[web] Documentation for the Dupuit implementation.

web/content/docs/benchmarks/liquid-flow/unconfined-aquifer.pandoc

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+date = "2017-02-17T14:33:45+01:00"
+title = "Unconfined Aquifer"
+weight = 171
+project = "Parabolic/LiquidFlow/Unconfined_Aquifer"
+author = "Thomas Kalbacher"
+
+[menu]
+  [menu.benchmarks]
+    parent = "liquid-flow"
+
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+
+{{< data-link >}}
+
+## Problem description
+
+An aquifer is called an unconfined or phreatic aquifer if its upper surface (water level) is accessible to the atmosphere through permeable material. In contrast to a confined aquifer, the groundwater level in an unconfined aquifer does not have a superimposed impermeable rock layer to separate it from the atmosphere.
+
+To simplify the problem of unconfined flow and to make it analytically writeable, Dupuit (1857) used the following assumptions, now commonly referred to as Dupuit-assumptions, in connection with unconfined aquifers:
+
+- the aquifer lower limit is a horizontal plane;
+- the groundwater flow is only horizontal and has no vertical hydraulic component;
+- the horizontal component of the hydraulic gradient is constant with the depth and equal to the slope of the groundwater level;
+- there is no Seepage.
+
+Using the assumptions of Dupuit, Forchheimer (1898) developed a differential equation for the unconfined steady-state case, and Boussinesq introduced the unconfined transient groundwater flow equation in 1904:
+
+$$
+\begin{eqnarray}
+\frac{∂}{∂x}(h\frac{∂h}{∂x})+\frac{∂}{∂y}(h\frac{∂h}{∂y}) = \frac{S_y}{K}\frac{∂h}{∂t}
+\label{Boussinesq}
+\end{eqnarray}
+$$
+
+
+
+where _h[m]_ is the hydraulic head, _S<sub>y</sub>_ is the specific yield and _K_ is the hydraulic conductivity. The Specific Yield _S<sub>y</sub>_, also known as the drainable porosity, is a quantity that is smaller or equal to the effective porosity in a coarse and porous medium. _S<sub>y</sub>_ indicates the volumetric water content that can flow out from the material under the influence of gravity.
+
+The examples shown here are horizontal 2D models parameterized by hydraulic head _h[m]_ .
+
+Since the formulation within OGS is basically based on pressure (_P_) and permeability (_k_), the following relationships must be considered during parameterization in order to be able to work head-based and with hydraulic conductivity (_K_):
+
+$$
+\begin{eqnarray}
+P= ρ g h
+\label{Pressure vs. head}
+\end{eqnarray}
+$$
+
+$$
+\begin{eqnarray}
+K=  \frac{kgh}{μ}
+\label{K & k}
+\end{eqnarray}
+$$
+
+For the model parameterization this means:
+
+- Gravity must be switched off:  		g=0
+- The density gets the value 1:			ρ=1
+- The viscosity gets the value 1:		μ=1
+
+Note: in such a case, the result is also an output of hydraulic head in [m] and not Pressure in [Pa].
+
+
+
+## Examples:
+
+The following simple examples, which have been compared with Modflow simulations, shall verify the result and demonstrate the basic parameterization.
+
+The basic scenario for the two-dimensional unconfined aquifer:
+
+- The area is 9800 m long, 5000 m wide.
+- In the East and West, the boundary is conditioned by &quot;no flow&quot; zones.
+- The material is a homogeneous coarse-grained sand with an isotropic hydraulic conductivity of 160 m/day (0.00185 m/s).
+- The aquifer thickness is 25m.
+
+### Scenario A:
+* Steady-state model.
+* In the north there is a fixed head boundary condition with 15 m.
+* The southern boundary has a fixed head boundary condition with 25 m.
+* the Specific Yield is set to _S<sub>y</sub>_ = 0.00
+{{< img src="../Dupuit_Scenario_A.jpg" >}}
+
+ 
+### Scenario B:
+* Like scenario A and additionally
+* with an average groundwater recharge rate = 3.54745E-09 m/s
+{{< img src="../Dupuit_Scenario_B.jpg" >}}
+
+
+### Scenario C:
+* like scenario A but 
+* with an inflow rate of 4.62963E-05 m3/s per meter at the southern boundary
+{{< img src="../Dupuit_Scenario_C.jpg" >}}
+ 
+
+### Scenario D:
+
+- like scenario A but transient and
+- with a Specific Yield _S<sub>y</sub>_ = 0.25.
+- Simulation time = 100 days.
+{{< img src="../Dupuit_Scenario_D.jpg" >}}
+
+### References:
+
+For more information see e.g.
+
+- _Boussinesq J. Recherches th´eoriques sur l’´ecoulement des nappes d’eau infiltr´ees dans le sol 445 et sur le d´ebit des sources. J. Math´ematiques Pures Appliqu´ees, 10(5–78):363–394, 1904._
+- _Diersch HJG. FEFLOW Finite Element Modeling of Flow, Mass and Heat Transport on Porous Media. Berlin, Heidelberg: Springer-Verlag; 2014. Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media 2014._
+- _Dupuit J. Mouvement de l’eau a travers le terrains permeables. C. R. Hebd. Seances Acad. Sci., 45:92–96, 1857._
+- _Forchheimer P. Über die Ergiebigkeit von Brunnenanalgen und Sickerschlitzen. Z. Architekt. Ing. Ver. Hannover, 32:539–563, 1886._
+- _Forchheimer P. Grundwasserspiegel bei Brunnenanlagen. Z. Osterreichhissheingenieur Architecten Ver, 44:629–635, 1898._
+- _Kolditz, O., Görke, U.-J., Shao, H., Wang, W.: Thermo-Hydro-Mechanical-Chemical Processes in Porous Media - Benchmarks and Examples, Lecture Notes in Computational Science and Engineering, 2012._
+- _Mishra P.K., Kuhlman K.L. (2013) Unconfined Aquifer Flow Theory: From Dupuit to Present. In: Mishra P., Kuhlman K. (eds) Advances in Hydrogeology. Springer, New York, NY 2013._
+
+