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IV.B General Formulation

Transport and relaxation processes in a three dimensional two-component porous medium 𝕊=𝕄 (defined above in chapter II) may be formulated very broadly as a system of partial differential equations

F𝕄r1,r2,,u1r,u2r,,uirrj,,2uirrjrk,=0r𝕄 (4.11)

for n unknown functions uir with i=1,,n and r=r1,,rdd. Here the unknown functions ui describe properties of the physical process (such as displacements, velocities, temperatures, pressures, electric fields etc.), and the given functions F,F𝕄 and F depend on a finite number of its derivatives. The function F provides a coupling between the processes in the pore space and in the matrix space. The main difficulty arises from the irregular structure of the boundary. The formulation may be generalized to porous media with more than two components. For a stochastic porous medium the solutions uir of (4.11) depend on the random realization, and one is usually interested the averages uir. Some authors [2, 41] have recently emphasized the difference between continuum descriptions such as (4.11) and discrete descriptions such as network models. The next section will show that discrete formulations arise as approximations and reduce to the continuum description of the same phenomenon in an appropriate limit.