Revision as of 11:36, 5 December 2007

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.

One-dimensional flow

Two-dimensional flow

Two-dimensional radially-symmetric flow

Solution #1 (Transient Theis Solution):

Boundary/initial conditions:

<math>\Phi(r,0)=\Phi_0\frac{}{}</math>

<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=QH(t)</math> 
(i.e., the total flow out at r=0 is equal to the total flow from the well, <math>Q</math>). 
Here, <math>H(t)</math> is the heaviside function, equal to 1 for <math>t>0</math>, zero otherwise.
 
<math>\underset{r\rightarrow \infty}{\lim}\Phi(r)=\Phi_0\frac{}{}</math>

Solution: 

<math>\Phi(r,t) = \Phi_0-\frac{Q}{4\pi}E_1\left(\frac{r^2 S_s}{4kt}\right)</math>

Where <math>E_1()</math> is the exponential integral (a.k.a. the "well function")

Three-dimensional flow

Three-dimensional spherically-symmetric flow

Three-dimensional axially-symmetric flow

Multi-layer systems

Dispersion

Density flow (interface)