Difference between revisions of "Library of analytic element solutions (under construction)"

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(analytical solutions (Bruggeman,1999 classification))
 
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= analytical solutions =
 
 
 
Categorization is loosely based upon scheme used by Bruggeman (1999)
 
Categorization is loosely based upon scheme used by Bruggeman (1999)
  
== Steady State Confined/Unconfined Flow ==
+
= Steady-state Confined/Unconfined Potential Flow =
  
 
Governing Equation (1D or 2D):  
 
Governing Equation (1D or 2D):  
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The integrated discharge vector may be obtained from the gradient of the potential.
 
The integrated discharge vector may be obtained from the gradient of the potential.
  
=== One-dimensional horizontal flow ===
+
== One-dimensional horizontal flow ==
  
 
Governing Equation:
 
Governing Equation:
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  <math>\Phi(x)=-\frac{1}{2}Nx^2+\left(-\beta+NL\right)x+\Phi_1</math>
 
  <math>\Phi(x)=-\frac{1}{2}Nx^2+\left(-\beta+NL\right)x+\Phi_1</math>
  
=== Two-dimensional radially-symmetric horizontal flow ===
+
== Two-dimensional radially-symmetric horizontal flow ==
  
 
Governing Equation:
 
Governing Equation:
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  <math>\Phi(r)=-\frac{Q}{2\pi}\ln\left(\frac{r}{R}\right)+\Phi_0</math>
 
  <math>\Phi(r)=-\frac{Q}{2\pi}\ln\left(\frac{r}{R}\right)+\Phi_0</math>
  
=== Two-dimensional general horizontal flow ===
+
== Two-dimensional general horizontal flow ==
  
 
Governing Equation:
 
Governing Equation:
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<math>\frac{\partial^2 \Phi}{\partial x^2}+\frac{\partial^2 \Phi}{\partial y^2}=-N</math>
 
<math>\frac{\partial^2 \Phi}{\partial x^2}+\frac{\partial^2 \Phi}{\partial y^2}=-N</math>
  
== Transient Phreatic (Unconfined) groundwater ==
+
= Transient Phreatic (Unconfined) groundwater =
  
 
Governing Equation (1D or 2D):  
 
Governing Equation (1D or 2D):  
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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).
 
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 ===
+
== Exact solutions ==
  
  
  
=== General two-dimensional horizontal flow ===
+
== General two-dimensional horizontal flow ==
  
== Transient Confined groundwater ==
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= Transient Confined groundwater =
  
  
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Where <math>\alpha=\frac{k}{S_s}</math> is the hydraulic diffusivity [<math>LT^{-1}</math>]of the aquifer.  
 
Where <math>\alpha=\frac{k}{S_s}</math> is the hydraulic diffusivity [<math>LT^{-1}</math>]of the aquifer.  
  
=== One-dimensional flow  ===
+
== One-dimensional flow  ==
  
=== Two-dimensional flow ===  
+
== Two-dimensional flow ==
  
=== Two-dimensional radially-symmetric flow ===
+
== Two-dimensional radially-symmetric flow ==
  
 
'''Solution #1 (Transient Theis Solution):'''
 
'''Solution #1 (Transient Theis Solution):'''
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  Where <math>E_1()</math> is the exponential integral (a.k.a. the "well function")
 
  Where <math>E_1()</math> is the exponential integral (a.k.a. the "well function")
  
=== Three-dimensional flow ===
+
== Three-dimensional flow ==
  
=== Three-dimensional spherically-symmetric flow ===
+
== Three-dimensional spherically-symmetric flow ==
  
=== Three-dimensional axially-symmetric flow ===
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== Three-dimensional axially-symmetric flow ==
  
== Multi-layer systems ==
+
= Multi-layer systems =
  
== Dispersion ==
+
= Dispersion =
  
==  Density flow (interface)==
+
=  Density flow (interface)=

Latest revision as of 08:24, 7 December 2007

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:

<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)