I. Information required to solve a groundwater flow problem
Problem domain (geometry)
Parameters (K and Ss or T and S)
IF problem is transient - Initial conditions
II. Examples solution for the 1 dimensional Laplace equation
Make the substitution of variables
and then separate variables to obtain
Integrate without limits of integration
where the C's are constants of integration
Separate variables and integrate a second time
which can be rearranged to yield the general solution
Use the boundary conditions of h = h0 at x=0 and h = hL at x=L to evaluate the constants A1 and A2.
III. Boundary conditions
As described in the text, there are three main types of boundary conditions.
The first type (Dirichlet) boundary condition specifies the value of the dependent variable at the boundary. For groundwater flow this is known as a specified head boundary condition. Special cases are uniform head, in which case the boundary is an equipotential, and constant head, in which case head does not change with time along the boundary.
A second type (Neumann) boundary condition specifies the first derivative of the dependent variable. For groundwater flow, this means specifying dh/dl normal (perpendicular) to the boundary. Specifying the gradient is equivalent to specifying a flux, q, normal to the boundary since q = -K dh/dl. This boundary condition is therefore known as a specified flux boundary. Special conditions include a flux of zero (no-flow boundary or a divide) and constant or uniform flux boundaries.
The third type (Cauchy) boundary condition specifies a function of the dependent variable and its first derivative. An example of how this might be used in a groundwater flow problem is illustrated in the text by a case of a layer of low permeability streambed sediment above an aquifer. Using a third-type boundary allows you to eliminate the low permeability unit from the problem domain, but still make use of the known head in the stream. Other applications of this boundary condition include the "general head" boundaries that are employed in numerical models.