## Groundwater flow equations

Here we will only consider groundwater flows that can be assumed to be well described by Darcy's law (see Chapter 5 for the relevant definitions). Flow will therefore take place from a point of high potential to a point of low potential. The constant of proportionality, which we call the hydraulic conductivity, is normally assumed to be a constant for saturated aquifers (whereas we saw earlier that it can vary over orders of magnitude for unsaturated conditions as a soil or rock dries out). Methods of estimating hydraulic conductivity and a table of typical values for different types of aquifers can be found in Chapter 5.

To address groundwater management issues we need to be able to model the patterns of potential and velocities in an aquifer in response to recharge and pumping. This requires a dynamic model that can be defined by combining Darcy's law with the Fig. 15.3 Definition diagram for flow in and out of a block of aquifer. Darcian fluxes indicated by qx, qy and qz. Block dimensions are Ax, Ay and Az.

Fig. 15.3 Definition diagram for flow in and out of a block of aquifer. Darcian fluxes indicated by qx, qy and qz. Block dimensions are Ax, Ay and Az.

continuity (mass balance) principle. Fig. 15.3 shows an element of saturated earth or rock with sides of length Ax, Ay and Az. Considering water movement in the x, y and z directions, and using the principle of continuity, the following mass balance equality can be written for the difference between inflow and outflow:

Inflow — outflow = Change in storage

Ah qxAyAz + qy AxAz + qz AxAy — qx+AX AyAz — qy+AyAxAz — qz+AZAxAy = Ss a AxAyAz q — qx+Ax) AyAz+(qy—qy+Ay)AxAz+(qz—qz+Az ) AxAy = Ss A AxAyAz

— AqxAyAz — AqyAxAz — Aqz AxAy = Ss AxAyAz Aqx Aqy Aqz_S Ah

Here qx is the Darcian velocity (flux per unit area of aquifer), Ss is called the specific storage, which relates a change in storage to a change in head and Ah!At is the change in potential during a short time step At. Note that if, in any direction (here, e.g. the x direction), the input flux is greater than the output flux, such that storage should be increasing, then the gradient term Aq/Ax is given a negative sign because the Darcian velocity is decreasing in the direction of increasing x, i.e. the gradient is negative. In confined aquifers, the specific storage, Ss, is small and is related to the compressibility of the porous medium and the density of water with changes in head. In unconfined aquifers it will be larger as a result of changes in storage in the unsaturated zone as the water table rises and falls with changes in head.

If the increments in space and time are reduced to infinitely small values, we can write the mass balance equation as a continuous partial differential equation in three dimensions: 