Steady-State One-Dimensional Flow: Materials in Parallel
--------------------------------------------------------

Introduction
~~~~~~~~~~~~

For one-dimensional, steady-state, flow through a saturated porous medium with constant properties, 
the general governing differential equation expressing mass conservation and Darcy's law becomes simply

	.. math:: \frac{d^2h}{dx^2} = 0
		:label: ode

where the total head (:math:`h`, [L]) is the sum of pressure head (:math:`P/\rho g`, [L]) 
and elevation (:math:`z`, [L])

	.. math:: h = \frac{P}{\rho g}+z

:math:`\rho` = density [M/L\ :sup:`3`\ ], :math:`g` = gravitational acceleration [L/T\ :sup:`2`\ ], 
and :math:`x` = horizontal distance [L]. The ordinary differential equation :eq:`ode` is easily solved by 
direct integration as

	.. math:: h = C_1 x + C_2
		:label: generalSoln

where the integration constants :math:`C_1` and :math:`C_2` depend on the boundary conditions.

For a simple heterogeneous porous medium composed of two constant-property materials in parallel, 
Equation :eq:`generalSoln` can be applied to each subregion separately. To analyze this 
special case, let the subscripts *1* and *2* denote the two subregions.


Analytic solution for prescribed inlet and outlet pressures
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

When hydraulic head is prescribed at both boundaries as

	.. math:: 
		h(0) &= h_0\\
		h(L) &= h_L
		:label: BCs

the analytic solution :eq:`generalSoln` for hydraulic head in each subregion (:math:`h_i`, [L]) becomes

	.. math:: 
		h_i &= (h_L - h_0) \frac{x}{L} + h_0, i=1,2
		:label: specificSoln

where :math:`L` = domain length [L]. The volumetric flowrate per unit area through a porous medium, 
or Darcy velocity (:math:`U`, [L/T]), is defined by Darcy's law as

	.. math:: U = -\frac{k}{\mu\rho g}\frac{dh}{dx} = -K\frac{dh}{dx}
		:label: DarcyVel

where :math:`k` = intrinsic permeability [L\ :sup:`2`\ ],
:math:`\mu` = viscosity [M/LT], and 
:math:`K` = hydraulic conductivity [L/T]. 
Applying Equation :eq:`DarcyVel` to each subregion using Equation :eq:`specificSoln` yields

	.. math:: 
		U_i &= K_i\frac{h_0 - h_L}{L}, i=1,2
		:label: specificDarcyVel

Note that the hydraulic head and Darcy velocity in each subregion are independent of the properties of
the other subregion.


Amanzi verification test problem
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The analytic solutions for hydraulic head and Darcy velocity can be used to test Amanzi
implementation of prescribed hydraulic head boundary conditions, Darcy's law, and mass conservation
on an elementary problem with discrete heterogeneity. 
To generate numerical results the following specifications are considered:

* Domain

	* :math:`x_{min} = y_{min} = z_{min} = 0`
	* :math:`x_{max} = 100 m, y_{max} = 2 m, z_{max} = 10 m`

* Horizontal flow in the x-coordinate direction

	* no-flow prescribed at the :math:`y_{min}, y_{max}, z_{min}, z_{max}` boundaries
	* prescribed hydraulic head at the x-coordinate boundaries: :math:`h(0) = 20m, h(L) = 19m`

* Material properties:

	* :math:`\rho = 998.2 \: kg/m^3, \mu = 1.002e-3 \: Pa\cdot s, g = 9.807 \: m/s^2` 
	* :math:`K_1 = 1.0 m/d` :math:`(k = 1.1847E-12 m^2)` for :math:`0 m \leqslant y \leqslant 1 m`
	* :math:`K_2 = 10 m/d` :math:`(k = 1.1847E-11 m^2)` for :math:`1 m \leqslant y \leqslant 2 m`

* Model discretization

	* :math:`\Delta x = 5 m, \Delta y = 1 m, \Delta z = 10 m`

For these input specifications, Amanzi simulation output is expected to closely match

	.. math:: h_i = 20m -\frac{x}{100m}, i=1,2
		:label: expectedH

and

	.. math:: 
		U_1 &= 0.01 m/d\\
		U_2 &= 0.1 m/d
		:label: expectedUs

following Equations :eq:`specificSoln` and :eq:`specificDarcyVel`.

Amanzi verification test results
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. include:: table_values.txt

.. plot:: amanzi_linear_materials_parallel_1d.py

