Porous media stochastic models

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

The classic and stochastic methods used for the analysis of liquid flow inside a porous medium are strongly related. These interactions are given by the relationships between the parameters of both types of models. We show here that the analysis of the flow of a liquid through a porous medium, using a stochastic model, can describe some of the parameters used in deterministic models such as ... 

In the scientific literature, we can find a large quantity of experimental results where the flow characterization inside a porous medium has shown that the value of the dispersion coefficient is not constant. Indeed, for the majority of porous structures the diffusion is frequently a function of the time or of the concentration of the diffusing species. As far as simple stochastic models cannot cover these situations, more complex models have been built to characterize these dependences. One of the first models that gives a response to this problem is recognized as the modd of motion with states having multiple vdodties. 

Network Modeling. Network models have been proposed such that the interconnectivity and the stochastic nature of the porous medium can be investigated [Sharma (80), Sharma and Yortsos (81), and Rege and Fogler (82)]. As mentioned previously, a network representation consists of a number of pore sites interconnected by bonds. In the model by Sharma and Yortsos (81), the bonds have negligible contribution toward the pore volume. Here, population balances are proposed for a... 

Because of the stochastic nature of the system it has long been attempted to model the flow and transport in the fractured rocks as were it taking place in a porous medium with average properties. The Advection-Dispersion equation, AD-equation that has been found useful in beds of small particles such as sand has been used extensively. However, there is increasing concern that it may not be applicable. Observations in drifts and tunnels show that even over distances of hundreds of meters the flowrate distribution is extremely uneven. (Abelin et al.l991). Also the basic., 00 assumption that dispersion can be characterised by a constant dispersion coefficient does not agree with observations. The dispersion coefficient seems to increases with observation distance up to km distances at least (Gelhar et al. 1992). 

---