Editing Library of analytic element solutions (under construction)
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| − | + | = analytic and analytic element solutions (Strack,1989 classification) = | |
| − | = | + | == horizontal confined flow == |
| − | + | === one-dimensional flow === | |
| − | + | === radial flow === | |
| − | + | === two-dimensional flow === | |
| − | + | == shallow unconfined flow == | |
| − | + | === one-dimensional flow === | |
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| − | + | === two-dimensional flow === | |
| − | + | == 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 == | ||
| − | + | == shallow interface flow == | |
| − | == | + | == aquifers with vertically varying hydraulic conductivity == |
| − | + | == shallow flow in aquifers with clay laminae == | |
| − | + | == shallow interface flow in aquifers with impermeable laminae == | |
| − | + | == shallow semiconfined flow == | |
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| − | + | == shallow flow in systems of aquifers separated by leaky layers == | |
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| − | == | + | == transient shallow flow == |
| − | + | == two-dimensional flow in the vertical plane == | |
| − | + | == three-dimensional flow == | |
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| − | = | + | = analytical solutions (Bruggeman,1999 classification) = |
| − | + | == Phreatic groundwater == | |
| − | <math>\ | + | 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>\ | + | <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 === | ||
| − | == | + | === Two-dimensional radial-symmetric horizontal flow === |
| − | = | + | === General two-dimensional horizontal flow === |
| + | == Confined groundwater == | ||
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| − | <math>\nabla \cdot \left(kH \nabla h\right)= | + | Governing Equation (1D or 2D): <math>\nabla \cdot \left(kH \nabla h\right)=S_s\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. | 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. | ||
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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 radial-symmetric flow === | |
| − | + | === General two-dimensional flow === | |
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| − | == Three-dimensional flow == | + | === Three-dimensional spherical flow === |
| − | == Three-dimensional | + | === Three-dimensional axial-symmetric flow === |
| − | == | + | === General three-dimensional flow === |
| − | = Multi-layer systems = | + | == Multi-layer systems == |
| − | = Dispersion = | + | == Dispersion == |
| − | = Density flow (interface)= | + | == Density flow (interface)== |
