Multidimensional Flow

Flow in a porous medium in two or three dimensions is important in situations such as the production of crude oil from reservoir formations. Thus, it is of interest to consider this situation briefly and to point out some characteristics of the governing equations.

Consider the flow of an incompressible fluid through a two-dimensional porous medium, as illustrated in Fig. 13-2. Assuming that the kinetic energy change is negligible and that the flow is laminar as characterized by Darcy's law, the Bernoulli equation becomes

where the density cancels out if the fluid is incompressible. Equation (13-22) can be applied in both the x and y directions, by taking L = Ax for the x direction and L = Ay for the y direction:

Figure 13-2 Two-dimensional flow in a porous medium.

Ax K 3x and

If Eq. (13-23) is differentiated with respect to x and Eq. (13-24) is differentiated with respect to y and the results are added, assuming p and K to be constant, we get

For an incompressible fluid, the term in parentheses is zero as a result of the conservation of mass (e.g., the microscopic continuity equation). Equation (13-25) can be generalized to three dimensions as

which is called the Laplace equation. The solution of this equation, along with appropriate boundary conditions, determines the potential (e.g., pressure) distribution within the medium. The derivatives of this potential then determine the velocity distribution in the medium [e.g., Eqs. (13-23) and (13-24)]. The Laplace equation thus governs the three-dimensional (potential) flow of an inviscid fluid. Note that the Laplace equation follows from Eq. (13-25) for either an incompressible viscous fluid, by virtue of the continuity equation, or for any flow with negligible viscosity effects (e.g., compressible flow outside the boundary layer near a solid boundary). It is interesting that the same equation governs both of these extreme cases.

The Laplace equation also applies to the distribution of electrical potential and current flow in an electrically conducting medium as well as the temperature distribution and heat flow in a thermally conducting medium. For example, if $ ) E, V ) i, and ยก/K ) re, where re is the electrical resistivity (re = RA/Ax), Eq. (13-22) becomes Ohm's law:

dx dx dy

Also, with $ ) T, V ) q, and K/i ) k, where k is the thermal conductivity, the same equations govern the flow of heat in a thermally conducting medium (e.g., Fourier's law):

By making use of these analogies, electrical analog models can be constructed that can be used to determine the pressure and flow distribution in a porous medium from measurements of voltage and current distribution in a conducting medium, for example. The process becomes more complex, however, when the local permeability varies with position within the medium, which is often the case.

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