Darcy diffusion

Darcy diffusion in shale is a complex issue because water exists in all states from tightly bound, to water of hydration of ions, to fully free water. Pore... 

In principle, all the processes described above can be expressed in the form of fully-coupled differential equations and solved mathematically. It is also possible to achieve coupling though hierarchally nested, iterative and convergent mathematical processes. For a simple Darcy diffusion process, the procedure could be ... 

From Darcy s equation we can determine a formula for the counterforce produced by the porous material to the flowing or diffusing component A, If this counterforce is found, it can be added to the diffusion resistance force caused by component B to component A hence the sum of these two forces represents the total diffusion resistance. 

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

Wilson, D.J., Gomez-Lahoz, C. and Rodriquez-Maroto, J.M., Soil cleanup by in-situ aeration. XVI. Solution and diffusion in mass-transport-limited operation and calculation of Darcy s constants. Sep. Sci. Technol., 29, 1133-1163, 1994. 

Leakage through a synthetic liner is controlled by Fick s first law, which applies to the process of liquid diffusion through the liner membrane. The diffusion process is similar to flow governed by Darcy s law except that it is driven by concentration gradients and not by hydraulic head. Diffusion rates in membranes are very low in comparison with hydraulic flow rates even in clays. In synthetic liners, therefore, the factor that most influences liner performance is penetrations. Synthetic liners may have imperfect seams or pinholes, which can greatly increase the amount of leachate that leaks out of the landfill. 

The two flux equations of importance to subsurface transport are Darcy s law for the advective flow of water and other liquids and Fick s law for the diffusive flow of molecules and gases. These laws are independently discussed below. 

It is often easy to measure the flux density, e.g., using a flowmeter, and then determine the hydraulic conductivity or diffusion coefficient by dividing the flux by the driving force. One of the most difficult problems is determining how to represent the driving force. The symbol V is called an operator, which signifies that some mathematical operation is to be performed upon whatever function follows. V means to take the gradient with respect to distance. For Darcy s law under saturated... 

Dissolution time, tdi (for tablet) Tablet mass, m Diffusivity, D Grain particle size, dp Tablet size, 77 Porosity, e Order-of-magnitude model derived from Fick s and Darcy s laws [6] 2m2 td x2d2pH4Ds(l-s)2... 

With these data and Darcy s law, the in-plane viscous permeabilihes were determined. Only the viscous permeability coefficient was determined because it was claimed that the inertial component was undetectable within the error limits of measuremenf for fhese fesfs. If is imporfanf to mention fhaf fhis technique could also be used to measure fhe permeabilify of diffusion layers wifh different fluids, such as liquid wafer. 

Relative permeability is defined as the ratio between the permeability for a phase at a given saturation level to the total (or single-phase) permeability of the studied material. This parameter is important when the two-phase flow inside a diffusion layer is investigated. Darcy s law (Equation 4.4) can be extended to two-phase flow in porous media [213] ... 

To determine the saturation for any of the models, the capillary pressure must be known at every position within a diffusion medium. Hence, the two-phase models must determine the gas and liquid pressure profiles. In typical two-phase flow in porous media, the movement of both liquid and gas is determined by Darcy s law for each phase and eq 47 relates the two pressures to each other. Many models utilize the capillary pressure functionality as the driving force for the liquid-water flow... 

In addition to the similarity between the heat conduction equation and the diffusion equation, erosion is often described by an equation similar to the diffusion equation (Culling, 1960 Roering et al., 1999 Zhang, 2005a). Flow in a porous medium (Darcy s law) often leads to an equation (Turcotte and Schubert, 1982) similar to the diffusion equation with a concentration-dependent diffu-sivity. Hence, these problems can be treated similarly as mass transfer problems. 

A related approach has been the introduction of maturation vessels constructed out of high density polyethylene (HDPE), in which the tank walls, of a certain thickness (typically 4 mm), are themselves used as the membrane to introduce O2 into the wine (Flecknoe-Brown, 2006). The term "permeation" is used in this context, rather than "diffusion," when the movement is controlled by a pressure difference and described by Darcy s law. "Flextank" maturation vessels have been designed to... 

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

The difference between the solution-diffusion and pore-flow mechanisms lies in the relative size and permanence of the pores. For membranes in which transport is best described by the solution-diffusion model and Fick s law, the free-volume elements (pores) in the membrane are tiny spaces between polymer chains caused by thermal motion of the polymer molecules. These volume elements appear and disappear on about the same timescale as the motions of the permeants traversing the membrane. On the other hand, for a membrane in which transport is best described by a pore-flow model and Darcy s law, the free-volume elements (pores) are relatively large and fixed, do not fluctuate in position or volume on the timescale of permeant motion, and are connected to one another. The larger the individual free volume elements (pores), the more likely they are to be present long enough to produce pore-flow characteristics in the membrane. As a rough rule of thumb, the transition between transient (solution-diffusion) and permanent (pore-flow) pores is in the range 5-10 A diameter. 

In practice we measure changes in pressure and salt concentration, rather than chemical potential, and it can be helpful to think of transport coefficients in terms of Darcy flow and diffusion. We define xs = ns/(nw + ns), where nw, ns are the number of moles of water and salt, and write the change in chemical potential of salt across the membrane as... 

The electrical conductivity of agarose gel increased with increasing water volume fraction, Figure 1. This is attributed primarily to the dependence of ion diffusivity on water volume fraction [14], An empirical model for relative ion diffusivity (D/Do) is proposed to be a function of the hydrodynamic radius ( rs) of solute and intrinsic Darcy permeability (k) of gel ... 

The proposed model consists of a biphasic mechanical description of the tissue engineered construct. The resulting fluid velocity and displacement fields are used for evaluating solute transport. Solute concentrations determine biosynthetic behavior. A finite deformation biphasic displacement-velocity-pressure (u-v-p) formulation is implemented [12, 7], Compared to the more standard u-p element the mixed treatment of the Darcy problem enables an increased accuracy for the fluid velocity field which is of primary interest here. The system to be solved increases however considerably and for multidimensional flow the use of either stabilized methods or Raviart-Thomas type elements is required [15, 10]. To model solute transport the input features of a standard convection-diffusion element for compressible flows are employed [20], For flexibility (non-linear) solute uptake is included using Strang operator splitting, decoupling the transport equations [9],... 

THEORETICAL ANALYSIS OF THE INFLUENCE OF A DIFFUSE DOUBLE-LAYER ON DARCY S LAW... 

Keywords Navier-Stokes equation, diffuse double-layer, Darcy s law... 

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