Groundwater motion equation
Substituting Equation (6.2) into Equation (6.64) gives:
Groundwater motion equation
If the groundwater is still and u = 0, there are
Groundwater motion equation
At this time, if there is an instantaneous point source at the origin, the analytical solution of equation (6.66) is
Groundwater motion equation
Where: b is the thickness of the aquifer.
If the radial flow of groundwater is a steady flow caused by water injection wells or pumping wells and conforms to the radial steady flow of island confined aquifers, the velocity can be expressed as follows by applying Equation (2.82) and Darcy's Law.
Groundwater motion equation
Where: q is well flow (taking water injection as positive). In this case, the longitudinal dispersion is also a function of the distance r, i.e.
Groundwater motion equation
Substituting Equation (6.68) and Equation (6.69) into Equation (6.64) and ignoring D0, we get
Groundwater motion equation
Next, we study the stable distribution characteristics of solute concentration in the confined aquifer around the island under the action of water injection wells. In this case, the left side of the equal sign of equation (6.70) has a zero change term with respect to time, and the following mathematical model is established:
Groundwater motion equation
Where: rw and R are the radius of water injection well and the radius of isolated island respectively; Cw and C0 are the solute concentrations at the borehole wall of the water injection well and at the boundary of the annular island respectively. The general solution of formula (6.7 1) is
Groundwater motion equation
According to the boundary condition (6.72) and formula (6.73), there are
Groundwater motion equation
If water is injected or pumped into an infinite confined aquifer, the head distribution does not conform to the results of the roundabout model. However, when studying solute transport phenomenon, in order to simplify the equation, we can consider that the velocity distribution when quasi-steady flow occurs satisfies the simple equation (6.68), so that the convection-dispersion process can be approximately described as equation (6.70). Introduce the following transformations
Groundwater motion equation
Equation (6.70) can be rewritten as follows
Groundwater motion equation
At the same time, the initial conditions and boundary conditions are transformed into
Groundwater motion equation
Bear( 1972) transforms equation (6.77) into Airy equation in the form of complex variable function in Laplace space, and derives the analytic solution of complex variable function of the above problem, but the boundary condition used is c (ξ = rw/α l, τ) = CW. Equation (6.79) is a special case of rw→0, and its corresponding analytical solution is
Groundwater motion equation
Where M(v) is a function containing a Bessel function. Bear( 1972) also introduced an approximate solution according to the previous literature, as shown below.
Groundwater motion equation
Where: ra is the average radius of injected water; G=Q/(2π？ B) yes.