Library of analytic solutions (under construction)

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

Steady-state Confined/Unconfined Potential 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 #2: Steady-state / One Dirichlet, One Neumann Boundary

Boundary conditions:

<math>\Phi(0)=\Phi_1; \frac{}{} Q_x(L)=\beta</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>)




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>\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>


<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

Solute Transport

Governing equation for single aqueous solute, with sorption:

<math>\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}</math>

Where <math>C</math> and <math>q</math> are the aqueous and sorbed concentrations of the solute, respectively, <math>v(\mathbf{x})</math> is the velocity vector, <math>\mathbf{D(\mathbf{x})}</math> is the dispersion tensor, <math>R(C)</math> is a general reaction term, <math>\rho_b</math> is the bulk dry density of the porous media, and <math>\theta</math> 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, <math>R=1+\frac{\rho_b}{\theta}\frac{\partial q}{\partial C}</math>, where <math>\frac{\partial q}{\partial C}=K_d</math>, and the reaction terms are limited to simple first or zeroth order decay and growth. This gives us:

<math>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 </math>

Where <math>\lambda </math> is the first-order decay constant and <math>\gamma</math> 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.

One-dimensional Transport

Solution #1: Ogata-Banks Solution

Boundary/initial conditions:

<math>\frac{\partial C}{\partial x}(\infty,t)=0\frac{}{}</math>


<math>C(x,t) = C_i+(C_0-C_i)A(x,t) \frac{}{}</math>  

Where <math>A(x,t)=\frac{1}{2}\mathbf{erfc}\left(\frac{Rx-vt}{2\sqrt{DRt}}\right)+\frac{1}{2}\exp\left(\frac{vx}{D}\right)\mathbf{erfc}\left(\frac{Rx+vt}{2\sqrt{DRt}}\right)</math>

note that if <math>D=0</math>, then 


Where <math>H(x)</math> is the Heaviside step function.

Two-dimensional Transport

Solution #1: Cleary and Ungs Solution

Density flow (interface)