Library of analytic element solutions (under construction)
Contents
- 1 analytic and analytic element solutions (Strack,1989 classification)
- 1.1 horizontal confined flow
- 1.2 shallow unconfined flow
- 1.3 combined shallow confined and unconfined flow
- 1.4 shallow unconfined flow with rainfall
- 1.5 shallow interface flow
- 1.6 aquifers with vertically varying hydraulic conductivity
- 1.7 shallow flow in aquifers with clay laminae
- 1.8 shallow interface flow in aquifers with impermeable laminae
- 1.9 shallow semiconfined flow
- 1.10 shallow flow in systems of aquifers separated by leaky layers
- 1.11 transient shallow flow
- 1.12 two-dimensional flow in the vertical plane
- 1.13 three-dimensional flow
- 2 analytical solutions (Bruggeman,1999 classification)
analytic and analytic element solutions (Strack,1989 classification)
horizontal confined flow
one-dimensional flow
radial flow
two-dimensional flow
shallow unconfined flow
one-dimensional flow
two-dimensional flow
combined shallow confined and unconfined flow
one-dimensional flow
radial flow
shallow unconfined flow with rainfall
shallow interface flow
aquifers with vertically varying hydraulic conductivity
shallow flow in aquifers with clay laminae
shallow interface flow in aquifers with impermeable laminae
shallow semiconfined flow
shallow flow in systems of aquifers separated by leaky layers
transient shallow flow
two-dimensional flow in the vertical plane
three-dimensional flow
analytical solutions (Bruggeman,1999 classification)
Phreatic (Unconfined) groundwater
Governing Equation (1D or 2D):
<math>\nabla \cdot \left(kh \nabla h\right)=S_y\frac{\partial h}{\partial t}-N </math>
Where <math>S_y</math> is the specific yield [-], <math>N</math> is the vertical influx to the aquifer (recharge/leakage) [<math>LT^{-1}</math>], and <math>h</math> [<math>L</math>] is both the head and the saturated thickness (i.e., the head is measured with the respect to a flat aquifer base).
A common transformation to linearize the steady state form of the equation is to solve the problem in terms of the Girinskii Potential:
<math>\Phi=\frac{1}{2}k h^2</math>
Leaving us with a revised governing equation (the Poisson equation):
<math>\nabla ^2\Phi=-N </math>
Exact solutions
One-dimensional horizontal flow
Governing Equation:
<math>\frac{\partial}{\partial x}\left(kh\frac{\partial h}{\partial x}\right)=S_s\frac{\partial h}{\partial t}-N</math>
Solution #1: Steady-state / Two Dirichlet Boundaries
Boundary conditions:
<math>h(0)=h_1; \frac{}{} h(L)=h_2</math>
Solution:
<math>\Phi(x)=-\frac{1}{2}Nx^2+\left(\frac{\Phi_2-\Phi_1}{L}+\frac{1}{2}NL\right)x+\Phi_1</math>
Where
<math>\Phi_1=\frac{1}{2}kh_1^2</math> and <math>\Phi_2=\frac{1}{2}kh_2^2</math>
Solution #2: Steady-state / One Dirichlet, One Neumann Boundary
Boundary conditions:
<math>h(0)=h_1; \frac{}{} Q_x(L)=\beta</math>
Solution:
<math>\Phi(x)=-\frac{1}{2}Nx^2+\left(-\beta+NL\right)x+\Phi_1</math>
Where
<math>\Phi_1=\frac{1}{2}kh_1^2</math>
Two-dimensional radial-symmetric horizontal flow
Governing Equation:
<math>\frac{\partial}{\partial r}\left(khr\frac{\partial h}{\partial r}\right)=S_s\frac{\partial h}{\partial t}-N</math>
Solution #1 (Steady State Thiem Solution):
Boundary conditions:
<math>\underset{r\rightarrow 0}{\lim}\int_0^{2\pi} rkh\frac{\partial h}{\partial r}d\theta=\underset{r\rightarrow 0}{\lim}\int_0^{2\pi} rQ_rd\theta=Q</math> (i.e., the total flow out at r=0 is equal to the total flow from the well, <math>Q</math>)
<math>\Phi(R)=\Phi_0\frac{}{}</math>
Solution:
<math>\Phi(r)=-\frac{Q}{2\pi}\ln\left(\frac{r}{R}\right)+\Phi_0</math>
General two-dimensional horizontal flow
Confined groundwater
Governing Equation (1D or 2D):
<math>\nabla \cdot \left(kH \nabla h\right)=S_sH\frac{\partial h}{\partial t}-N </math>
Where <math>S_s</math> is the specific storage [-], <math>N</math> is the vertical influx to the aquifer (recharge/leakage) [<math>LT^{-1}</math>], <math>H</math> is the saturated thickness of the aquifer and <math>h</math> [<math>L</math>] is the head, measured with respect to an arbitrary datum.
For piecewise constant properties, we can substitute in the confined Girinskii potential (<math>\Phi=kHh</math>) to obtain the following governing equation:
<math>\nabla ^2\Phi=\frac{1}{\alpha}\frac{\partial \Phi}{\partial t}-N </math>
Where <math>\alpha=\frac{k}{S_s}</math> is the hydraulic diffusivity [<math>LT^{-1}</math>]of the aquifer.