Theoretical Analysis of Transport in Porous Media: Multi-Equation and Hybrid Models for a Generic Transport Problem with Nonlinear Source Terms
Kambiz Vafai · 2015
Many challenges we face today in porous media sciences relate to a crucial need to better describe transport phenomena in heterogeneous multiscale systems. A typical multiscale problem is illustrated in Figure 7.1a, where the pore-scale properties, such as the indicator eld describing the phase geometry, vary rapidly with the spatial coordinates relative to the length scale of the macroscopic domain. This can be interpreted as L. (7.1) This inequality (7.1) implies a signicant numerical and physical complexity. The numerical complexity arises from the necessity to compute coupled processes occurring over a broad spectrum of spatial and temporal scales. The physical complexity follows from the scale dependence of the partial differential equations that are used to describe transport phenomena. For example, Stokes equations at the pore-scale transition to Darcy’s law at the macroscale. Another example is solute advection and diffusion at the pore-scale, which yield dispersion effects at a coarser scale. During solute biodegradation in soils, a stochastic reaction rate at the molecular level may be described via a Monod reaction rate at the cellular scale (due to metabolic limitations), which may become a Monod reaction rate at the biolm scale (but with different parameters encompassing diffusion limitations), which in turn may be described by a rst-order reaction rate at the Darcy scale (e.g., due to low solute concentration). This cascade of reaction rates illustrates the complexity of the problem and is schematized in Figure 7.2.