Girinskii potential function of uniform confined aquifer is defined as
Groundwater motion equation
Where: k is the permeability coefficient of aquifer; B is its thickness; H is the head. The reference datum of water head is the bottom of aquifer. If the water head of the aquifer is lower than its top, it will become an unconfined aquifer with groundwater level, and the Girinskii potential function at this time is defined as
Groundwater motion equation
Where: h is the height of the groundwater level relative to the floor. On the interface of pressureless conversion, there are obviously
Groundwater motion equation
The Girinskii potential function defined in this way satisfies Laplace equation whether there is pressure or no pressure.
Groundwater motion equation
Girinskii potential function is used to solve one-dimensional pressureless steady flow problem. As shown in fig. 2. 10, the problem of pressureless steady flow controlled by fixed head boundary on both sides can be described as follows.
Figure 2. 10 One-dimensional steady flow of confined-unconfined groundwater
Groundwater motion equation
The general solution of the governing equation is
Groundwater motion equation
According to the boundary conditions are
Groundwater motion equation
Therefore, the solution to this pressureless problem is
Groundwater motion equation
The position of pressureless interface is x = xs, and the following conditions shall be met:
Groundwater motion equation
The solution of equation (2.204) is
Groundwater motion equation
This shows that the position of pressureless interface has nothing to do with permeability coefficient.
Girinskii potential function can also be used to solve the problem of axisymmetric confined steady flow, and then the axisymmetric form of equation (2. 199) is needed.
Groundwater motion equation
If there is a pumping well at the origin, there may be a pressure-free zone around the well, and the boundary conditions can be written as follows
Groundwater motion equation
Where: Qw is the production flow of the well; Rw is the radius of the borehole. The general solution of the point source problem can be obtained from equations (2.206) and (2.207).
Groundwater motion equation
Where: C 1 is an integral constant that depends on another boundary condition. Chen et al. (2006) used equation (2.208) to solve the problem of non-pressure steady flow caused by multi-well pumping near the constant head boundary.
For the unconstrained surface flow problem shown in Figure 2. 1 1, you can also refer to the superposition method of real wells and virtual wells shown in Figure 2.9, and express the flow field as
Figure 2. 1 1 Unsteady steady flow caused by pumping well
Groundwater motion equation
Where: r 1 and r2 are the distances between the observation point and the actual well and the virtual well respectively; φ0 is an undetermined constant. On the boundary of constant head, there are
Groundwater motion equation
Therefore, Equation (2.209) can be rewritten as
Groundwater motion equation
Obviously, at the interface position (xs, ys) of the pressureless diaphragm, there are
Groundwater motion equation
According to formula (2.2 12), the boundary of confined-unconfined zoning is a circle on the plane, and its analytical geometric equation is as follows
Groundwater motion equation
In ...
Groundwater motion equation