Difference between revisions of "Library of analytic element solutions (under construction)"
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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>) | + | <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> | <math>\Phi(R)=\Phi_0\frac{}{}</math> |
Revision as of 11:10, 5 December 2007
Contents
analytical solutions (Bruggeman,1999 classification)
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).
A common transformation to linearize the steady state form of the equation is to solve the problem in terms of the Girinskii Potential:
<math>\Phi=\frac{1}{2}k h^2</math>
Leaving us with a revised governing equation (the Poisson equation):
<math>\nabla ^2\Phi=-N </math>
Exact solutions
One-dimensional horizontal flow
Governing Equation:
<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
Boundary conditions: <math>h(0)=h_1; \frac{}{} h(L)=h_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> Where <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
Boundary conditions: <math>h(0)=h_1; \frac{}{} Q_x(L)=\beta</math> Solution: <math>\Phi(x)=-\frac{1}{2}Nx^2+\left(-\beta+NL\right)x+\Phi_1</math> Where <math>\Phi_1=\frac{1}{2}kh_1^2</math>
Two-dimensional radial-symmetric horizontal flow
Governing Equation:
<math>\frac{\partial}{\partial r}\left(khr\frac{\partial h}{\partial r}\right)=S_s\frac{\partial h}{\partial t}-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>
General two-dimensional horizontal flow
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.