# 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):

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

Where $\phi$ is the saturated thickness of the aquifer [$L$], $N$ is the vertical influx to the aquifer (recharge/leakage) [$LT^{-1}$], 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 ($\phi=H$) and unconfined flow ($\phi=h$):

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

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.

## One-dimensional horizontal flow

Governing Equation:

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

Solution #1: Steady-state / Two Dirichlet Boundaries

Boundary conditions:

$\Phi(0)=\Phi_1; \frac{}{} \Phi(L)=\Phi_2$

Solution:

$\Phi(x)=-\frac{1}{2}Nx^2+\left(\frac{\Phi_2-\Phi_1}{L}+\frac{1}{2}NL\right)x+\Phi_1$


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

Boundary conditions:

$\Phi(0)=\Phi_1; \frac{}{} Q_x(L)=\beta$

Solution:

$\Phi(x)=-\frac{1}{2}Nx^2+\left(-\beta+NL\right)x+\Phi_1$


## 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:

$\underset{r\rightarrow 0}{\lim}\int_0^{2\pi} rkh\frac{\partial h}{\partial r}d\theta=\underset{r\rightarrow 0}{\lim}\int_0^{2\pi} rQ_rd\theta=Q$
(i.e., the total flow out at r=0 is equal to the total flow from the well, $Q$)

$\Phi(R)=\Phi_0\frac{}{}$

Solution:

$\Phi(r)=-\frac{Q}{2\pi}\ln\left(\frac{r}{R}\right)+\Phi_0$


## Two-dimensional general horizontal flow

Governing Equation:

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

# 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) [$LT^{-1}$], 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).

# 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) [$LT^{-1}$], $H$ 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 ($\Phi=kHh$) 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 [$LT^{-1}$]of the aquifer.

## Two-dimensional radially-symmetric flow

Solution #1 (Transient Theis Solution):

Boundary/initial conditions:

$\Phi(r,0)=\Phi_0\frac{}{}$

$\underset{r\rightarrow 0}{\lim}\int_0^{2\pi} rkH\frac{\partial h}{\partial r}d\theta=\underset{r\rightarrow 0}{\lim}\int_0^{2\pi} rQ_rd\theta=QH(t)$
(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.

$\underset{r\rightarrow \infty}{\lim}\Phi(r)=\Phi_0\frac{}{}$

Solution:

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

Where $E_1()$ is the exponential integral (a.k.a. the "well function")


# 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, $\rho_b$ is the bulk dry density of the porous media, and $\theta$ 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 $\lambda$ is the first-order decay constant and $\gamma$ 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:

$C(x,0)=C_i\frac{}{}$
$C(0,t)=C_0\frac{}{}$
$\frac{\partial C}{\partial x}(\infty,t)=0\frac{}{}$
$\lambda=\gamma=0$

Solution:

$C(x,t) = C_i+(C_0-C_i)A(x,t) \frac{}{}$

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

note that if $D=0$, then

$A(x,t)=H(Rx-vt)\frac{}{}$

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


## Two-dimensional Transport

Solution #1: Cleary and Ungs Solution