the momentum equations, while the pressure correction method based on the SIMPLE technique is adopted to satisfy the continuity equation. A new upwind method is developed for the triangular and unstructured mesh, which requires information only from two neighbouring cells but is of order of accuracy higher than linear. A new procedure is also introduced to deal with the nonorthogonal term. The pressure on the body surface required in solving the momentum equation is obtained through the Poisson equation in the local cell. Results including flow field, pressure distribution and force are provided for fixed single and multiple cylinders and for an unrestrained cylinder in steady incoming flow with Reynolds numbers at 200 and 500 and in unsteady flow with Keulegan–Carpenter numbers at 5 and 10.     

An explicit transient algorithm for predicting incompressible viscous flows in Arbitrary Geometry

A numerical method for predicting viscous flows in complex geometries has been presented. Integral mass and momentum conservation equations are deploved and these are discretized into algebraic form through numerical quadrature. The physical domain is divided into a number of non-orthogonal control volumes which are isoparametrically mapped on to standard rectangular cells. Numerical integration for unsteady mementum equations is performed over such non-orthogonal cells. The explicitly advanced velocity components obtained from unsteady momentum equations may not necessarily satisfy the mass conservation condition in each cell. Compliance of the mass conservation equation and the consequent evolution of correct pressure distribution are accomplished through an iterative correction of pressure and velocity till divergence-free condition is obtained in each cell. The algorithm is applied on a few test problems, namely, lid-driven square and oblique cavities, developing flow in a rectangular channel and flow over square and circular cylinders placed in rectangular channels. The results exhibit good accuracy and justify the applicability of the algorithm. This Explicit Transient Algorithm for Flows in Arbitrary Geometry is given a generic name EXTRAFLAG.     

Effects of heat generation on g-Jitter induced natural convection flow in a channel with isothermal or isoflux walls

This paper discusses the behavior of g-jitter induced free convection in microgravity under the influence of a transverse magnetic field and in the presence of heat generation or absorption effects for a simple system consisting of two parallel impermeable infinite plates held at four different thermal boundary conditions. The governing equations for this problem are derived on the basis of the balance laws of mass, linear momentum, and energy modified to include the effects of thermal buoyancy, magnetic field and heat generation or absorption as well as Maxwell's equations. The fluid is assumed to be viscous, Newtonian and have constant properties except the density in the body force of the balance of linear momentum equation. The governing equations are solved analytically for the induced velocity and temperature distributions as well as for the electric field and total current for electrically-conducting and insulating walls. This is done for isothermal–isothermal, isoflux–isothermal, isothermal–isoflux and isoflux–isoflux thermal boundary conditions. Graphical results for the velocity amplitude and distribution are presented and discussed for various parametric physical conditions.     

A Phenomenological Approach to Modeling Transport in Porous Media

The objective of this article is to make use of the phenomenological approach to construct models for the transport of extensive quantities, such as mass of a fluid phase, mass of a component of a fluid phase, momentum of a phase and energy, in porous medium domains. Special attention is devoted to express the fluxes of these extensive quantities, especially the non-advective ones, as functions of their relevant driving forces, obeying the principle of minimum entropy production. It is shown that for each extensive quantity, we have a linear diffusive flux term, a non-linear diffusive term, and a dispersive flux term. The latter is shown to be proportional to the velocity squared. In each case, the number of moduli that describe fluid and porous matrix properties is determined. The momentum balance equation for a porous medium domain, which is the “motion equation,” is analyzed and simplified for special cases, leading to Darcy’s law and to Brinkman’s equation.