The continuity or mass balance equation for a solute species is a general representation of solute transport in the soil system and accounts for changes in solute concentration with time at any location in the soil. To derive the continuity equation, let us examine the transport of a solute species through a small volume element of a soil. For simplicity, we consider the volume element to be a small rectangular parallepiped with dimensions Δx, Δy, and Δz as shown in Figure 3.1. Assume that Jx is the flux or rate of movement of solute species i in the x direction, that is, the mass of solute entering the face ABCD of the volume element per unit area and time. Therefore, the solute inflow rate, or total solute mass entering into ABCD per unit time, is

Solute inflow rate = J y zx (3.2)

Similarly, if Jx+Δx is the solute flux in the x direction for solute leaving the face EFGH, the total mass of solute leaving EFGH per unit time, that is, the solute outflow rate, is

Solute ouflow rate = J y zx+ x (3.3)

From elementary calculus Jx+Δx can be evaluated (approximately) from

J = J +

J x

∂ (3.4)

where ΜJx/Μx is the rate of change of Jx in the x direction. From Equation 3.4, the net mass of solute flow (inflow minus outflow) per unit time in the volume element from solute movement in the x direction is

Solute inflow rate Solute outflow rate = ( J J ) y z

= J x

x y z

− −

−

∂ ∂

(3.5)

Similarly, the net mass of solute flow per unit time from solute movement in the y direction is

J

y x y zy−

∂ ∂ (3.6)

and from solute movement in the z direction is

J z

x y zz− ∂ ∂ (3.7)

Adding Equations 3.5, 3.6, and 3.7 yields the net mass of solute (inflowoutflow) per unit time for the entire volume element as a result of solute movement in the x, y, and z directions.