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)\[U = 1.0 \text{ [m/d]}\]

following Equations (4) and (5).

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

Dar56

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