Steady-State One-Dimensional Flow with Flux and Head Boundary Conditions#
Capabilities Tested#
This one-dimensional, steady-state flow problem tests the Amanzi saturated flow process kernel to simulate flow through a homogeneous, saturated porous medium with constant properties. The analytical solutions for hydraulic head and volumetric flux (derived from Darcy’s Law) can be used to test the Amanzi implementation of prescribed hydraulic head boundary conditions, prescribed flux boundary conditions, Darcy’s law [Dar56], and mass conservation for an elementary problem. Capabilities tested include:
single-phase, one-dimensional flow
steady-state flow
saturated flow
constant-head (Dirichlet) boundary conditions
exactness of numerical scheme for linear head
specified volumetric flux (Neumann) boundary conditions
homogeneous porous medium
uniform mesh
For details on this test, see About.
Background#
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:
(1)#\[\frac{d^2h}{dx^2} = 0\]
where the total head (\(h\), [L]) is the sum oh5ohli& of pressure head (\(P/\rho g\), [L]) and elevation (\(z\), [L]):
\[h = \frac{P}{\rho g}+z\]
where \(\rho\) = density [M/L3], \(g\) = gravitational acceleration [L/T2], and \(x\) = horizontal distance [L]. The ordinary differential equation (1) is easily solved by direct integration as:
(2)#\[h = C_1 x + C_2\]
where the integration constants \(C_1\) and \(C_2\) depend on the boundary conditions.
Model#
The analytic solution for prescribed inlet flow and outlet pressure is given below.
When volumetric flux is prescribed at the inlet boundary \(x = 0\) and hydraulic head at the outlet \(x = L\) as:
(3)#\[\begin{split}U(0) &= U_0\\ h(L) &= h_L\end{split}\]
the analytic solution (2) for hydraulic head becomes:
(4)#\[h = \frac{U_0L}{K} (1 - \frac{x}{L}) + h_L\]
where \(L\) = domain length [L]. For these boundary conditions the volumetric flow rate per unit area, or volumetric flux (\(U\), [L3/L2T]), is constant and defined by Darcy’s law as:
(5)#\[U = -\frac{k}{\mu}\rho g \frac{dh}{dx} = -K\frac{dh}{dx} = -K\frac{-U_0}{K} = U_0\]
where \(k\) = intrinsic permeability [L2], \(\mu\) = viscosity [M/LT], and \(K\) = hydraulic conductivity [L/T].
Problem Specification#
The analytic solutions for hydraulic head and volumetric flux can be used to test Amanzi implementation of prescribed hydraulic head boundary conditions, Darcy’s law, and mass conservation on an elementary problem.
Schematic#
The domain is shown in the following schematic.
One-dimensional, steady-state flow through a saturated porous medium.#
Mesh#
The numerical mesh has dimensions \(100 \text{ m } \times 2 \text{ m } \times 10 \text{ m}\). The mesh is comprised of 20 cells with uniform discretization such that it contains 20 cells in the x-direction, 1 cell in the y-direction, and 1 cell in the z-direction (\(\Delta x = 5 \text{ m}, \: \Delta y = 2 \text{ m}, \: \Delta z = 10 \text{ m}\)).
Variables#
To generate numerical results the following specifications are considered:
Domain:
\(x_{min} = y_{min} = z_{min} = 0\)
\(x_{max} = 100\) m, \(y_{max} = 2\) m, \(z_{max} = 10\) m
Horizontal flow in the x-coordinate direction:
no-flow prescribed at the \(y_{min}, \: y_{max}, \: z_{min}, \: z_{max}\) boundaries
prescribed volumetric flux at the x-coordinate inlet: \(U(0) = 0.01 \text{ [m/d]}\)
prescribed hydraulic head at the x-coordinate outlet: \(h(L) = 19 \text{ [m]}\)
Material properties:
fluid density: \(\rho = 998.2 \: \text{[kg/m}^3\text{]}\)
dynamic viscosity: \(\mu = 1.002 \times 10^{-3} \: \text{[Pa} \cdot \text{s]}\)
gravitational acceleration: \(g = 9.807 \: \text{[m/s}^2\text{]}\)
hydraulic conductivity: \(K = 1.0 \: \text{[m/d]}\) * (permeability: \(k = 1.1847 \times 10^{-12} \text{ [m}^2\text{]})\)
For these input specifications, Amanzi simulation output is expected to closely match
(6)#\[h = 20 -\frac{x}{100m} \text{ [m]}\]
and
(7)#\[U = 1.0 \text{ [m/d]}\]
Results and Comparison#
The discretization is exact for linear solutions, and it is clear in the figure that Amanzi has reproduced the exact solution inside the computationla domain. On the boundary the observation is taken form nearby cell. This will be fixed in the future.
(Source code, png, hires.png, pdf)
This is also visible in the following table:
x [m] |
z [m] |
Analytic [Pa] |
Amanzi [Pa] |
0.0 |
5.0 |
248159.9321 |
247915.2081 |
2.5 |
5.0 |
247915.2081 |
247915.2081 |
47.5 |
5.0 |
243510.1763 |
243510.1763 |
50.0 |
5.0 |
243265.4523 |
243265.4523 |
52.5 |
5.0 |
243020.7283 |
243020.7283 |
97.5 |
5.0 |
238615.6964 |
238615.6964 |
100.0 |
5.0 |
238370.9724 |
238615.6964 |
References#
H. Darcy. Les fontaines publiques de la ville de Dijon: exposition et application des principes a suivre et des formules a employer. Dalmont, 1856.
About#
Directory: testing/verification/flow/saturated/steady-state/linear_flux_head_1d
Authors: Greg Flach (SRNL) and Marc Day (LBNL)
Maintainer(s): David Moulton, moulton@lanl.gov
Input Files:
amanzi_linear_flux_head_1d-u.xml
Spec Version 2.3.0, unstructured mesh framework
mesh: generated internally
Analytic solution computed with golden output
Subdirectory: golden_output
Input Files:
steady-flow_data.h5
Todo
keb: find a schematic for this test
keb: amanzi_linear_flux_head_1d-s.xlm seems like an older format
keb: fix observations on the boundary