Library of analytic solutions (under construction)

From Aemwiki

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

1 Steady-state Confined/Unconfined Potential Flow
2 Transient Phreatic (Unconfined) groundwater
- 2.1 Exact solutions
- 2.2 General two-dimensional horizontal flow
3 Transient Confined groundwater
4 Multi-layer systems
5 Solute Transport
- 5.1 One-dimensional Transport
- 5.2 Two-dimensional Transport
6 Density flow (interface)

[edit] Steady-state Confined/Unconfined Potential Flow

Governing Equation (1D or 2D):

$\nabla \cdot \left(k\phi \nabla h\right)=-N$

Where $φ$ is the saturated thickness of the aquifer [ $L$ ], $N$ is the vertical influx to the aquifer (recharge/leakage) [ $L T - 1$ ], and $h$ [ $L$ ] 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 ( $φ = H$ ) and unconfined flow ( $φ = h$ ):

$\Phi=\frac{1}{2}k h^2$

if $h < H$ , and

$\Phi=kHh-\frac{1}{2}kH^2$

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

$\nabla ^2\Phi=-N$

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

[edit] One-dimensional horizontal flow

Governing Equation:

$\frac{\partial^2 \Phi}{\partial x^2}=-N$

Solution #1: Steady-state / Two Dirichlet Boundaries

Boundary conditions: 



Solution:

Solution #2: Steady-state / One Dirichlet, One Neumann Boundary

Boundary conditions:



Solution:

[edit] Two-dimensional radially-symmetric horizontal flow

Governing Equation:

$\frac{\partial}{\partial r}\left(r\frac{\partial \Phi}{\partial r}\right)=-N$

Solution #1 (Steady State Thiem Solution):

Boundary conditions: 

 
(i.e., the total flow out at r=0 is equal to the total flow from the well,  $Q$ )



Solution:

[edit] Two-dimensional general horizontal flow

Governing Equation:

$\frac{\partial^2 \Phi}{\partial x^2}+\frac{\partial^2 \Phi}{\partial y^2}=-N$

[edit] Transient Phreatic (Unconfined) groundwater

Governing Equation (1D or 2D):

$\nabla \cdot \left(kh \nabla h\right)=S_y\frac{\partial h}{\partial t}-N$

Where $S y$ is the specific yield [-], $N$ is the vertical influx to the aquifer (recharge/leakage) [ $L T - 1$ ], and $h$ [ $L$ ] is both the head and the saturated thickness (i.e., the head is measured with the respect to a flat aquifer base).

[edit] Exact solutions

[edit] General two-dimensional horizontal flow

[edit] Transient Confined groundwater

Governing Equation (1D or 2D):

$\nabla \cdot \left(kH \nabla h\right)=S_sH\frac{\partial h}{\partial t}-N$

Where $S s$ is the specific storage [-], $N$ is the vertical influx to the aquifer (recharge/leakage) [ $L T - 1$ ], $H$ is the saturated thickness of the aquifer and $h$ [ $L$ ] is the head, measured with respect to an arbitrary datum.

For piecewise constant properties, we can substitute in the confined Girinskii potential ( $Φ = k H h$ ) to obtain the following governing equation:

$\nabla ^2\Phi=\frac{1}{\alpha}\frac{\partial \Phi}{\partial t}-N$

Where $\alpha=\frac{k}{S_s}$ is the hydraulic diffusivity [ $L T - 1$ ]of the aquifer.

[edit] One-dimensional flow

[edit] Two-dimensional flow

[edit] Two-dimensional radially-symmetric flow

Solution #1 (Transient Theis Solution):

Boundary/initial conditions:



 
(i.e., the total flow out at r=0 is equal to the total flow from the well,  $Q$ ). 
Here,  $H (t)$  is the heaviside function, equal to 1 for  $t > 0$ , zero otherwise.
 


Solution: 



Where  $E 1 ()$  is the exponential integral (a.k.a. the "well function")

[edit] Three-dimensional flow

[edit] Three-dimensional spherically-symmetric flow

[edit] Three-dimensional axially-symmetric flow

[edit] Multi-layer systems

[edit] Solute Transport

Governing equation for single aqueous solute, with sorption:

$\frac{ \partial \theta C}{\partial t}=-v(\mathbf{x})\theta\nabla C+\nabla \cdot (\mathbf{\theta D(\mathbf{x})} \nabla C) - R(C) -\frac{\rho_b}{\theta}\frac{\partial q}{\partial t}$

Where $C$ and $q$ are the aqueous and sorbed concentrations of the solute, respectively, $v(\mathbf{x})$ is the velocity vector, $\mathbf{D(\mathbf{x})}$ is the dispersion tensor, $R (C)$ is a general reaction term, $ρ b$ is the bulk dry density of the porous media, and $θ$ is the porosity of the media.

For development of analytical solutions, it is typically assumed that the flow is uniform in the x-direction, the porosity is uniform and constant, sorption may be modeled using a linear retardation factor, $R=1+\frac{\rho_b}{\theta}\frac{\partial q}{\partial C}$ , where $\frac{\partial q}{\partial C}=K_d$ , and the reaction terms are limited to simple first or zeroth order decay and growth. This gives us:

$R\frac{\partial C}{\partial t}=-v_x\frac{\partial C}{\partial x}+D_{xx}\frac{\partial^2 C}{\partial x^2}+D_{yy}\frac{\partial^2 C}{\partial y^2} +D_{zz}\frac{\partial^2 C}{\partial z^2}- \lambda C + \gamma$

Where $λ$ is the first-order decay constant and $γ$ is a zeroth-order growth term. This equation is linear with constant coefficients, and thus amenable to analytical solution using a wide variety of methods.

[edit] One-dimensional Transport

Solution #1: Ogata-Banks Solution

Boundary/initial conditions:




 $λ = γ = 0$ 

Solution: 

  

Where 

note that if  $D = 0$ , then 



Where  $H (x)$  is the Heaviside step function.