Fig. 12.2 Elementary control volume

Consider the control volume as shown in the Fig. 12.1. The dimension of the rectangular parallelepiped box is dx, dy and dz. Let q_x, q_y and q_z  are the volumetric flow per unit area entering in to the control volume through (y - z), (x - z) and (x - y) faces respectively.

The mass flux inflow along x direction is ρqxdydz.

Outflow in the x - direction will be
The excess of inflow over outflow of mass during time interval can be expressed as,

(12.1) (12.2)

Similarly in y and z direction, we have

(12.3)

Total excess of mass inflow over outflow can be expressed as

(12.4)

Now, as per the principle of mass conservation, the excess of mass must be equal to the change in mass during dt.
Change in storage can be written as .
Now,

(12.5)

Where S₀ is the specific storage,
            V_T is the total volume of the porous matrix

Then,

(12.6)

Now rate of change of storage can be written as

(12.7)

Putting this in the flow equation, one can have

(12.8)

Considering water is incompressible (ρ = constant)

(12.9) (12.10)

Now, as per Darcy’s law q_x, q_y and q_z can be written as

(12.11) (12.12) (12.13)

Putting in flow equation

(12.14) (12.15) (12.16)

In case of homogeneous aquifer,

(12.17)