Editing Library of analytic element solutions (under construction)

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Categorization is loosely based upon scheme used by Bruggeman (1999)
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= analytic and analytic element solutions (Strack,1989 classification) =
  
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= Steady-state Confined/Unconfined Potential Flow =
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== horizontal confined flow ==
  
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Governing Equation (1D or 2D):
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=== one-dimensional flow ===
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=== radial flow ===
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=== two-dimensional flow ===
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== shallow unconfined flow ==
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=== one-dimensional flow ===
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=== two-dimensional flow ===
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== combined shallow confined and unconfined flow ==
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=== one-dimensional flow ===
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=== radial flow ===
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== shallow unconfined flow with rainfall ==
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== shallow interface flow ==
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== aquifers with vertically varying hydraulic conductivity ==
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== shallow flow in aquifers with clay laminae ==
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== shallow interface flow in aquifers with impermeable laminae ==
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== shallow semiconfined flow ==
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== shallow flow in systems of aquifers separated by leaky layers ==
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== transient shallow flow ==
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== two-dimensional flow in the vertical plane ==
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== three-dimensional flow ==
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= analytical solutions (Bruggeman,1999 classification) =
  
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<math>\nabla \cdot \left(k\phi \nabla h\right)=-N </math>
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== Phreatic (Unconfined) groundwater ==
  
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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.
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Governing Equation (1D or 2D):
  
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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>):
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<math>\nabla \cdot \left(kh \nabla h\right)=S_y\frac{\partial h}{\partial t}-N </math>
  
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<math>
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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).
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\Phi=\frac{1}{2}k h^2
 
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</math>
 
  
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if <math>h<H</math>, and
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A common transformation to linearize the steady state form of the equation is to solve the problem in terms of the Girinskii Potential:
  
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<math>
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<math>\Phi=\frac{1}{2}k h^2</math>
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\Phi=kHh-\frac{1}{2}kH^2
 
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</math>
 
  
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if <math>h>H</math>. This results in the Laplace/Poisson equation:
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Leaving us with a revised governing equation (the Poisson equation):
  
 
<math>\nabla ^2\Phi=-N </math>
 
<math>\nabla ^2\Phi=-N </math>
  
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The integrated discharge vector may be obtained from the gradient of the potential.
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=== Exact solutions ===
  
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== One-dimensional horizontal flow ==
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=== One-dimensional horizontal flow ===
  
 
Governing Equation:
 
Governing Equation:
  
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<math>\frac{\partial^2 \Phi}{\partial x^2}=-N</math>
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<math>\frac{\partial}{\partial x}\left(kh\frac{\partial h}{\partial x}\right)=S_s\frac{\partial h}{\partial t}-N</math>
  
 
'''Solution #1: Steady-state / Two Dirichlet Boundaries'''  
 
'''Solution #1: Steady-state / Two Dirichlet Boundaries'''  
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Boundary conditions:  
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Boundary conditions:
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<math>\Phi(0)=\Phi_1; \frac{}{} \Phi(L)=\Phi_2</math>
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<math>h(0)=h_1; \frac{}{} h(L)=h_2</math>
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Solution:
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Solution:
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<math>\Phi(x)=-\frac{1}{2}Nx^2+\left(\frac{\Phi_2-\Phi_1}{L}+\frac{1}{2}NL\right)x+\Phi_1</math>
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<math>\Phi(x)=-\frac{1}{2}Nx^2+\left(\frac{\Phi_2-\Phi_1}{L}+\frac{1}{2}NL\right)x+\Phi_1</math>
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Where
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<math>\Phi_1=\frac{1}{2}kh_1^2</math> and <math>\Phi_2=\frac{1}{2}kh_2^2</math>
  
 
'''Solution #2: Steady-state / One Dirichlet, One Neumann Boundary'''  
 
'''Solution #2: Steady-state / One Dirichlet, One Neumann Boundary'''  
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Boundary conditions:
 
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<math>\Phi(0)=\Phi_1; \frac{}{} Q_x(L)=\beta</math>
 
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Solution:
 
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<math>\Phi(x)=-\frac{1}{2}Nx^2+\left(-\beta+NL\right)x+\Phi_1</math>
 
  
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== Two-dimensional radially-symmetric horizontal flow ==
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Boundary conditions:
  
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Governing Equation:
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<math>h(0)=h_1; \frac{}{} Q_x(L)=\beta</math>
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Solution:
  
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<math>\frac{\partial}{\partial r}\left(r\frac{\partial \Phi}{\partial r}\right)=-N</math>
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<math>\Phi(x)=-\frac{1}{2}Nx^2+\left(-\beta+NL\right)x+\Phi_1</math>
  
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'''Solution #1 (Steady State Thiem Solution):'''
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Where
  
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Boundary conditions:
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<math>\Phi_1=\frac{1}{2}kh_1^2</math>
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<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>
 
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(i.e., the total flow out at r=0 is equal to the total flow from the well, <math>Q</math>)
 
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<math>\Phi(R)=\Phi_0\frac{}{}</math>
 
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Solution:
 
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<math>\Phi(r)=-\frac{Q}{2\pi}\ln\left(\frac{r}{R}\right)+\Phi_0</math>
 
  
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== Two-dimensional general horizontal flow ==
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=== Two-dimensional radial-symmetric 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>
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<math>\frac{\partial}{\partial r}\left(khr\frac{\partial h}{\partial r}\right)=S_s\frac{\partial h}{\partial t}-N</math>
  
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= Transient Phreatic (Unconfined) groundwater =
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'''Solution #1 (Steady State Thiem Solution):'''
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Governing Equation (1D or 2D):  
 
  
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<math>\nabla \cdot \left(kh \nabla h\right)=S_y\frac{\partial h}{\partial t}-N </math>
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Boundary conditions:
  
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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).
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<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>)
  
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== Exact solutions ==
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<math>\Phi(R)=\Phi_0\frac{}{}</math>
  
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Solution:
  
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<math>\Phi(r)=-\frac{Q}{2\pi}\ln\left(\frac{r}{R}\right)+\Phi_0</math>
  
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== General two-dimensional horizontal flow ==
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=== General two-dimensional horizontal flow ===
  
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= Transient Confined groundwater =
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== 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.  
  
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== One-dimensional flow  ==
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=== One-dimensional flow  ===
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== Two-dimensional flow ==
 
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== Two-dimensional radially-symmetric flow ==
 
  
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'''Solution #1 (Transient Theis Solution):'''
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=== Two-dimensional flow ===
  
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Boundary/initial conditions:
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=== Two-dimensional radially-symmetric flow ===
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<math>\Phi(r,0)=\Phi_0\frac{}{}</math>
 
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<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>
 
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(i.e., the total flow out at r=0 is equal to the total flow from the well, <math>Q</math>).
 
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Here, <math>H(t)</math> is the heaviside function, equal to 1 for <math>t>0</math>, zero otherwise.
 
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<math>\underset{r\rightarrow \infty}{\lim}\Phi(r)=\Phi_0\frac{}{}</math>
 
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Solution:
 
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<math>\Phi(r,t) = \Phi_0-\frac{Q}{4\pi}E_1\left(\frac{r^2 S_s}{4kt}\right)</math>
 
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Where <math>E_1()</math> is the exponential integral (a.k.a. the "well function")
 
  
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== Three-dimensional flow ==
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=== Three-dimensional flow ===
  
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== Three-dimensional spherically-symmetric flow ==
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=== Three-dimensional spherically-symmetric flow ===
  
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== Three-dimensional axially-symmetric flow ==
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=== Three-dimensional axially-symmetric flow ===
  
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= Multi-layer systems =
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== Multi-layer systems ==
  
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= Dispersion =
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== Dispersion ==
  
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=  Density flow (interface)=
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==  Density flow (interface)==

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