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Fluid flow

Fluid flow in porous materials is of great interest for many practical reasons, including the service life of building materials, petroleum recovery, waste containment, catalysis, filtering, and others [6,25,64]. The continuum equation for calculating flow properties is the Navier-Stokes equation [64]. The Stokes equation is the slow-flow linearized version of the full Navier-Stokes equations. For flow through porous materials, in almost all cases, one is just interested in this slow-flow limit. The Stokes equation, in the steady-state limit, is given by [64]

applies.

There are many different ways to solve the Stokes equations [65], including both finite difference and finite element methods. One way of solving the Stokes equation that is well-adapted to a digital image uses a "marker-and-cell " (MAC) mesh [65]. Figure 8 shows the same image as in Figs. 6 and 7, where the pore phase through which the fluid flows is colored. The nodes indicate where the pressures are determined, and arrows show where the fluid velocities are determined, in the middle of pixel sides [15]. All fluid velocities right at a color-white (pore-solid) boundary are set to zero, so no arrows are shown at these points in Fig. 8. This algorithm is similarly constructed in 3-D [56].

Figure 8: Showing a part of the MAC mesh for the same digital image as was used in Figs. 6 and 7. The pressure is evaluated at the nodes, and the fluid velocities are evaluated at the arrow tips, at pixel boundaries. The colored area is pore space and has fluid in it, and the white areas are solid. Fluid velocities are forced to be zero (no arrows shown) at fluid-solid boundaries.

Darcy's law [64] is found to describe macroscopic flow through porous media:

where k, the permeability, has dimensions of length squared, is the average fluid velocity in the entire volume of the sample (not just the pores), and P is the pressure drop over the sample length L. Darcy's law is a macroscopic equation, obtainable from the Stokes' equation [67], which treats the porous material as a homogeneous material defined by a certain bulk resistance to fluid flow through it. Darcy's law is mathematically analogous to Ohm's law, with µ / k playing the role of the resistivity.

Since permeability has units of length squared, and the conductivity, normalized by the conductivity of the conducting phase, is something like a dimensionless tortuosity, there have been many attempts to generate a length scale from the pore space that can relate the two quantities. The most widely used of these length scales are based on the specific surface area [64], an electrically weighted specific surface area that comes from solutions of Laplace's equation in the conducting pore phase [68], and a length scale based on mercury injection [69]. A common idea has been to find a length scale that correctly weights the parts of the pore space where the fluid actually goes. These length scales are all reviewed and compared on the same set of digital images in Ref. [15].