Editing Library of analytic element solutions (under construction)
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− | + | = analytical solutions (Bruggeman,1999 classification) = | |
− | = | + | == Phreatic (Unconfined) groundwater == |
Governing Equation (1D or 2D): | Governing Equation (1D or 2D): | ||
− | <math>\nabla \cdot \left( | + | <math>\nabla \cdot \left(kh \nabla h\right)=S_y\frac{\partial h}{\partial t}-N </math> |
− | Where <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). |
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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: | |
− | <math> | + | <math>\Phi=\frac{1}{2}k h^2</math> |
− | \Phi= | ||
− | </math> | ||
− | + | Leaving us with a revised governing equation (the Poisson equation): | |
<math>\nabla ^2\Phi=-N </math> | <math>\nabla ^2\Phi=-N </math> | ||
− | + | === Exact solutions === | |
− | == One-dimensional horizontal flow == | + | === One-dimensional horizontal flow === |
Governing Equation: | Governing Equation: | ||
− | <math>\frac{\partial | + | <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: | Boundary conditions: | ||
− | <math> | + | <math>h(0)=h_1; \frac{}{} h(L)=h_2</math> |
Solution: | 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> | <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: | Boundary conditions: | ||
− | <math> | + | <math>h(0)=h_1; \frac{}{} Q_x(L)=\beta</math> |
Solution: | Solution: | ||
<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> | ||
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+ | Where | ||
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+ | <math>\Phi_1=\frac{1}{2}kh_1^2</math> | ||
− | == Two-dimensional | + | === Two-dimensional radial-symmetric horizontal flow === |
Governing Equation: | Governing Equation: | ||
− | <math>\frac{\partial}{\partial r}\left( | + | <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):''' | '''Solution #1 (Steady State Thiem Solution):''' | ||
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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> | ||
− | == | + | === General two-dimensional horizontal flow === |
− | + | == 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 === |
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− | + | === Two-dimensional flow === | |
− | + | === Two-dimensional radially-symmetric flow === | |
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− | == Three-dimensional flow == | + | === Three-dimensional flow === |
− | == Three-dimensional spherically-symmetric flow == | + | === Three-dimensional spherically-symmetric flow === |
− | == Three-dimensional axially-symmetric flow == | + | === Three-dimensional axially-symmetric flow === |
− | = Multi-layer systems = | + | == Multi-layer systems == |
− | = Dispersion = | + | == Dispersion == |
− | = Density flow (interface)= | + | == Density flow (interface)== |