Editing Library of analytic element solutions (under construction)

= analytical solutions =

Categorization is loosely based upon scheme used by Bruggeman (1999)

== Steady State Confined/Unconfined Flow ==

Governing Equation (1D or 2D): 

<math>\nabla \cdot \left(k\phi \nabla h\right)=-N </math>

Where <math>\phi</math> is the saturated thickness of the aquifer [<math>L</math>], <math>N</math> is the vertical influx to the aquifer (recharge/leakage) [<math>LT^{-1}</math>], and <math>h</math> [<math>L</math>] is the head  measured with the respect to a flat aquifer base.

A common transformation (the Girinskii potential) can be used to simplified the governing equation, and make it equally valid for both confined flow (<math>\phi=H</math>) and unconfined flow (<math>\phi=h</math>):

<math>
\Phi=\frac{1}{2}k h^2
</math>

if <math>h<H</math>, and 

<math>
\Phi=kHh-\frac{1}{2}kH^2
</math>

if <math>h>H</math>. This results in the Laplace/Poisson equation:

<math>\nabla ^2\Phi=-N </math>

The integrated discharge vector may be obtained from the gradient of the potential.

=== One-dimensional horizontal flow ===

Governing Equation:

<math>\frac{\partial^2 \Phi}{\partial x^2}=-N</math>

'''Solution #1: Steady-state / Two Dirichlet Boundaries''' 
 
 Boundary conditions: 
 
 <math>\Phi(0)=\Phi_1; \frac{}{} \Phi(L)=\Phi_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>

'''Solution #2: Steady-state / One Dirichlet, One Neumann Boundary''' 
 
 Boundary conditions:
 
 <math>\Phi(0)=\Phi_1; \frac{}{} Q_x(L)=\beta</math>
 
 Solution:
 
 <math>\Phi(x)=-\frac{1}{2}Nx^2+\left(-\beta+NL\right)x+\Phi_1</math>

=== Two-dimensional radially-symmetric horizontal flow ===

Governing Equation:

<math>\frac{\partial}{\partial r}\left(r\frac{\partial \Phi}{\partial r}\right)=-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>

=== Two-dimensional general horizontal flow ===

Governing Equation:

<math>\frac{\partial^2 \Phi}{\partial x^2}+\frac{\partial^2 \Phi}{\partial y^2}=-N</math>

== Transient 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).

=== Exact solutions ===



=== General two-dimensional horizontal flow ===

== Transient 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)==