Page 142 - Analysis and Design of Energy Geostructures

114 Analysis and Design of Energy Geostructures

The above indicates that the velocity field for an incompressible fluid is a solenoi-
dal field, that is a field in which the divergence of the considered variable is equal to
zero at all points in space.

3.12.3 Laplace’s equation
Recalling that Darcy’s law expresses a relationship between the seepage velocity and
the hydraulic gradient, Eq. (3.42) can be rewritten as
r krhÞ 5 0 ð3:43Þ
ð
Assuming the medium to be isotropic allows the writing of the following form of
Laplace’s equation (e.g. for the piezometric head)
2
r h 5 0 ð3:44Þ
2
The quantity r h is as follows in various coordinate systems.
• Cartesian coordinates x, y, z:
2 2 2
2
r h 5 @ h 1 @ h 1 @ h ð3:45Þ
2 2 2
@x @y @z
• Cylindrical coordinates r, θ, z:

2 2 2
2
r h 5 @ h 1 1 @h 1 1 @ h 1 @ h ð3:46Þ
2
2 r @θ 2 2
@r r @r @z
• Spherical coordinates r, θ, φ:
2 1 @ 1 2
2
r h 5 @ h 1 2 @h 1 sinθ @h 1 @ h ð3:47Þ
2
2 r sinθ @θ @θ r sin θ @φ 2
2
2
@r r @r
Eq. (3.44) is associated with steady-state conditions and often represents the basis
for analysis and design considerations.

3.13 Initial and boundary conditions for mass conservation

Analogous considerations to those presented for characterising heat transfer problems
hold for describing mass transfer problems with reference to the initial and boundary
conditions. In this case, the boundary conditions are generally expressed either as a
function of a hydraulic head, H (Dirichlet’s condition) or a flux, @H=@n i (Neumann’s
condition).