in which p is density (massholume); R is the volumetric injection r a t e h i t volume (l/time) ; and 4 is porosity (dimensionless).

terms gives Dividing (2.5) by AxAyAz and rearranging

[(QP>,+A~ - ( Q d X l [(Qdy+ny - (Qdy1 AxAyAz AxAyAz

The Darcy velocity is the flow volume across the element face divided by the area of the face, or

. (2.7) Qx QY QZ %==' ' Q Y = K ' q z = , x n y

Using equations (2.7), and choosing smaller and smaller values of Ax, Ay, A Z and A t , and using the definition of a derivative, equation (2.6) becomes

Final Equations

density changes are small. Under this condition, the spatial derivatives of density on the left side of the equation are generally negligible; whereas, the time derivative on the right side may be related to hydraulic head (see, for example, Davis and Dewiest, 1966). The resulting equation is

For a slightly compressible fluid such as water,

where Ss is the specific storage. Substitution of equations (2.3) into (2.9) gives

a ah a ah a ah ah - (Kxx - ) + - (Kyy - ) + - (Kzz - ) + R = Ss - , ax ax ay ay az az a t

. . . . . (2.10)

which is the unsteady or transient, three-dimensional

ground-water flow equation. It is sometimes called the diffusion equation, and equations of the same form occur in the theories of unsteady flow of heat and electricity. In mathematics, it is classified as a parabolic partial differential equation. Equation (2.10) states that the flow components in the x-, y-, and z-directions plus the sourcehink term must balance the change in storage. Equation (2.10) is often written using a type of mathematical shorthand as

(2.1 1 )

where 0 is the differential operator, the single bar indicates a vector quantity, and the double bar indicates a second-order tensor quantity.

For many problems, the velocity distribution, and hence the hydraulic head distribution, does not change with time; that is, the problem is steady- state. Many regional ground-water flow systems can be represented as a steady-state boundary value problem. For steady flow, a h/a t = 0, so equation (2.10) becomes

a ah a ah a ah - MXx - ) +- ( K ~ ~ - ) + --(K,~ - ) = o , (2.12) ax ax ay ay az az

where for convenience the sourcehink term has been dropped. In mathematics, the steady-state equation is classified as an elliptic partial differential equation.

reduces to For a homogeneous medium, equation (2.12)

Kxx a2h 7 + Kyy a2h - + Kzz - a2h = 0 1

ax aY a z2 (2.13)

and for an isotropic medium, K,, = Kyy = K,, = K, and

a2h a2h a2h ax2 ay2 az2

K ( - + - + - ) = O ,

or dividing (2.14) by hydraulic conductivity

a2h a2h a2h - +-+-=(I, ax2 ay2 a 2

(2.14)

(2.15)

which is Laplace's equation. Note that if we had used the transient equation we would have obtained an Ss/K term, which is called the hydraulic diffusivity .

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