diff --git a/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_A.jpg b/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_A.jpg new file mode 100644 index 0000000000000000000000000000000000000000..9b705eaec67c60deb82e2ce413deca899dc7eb3a --- /dev/null +++ b/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_A.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4a9f74252b47cf5fd7c7265964b733ea44b81f987b4e705918bf8c4e4e7018d1 +size 46401 diff --git a/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_B.jpg b/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_B.jpg new file mode 100644 index 0000000000000000000000000000000000000000..ab4d983de7bc36c80258fee455eeb18f01f8c148 --- /dev/null +++ b/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_B.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:681deaba2b15af4b99ca031521d10ce175523e17a411a5a6a45855cc20ed22e3 +size 38738 diff --git a/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_C.jpg b/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_C.jpg new file mode 100644 index 0000000000000000000000000000000000000000..37e154b2297cc65eb6ac574dcb291a8e30ca45fd --- /dev/null +++ b/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_C.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:59cbde9560b874a452d2d74a41971b01b4943dd2e8ff762601ad7d026afe7064 +size 39579 diff --git a/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_D.jpg b/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_D.jpg new file mode 100644 index 0000000000000000000000000000000000000000..5a9a8fd9814ad6cfccf417346b71fa6e5aefafd6 --- /dev/null +++ b/web/content/docs/benchmarks/liquid-flow/Dupuit_Scenario_D.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:78d92e328122d3ced975810c49a542a3d1ecb3072ae901d05d932d21b2feefc0 +size 39921 diff --git a/web/content/docs/benchmarks/liquid-flow/unconfined-aquifer.pandoc b/web/content/docs/benchmarks/liquid-flow/unconfined-aquifer.pandoc new file mode 100644 index 0000000000000000000000000000000000000000..39ca6ab967892caf7258e308194f92443baf6c0e --- /dev/null +++ b/web/content/docs/benchmarks/liquid-flow/unconfined-aquifer.pandoc @@ -0,0 +1,118 @@ ++++ +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" + ++++ + +{{< 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_): + +$$ +\begin{eqnarray} +P= Ï g h +\label{Pressure vs. head} +\end{eqnarray} +$$ + +$$ +\begin{eqnarray} +K= \frac{kgh}{μ} +\label{K & k} +\end{eqnarray} +$$ + +For the model parameterization this means: + +- Gravity must be switched off: g=0 +- The density gets the value 1: Ï=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 "no flow" 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. Ãœber 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._ + +