Liquid Flow (LIQUID_FLOW)

Introduction

The Liquid-Flow-Process models saturated single-phase flow, which can be of variable density, in porous and fractured media. It may be used for modeling water flow but can be equally used to model flow of other liquids and also gas as long as flow follows the Darcy regime. However, the natural application of the Liquid-Flow-Process is in modeling groundwater flow. As such, the documentation above specifically targets hydrogeologists. In the context of groundwater modeling, it is important to note that the pressure-based variant of the groundwater flow equation is implemented in OGS, which may not be the most intuitive for hydrogeologists. To this end, the documentation above aims, among other things, to give conversions between the pressure-based and the hydraulic-head based formulations of the groundwater flow equation as well as it outlines how to use the Liquid-Flow-Process to natively model the hydraulic head-based formulation of the groundwater flow equation commonly used in hydrogeology.

Physical key features

• Fluid flow: liquid (incompressible or compressible) flow; gas (compressible) flow is also supported when it follows the Darcy regime.
• Primary variable: Pressure (in most applications liquid-phase pressure, i.e. pressure of the water phase), in alternative parameterization the pressure can be converted to hydraulic head.
• Secondary variable: Darcy velocity (aka “specific discharge”).

Theoretical background

The LiquidFlow process can solve the fluid-flow equation in two variants.

Mass balance formulation

The more general formulation is the mass-balance version of the equation:

This variant accounts for compressible (density-dependent) flow. It is selected by including the following tag in the project file:

<equation_balance_type>mass</equation_balance_type>

Volume balance formulation

Dividing the mass balance equation by $\rho_w$ and assuming density is constant in space (such that $\nabla \cdot (\rho_w \, \cdot) \approx \rho_w \, \nabla \cdot (\cdot)$) gives the volume-balance form. Substituting the OGS storage coefficient $\beta_s = \partial n_e / \partial p_w$ yields:

In the code, the volume balance mode enforces a constant (pressure-independent) fluid density, i.e. $\partial \rho_w / \partial p_w \equiv 0$. The equation therefore simplifies to:

This is the default formulation. To explicitly select it, set the following tag in the project file:

<equation_balance_type>volume</equation_balance_type>

with the Darcy velocity as secondary variable:

where the subscript $w$ denotes “water”. The following table lists all variables used in the equations:

Please note that in OGS the gravitational acceleration vector is defined as $\mathbf{b} = - \mathbf{g}$. Thus, if one wants to work with the pressure-based variant of groundwater flow, the input for the gravitational acceleration in 3D is:

<specific_body_force> 0 0 -9.81</specific_body_force>

Hydraulic head parameterisation

It is important to note that OGS also has a “storage” coefficient, which is defined as such: $\beta_s = \partial n_e / \partial p_w$. One can show that for equation (1) under the assumption $\partial \rho_w / \partial p_w = 0$ the following relation holds: $\beta_s = S_0/(\rho_w g)$, relating the specific storage of hydrogeology to the storage implemented in OGS.

Consequently, the hydrogeological variant of the groundwater equation with hydraulic head $h$ as process variable can be achieved by applying the following measures:

• Provide $(0,0,0)^T$ as gravitational vector $g$ using the tag <specific_body_force>
• Set $\rho_w = 1$. Please note that OGS will automatically ignore the partial derivative $\partial \rho_w / \partial p_w$ as $\rho_w$ is set to a constant value.
• Set $\mu_w=1$

Then:

• The liquid-phase pressure $p_w$ will be equal to hydraulic head $h$
• The intrinsic permeability $\mathbf{K}_0$ will be the hydraulic conductivity $\mathbf{K}$
• The storage of OGS $\beta_s$ will be equal to the specific storage $S_0$

Thus, one effectively models the groundwater flow equation in the following form:

A detailed derivation of the Liquid-Flow equations and further information on its various formulations can be found in Governing Equations for the LiquidFlow-Process in OpenGeoSys.

Finite Element Discretization

The Liquid-Flow equation is discretized using the finite element method. The discrete system takes the form:

where the element matrices depend on the chosen equation balance type as follows.

Element mass matrix

Element conductance matrix

Element load vector

OGS project prj file

Liquid Flow process has to be declared in the project file prj in the processes block. For example in the following way:

<process>
    <name>LiquidFlow</name>
    <type>LIQUID_FLOW</type>
    <integration_order>2</integration_order>
    <process_variables>
        <process_variable>pressure</process_variable>
    </process_variables>
    <secondary_variables>
        <secondary_variable internal_name="darcy_velocity" output_name="v"/>
    </secondary_variables>
    <specific_body_force>0.0 0.0</specific_body_force>
</process>

Media properties

For Liquid-Flow fluid and porous medium properties need to be specified.

Fluid properties

$\large^{\star}$ functional dependencies can be specified. See the OGS User Guide

Medium properties

$\large^{\star}$ functional dependencies can be specified. See the OGS User Guide

Benchmarks

See the OGS benchmark gallery for Liquid-Flow examples.

Workflows

See the OGSTools documentation for complete workflows (Jupyter Notebooks) covering geometric description, meshing, project generation, simulation, and visualization of results.