Appendix II

Appendix II
Equation of Flow for Transient Flow Through Deforming Saturated Media

A rigorous development of the equation of flow for transient flow in saturated media must recognize the fact that transient changes in fluid pressure lead to deformations in the granular skeleton of a porous medium, and these deformations imply that the medium, as well as the water, is in motion. This realization creates the need for two refinements to the classical derivation presented by Jacob (1940) and followed in Section 2.11 of this text. First, as recognized by Biot (1955), it is necessary to cast Darcy’s law in terms of the relative velocity of fluid to grains. And second, as recognized by Cooper (1966), it is necessary to consider the conservation of mass for the medium as well as for the fluid in the elemental control volume. One can develop the continuity relationships in one of three ways: (1) by considering a deforming elemental volume in deforming coordinates, (2) by considering a deforming elemental volume in fixed coordinates, or (3) by considering a fixed elemental volume in fixed coordinates. Following Gambolati and Freeze (1973), we will use a fixed elemental volume in fixed coordinates. The approach requires the use of vector notation and the material derivative (total derivative, substantial derivative). If these concepts are not familiar, Aris (1962) and Wills (1958) provide introductory treatments. The development will be presented here for a homogeneous, isotropic medium with hydraulic conductivity K, porosity n, and vertical compressibility \alpha. The same approach is easily adapted to heterogeneous and anisotropic media.

In vector notation, the three-dimensional form of Darcy’s law [Eq. (2.34)] is

\vec{v} = -K\nabla h (A2.1)

where \vec{v} = (v_x, v_y, v_z) is the relative velocity of fluid to grains, and \nabla h = (\partial h/\partial x, \partial h/\partial y, \partial h/\partial z) is the hydraulic gradient. We can expand the vector \vec{v} as

\vec{v} = n(\vec{v}_w - \vec{v}_s) (A2.2)

where \vec{v}_w is the fluid velocity and \vec{v}_s the velocity of the deforming medium.

The equation of state for the water [Eq. (2.47)] is

\rho = \rho_0 e^\beta^p (A2.3)

and that for the soil grains, which are incompressible, is

\rho_s = constant (A2.4)

The equation of continuity for the water is

-\nabla \cdot [np\vec{v}_w] = \frac{\partial}{\partial t}[np] (A2.5)

and that for the soil is

-\nabla \cdot [(1 - n)\rho_s\vec{v}_s] = \frac{\partial}{\partial t}[(1 - n)\rho_s] (A2.6)

In these equations \nabla \cdot is the divergence operator:

\nabla \cdot = \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z}

Expanding Eq. (A2.5), We arrive at

-\rho\nabla \cdot (n\vec{v}_w) - n\vec{v}_w \cdot \nabla\rho = n\frac{\partial\rho}{\partial t} + \rho\frac{\partial n}{\partial t} (A2.7)

If we cancel \rho_s from Eq. (A2.6) and rearrange that equation, we obtain an expression for \partial n/\partial t. This can be substituted in Eq. (A2.7) together with an expression for n\vec{v}_w obtained from Eq. (A2.2). After dividing through by \rho and rearranging, Eq. (A2.7) becomes

-\nabla \cdot \vec{v} - (\frac{v}{\rho}) \cdot \nabla\rho = (\frac{n}{\rho})(\frac{\partial\rho}{\partial t} + \vec{v}_s \cdot \nabla\rho) + \nabla \cdot \vec{v}_s (A2.8)

If we use the material derivative, D/Dt = \partial/\partial t + \vec{v}_w \cdot \nabla, Eq. (A2.8) can be written as

-\nabla \cdot \vec{v} - (\frac{v}{\rho}) \cdot \partial\rho = \frac{n}{\rho}\frac{Dp}{Dt} + \nabla \cdot \vec{v}_s (A2.9)

The first term on the right-hand side of Eq. (A2.9) can be related to the compressibility of water \beta by the relation

\frac{D\rho}{Dt} = \rho\beta \frac{Dp}{Dt} (A2.11)

The material derivative on the right-hand side of Eq. (A2.10) can be replaced by a partial derivative only if the following inequality is satisfied:

\vec{v}_s \cdot \nabla p \ll \frac{\partial p}{\partial t} (A2.12)

Then, substituting Eqs. (A2.1) and (A2.10) in Eq. (A2.9) yields

\nabla \cdot (K\nabla h) = n\beta\frac{\partial p}{\partial t} + \nabla \cdot \vec{v}_s (A2.13)

In a three-dimensional stress field, the grain velocity vector \vec{v}_s = (v_s_x, v_s_y, v_s_z) is related to the deformation (or soil displacement) vector \vec{u} = (u_x, u_y, u_z) by

\vec{v}_s = \frac{D\vec{u}}{Dt} (A2.14)

In a one-dimensional vertical stress field,

v_s_x = v_s_y = u_x = u_y = 0 (A2.15)

If the conditions of Eq. (A2.15) are satisfied, the final term of Eq. (A2.13) can be expanded (Cooper, 1966; Gambolati and Freeze, 1973; Gambolati, 1973a) as

\nabla \cdot \vec{v}_s = \frac{\partial v_s_z}{\partial z} = \frac{\partial}{\partial z}(\frac{Du_z}{Dt}) = \frac{D}{Dt}(\frac{\partial u_z}{\partial z}) = \alpha\frac{Dp}{Dt} (A2.16)

where \alpha is the vertical compressibility of the porous medium. The change of derivative around the central equality is valid for a position vector but not in general. The material derivative in the right-hand expression of Eq. (A2.16) can be replaced by the partial derivative if Eq. (A2.11) is satisfied. In that case, Eq. (A2.13) becomes

\nabla \cdot (K\nabla h) = n\beta\frac{\partial p}{\partial t} + \alpha\frac{\partial p}{\partial t} (A2.17)

Since \rho = \rho g(h - z) and K is a constant, Eq. (A2.17) simplifies to

\nabla^2h = \frac{\rho g(\alpha +n\beta}{K}\frac{\partial h}{\partial t} (A2.18)

Or, recalling that S_s = \rho g(\alpha + n\beta) and expanding the vector notation,

\frac{\partial^2h}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} = \frac{S_s}{K}\frac{\partial h}{\partial t} (A2.19)

Equation (A2.19) is identical to Eq. (2.75) developed by Jacob (1940). The more rigorous development makes it clear that the validity of the classical equation of flow rests on the satisfaction of the inequalities of Eqs. (A2.11) and (A2.12) and the stress condition of Eq. (A2.15). It is unlikely that these conditions are always satisfied. Gambolati (1973b) shows that where the rate of consolidation \vec{v}_s exceeds the rate of percolating fluid \vec{v}_w, as it can in thick clay layers of low permeability and high compressibility, the inequalities may not be satisfied. With regard to the stress condition, the \nabla \cdot \vec{v}_s, term at the end of Eq. (A2.13) is really the protruding tip of an iceberg that relates the three-dimensional flow field to the three-dimensional stress field. Biot (1941, 1955) first exposed the interrelationships and Verruijt (1969) provides a clear derivation. Schifiman et al. (1969) provides a comparison of the classical and Biot approaches, and Gambolati (1974) analyses the range of validity of the classical equation of flow.