Seepage equation

When we solve the diffusion problem which is given by, e.g., (5.29), we encounter a problem that relates to the evaluation of the mean velocity v (note that dca/dt = dcoi/dt + v gradca,). It may be possible to solve a microscale problem based on the Navier-Stokes equation however, in classical soil mechanics we commonly use the seepage equation to determine v. Using the assumption of incompressibility of a fluid, we can derive the seepage equation from (5.18) and (5.37). 

By substituting (5.54) into (5.51), we finally obtain a seepage equation that includes the volumetric deformation of the solid phase tr Z) as follows ... 

Using an equation of equilibrium or motion, which determines the deformation of the solid skeleton, and (5.58), a system of differential equations for specifying the mean velocity v (i.e., the conventional consolidation problem) is achieved. Note that in (5.58)

total head excluding the velocity potential), k is the hydraulic conductivity tensor, p is the pore pressure of the fluid, g is the gravity constant, and is the datum potential. Thus by starting with the mass conservation laws for both fluid and solid phases, we can simultaneously obtain the diffusion equation and the seepage equation which includes a term that accounts for the volumetric deformation of the porous skeleton. 

Note 5.4 (On the permeability andflow in a porous medium). The seepage equation can be obtained by substituting Darcy s law into the mass conservation equations of fluid and solid phases, as described above. The effects of the micro-structure and microscale material property are put into the hydraulic conductivity k, which is fundamentally specified through experiments. It is not possible to specify the true velocity field by this theory, whereas by applying a homogenization technique, we can determine the velocity field that will be affected by the microscale characteristics. In Chap. 8 we will outline the homogenization theory, which is applied to the problem of water flow in a porous medium, where the microscale flow field is specified. 

By substituting (5.53) into (5.52), we obtain (5.54), which, when substituted into (5.62), gives the following seepage equations for a compressible fluid ... 

As described in Sect. 5.2, the fluid flow problem in a porous medium is given by a seepage equation, which we again show here together with Darcy s law, assuming that the intrinsic solid particles and fluid are incompressible. Note that we will not consider diffusive processes. 

A mass conservation law for water together with the incompressibility condition results in the following seepage equation ... 

By substituting (6.2) and (6.3) into the mass conservation equation (6.1), and considering the incompressibiUty of water (p = constant), we have the following seepage equation, which includes the effect of the volume change of the skeleton ds /dt = trD ... 

This is combined with the seepage equation (6.5), giving Biot s consolidation equation. 

We now introduce incompressibility conditions for the fluid phase and the solid phase (p = constant, p = constant). Under these conditions we have (5.55) and (5.56). Since trO = divv, we can apply Nanson s formula to (5.55), and obtain the following seepage equation for consolidation in the reference configuration ... 

If a homogenization analysis (HA) is applied to porous media flow, which is described by the Stokes equation, we can immediately obtain Darcy s formula and the seepage equation in a macroscale field while in the microscale field the distributions of velocity and pressure are specified (Sanchez-Palencia 1980). We can also apply HA for a problem with a locally varying viscosity. 

Darcy s law (8.22) into the averaged (8.17), we obtain the following homogenized seepage equation, which gives the macroscale incompressibility condition ... 

Furthermore, if we introduce an integral averaging (- j in the micro-domain S2i, we obtain the following macroscale seepage equation ... 

Equation [3-17] does not hold at very low seepage velocities because mechanical dispersion no longer dominates Fickian mass transport. When the mechanical dispersion coefficient becomes less than the effective molecular diffusion coefficient, the longer travel times associated with lower velocities do not result in further decreases in Fickian mass transport. 

The jointed rock is treated as a hydraulically discontinuous mass. According to this model the rock matrix is not permeable and the seepage flow is limited in fractures. Taking into account the contribution of water saturation and seepage pressure, the total strain and constitutive equations of the rock mass was obtained. 

A number of works are devoted to coupling analysis in the tunnel engineering. Huang Tao et al. (1999) reported a study of seepage, temperature and stress field of slope, in which the Qinglin tunnel was excavated. The main equations used are as follows. 

The diffusion in a porous media is affected by the flow field, so we treat here the seepage flow problem in the bentonite starting with the Stokes equation and applying the multiscale HA method. 

Coupling relations between seepage field and deformation field include relations among porosity and permeability change versus stress and pore pressure, as expressed in a exponential function in equation (11) and the effects of flow on mechanical... 

If the dam and its base rock are affected only by gravity and seepage force, which are considered as body forces, and denoting that the tensile stress is positive, the balance differential equations expressed by displacements and general water heads can be given as... 

According to the law of mass conservation, the continuity equation of the water are derived as given in eq. (2). For simplify, the equation ie expressed along the main seepage directions denoted as coordinate axes x, y, z ... 

The basic equations for the coupling analysis of stress and seepage fields are composed of equations (I) and (2), whose boundary conditions include (a) displacement boundary condition < ) = < , (b) stress boundary condition = cr, (where... 

The FEM solution scheme for the coupled equations of displacement and seepage fields is... 

Short-time introduction of a foreign component. Let us assume that. moles of the component i were introduced in water flow momentarily at tg = 0. From this moment on the introduction location, i.e., point = 0, is migrating together with groimd water in the direction of filtration at the seepage velocity V, and the excess of the component i is being dispersed from this mobile origin proportionately to dispersion coefficients. That is why at any moment of time t > in equation (3.45) j4 is equal not 0 but corresponds to the position of point, i.e., equal to VJ . 

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