Flow and transport in porous media

Table of Contents

• Porous medium properties
• Mixture properties
• Fluid properties
• Fluid-fluid and Fluid-solid interactions
• Transport processes in porous media
• Gas mixing laws
• Porous medium flow models

This section gives a brief introduction to the general physics concepts used for single and multi-phase flow in porous media in many DuMux models. The following description mostly concerns models on the Darcy (homogenized) continuum scale and assumes the existence of a representative elementary volume (REV) for which average quantities are defined. For a more comprehensive treatment of the mathematical modeling and description of the physical processes, we recommend the references [24], [40]. (Before we start, we remark that DuMux can be used to solve general conservation equations such as the Navier-Stokes equations and also support other modeling concepts such as pore-networks models to simulate pore-scale processes. Here, we focus on models in the scope of Darcy's law.)

We start with a few basic definitions and introduce some notation.

Phase. A phase is defined as a continuum having distinct properties (e.g. density and viscosity). If phases are miscible, they contain dissolved portions of the substance of the other phase. Fluid and solid phases are distinguished. The fluid phases have different affinities to the solid phases. The phase, which has a higher affinity to the solid phases is referred to as the (more) wetting phase. In the case of two phases, the less wetting one is called the nonwetting phase.

For compositional multi-phase models, fluid phases may be composed of several components, while the solid phases are assumed to consist exclusively of a single component.

Component. The term component stands for constituents of the phases which can be associated with a unique chemical species or, more generally, with a group of species exploiting similar physical behavior. For example, the figure at the beginning of the section shows shows a water-gas-NAPL system composed of the phases water (subscript $\text{w}$), gas ( $\text{g}$), and NAPL ( $\text{n}$). These phases are composed of the components water (superscript $\text{w}$), the pseudo-component air ( $\text{a}$), and an organic contaminant ( $\text{c}$).

The composition of the components in a phase can influence the phase properties. Furthermore, for mass transfer, the phase behavior is quite different from the component behavior.

Thermodynamic equilibrium. For the non-isothermal, multi-phase, multi-component processes in porous media we state the assumption of local thermodynamic equilibrium. Chemical equilibrium means that the mass/mole fractions of a component in different phases are in equilibrium. Thermal equilibrium assumes the same temperature for all considered phases. Mechanical equilibrium means that the forces at the fluid-fluid and solid-fluid phase interfaces are in balance and the interfaces are not moving. Assuming that one of these conditions hold locally (within an REV) can be used to simplified the governing equations. For instance, it can often be assumed due to slow transport processes in porous media that all phases have the same temperature. In other words, reaching thermal equilibrium in an REV is a process much faster than time scales of interest.

Notation (convention). The subscript index $\alpha$, e.g. $\text{w}$, $\text{n}$, and $\text{g}$ refers to the phase, while the superscript $\kappa$, e.g. $\text{w}$, $\text{a}$, and $\text{c}$ refers to the component. The following symbols are often used in the DuMux documentation of the mathemetical models for flow in porous media.

Symbol	Description	Symbol	Description
$p_{\alpha }$	phase pressure	$\varphi$	porosity
$T$	temperature	$K$	intrinsic permeability tensor
$S_{\alpha }$	phase saturation	$\tau$	tortuosity
$x_{\alpha }^{\kappa }$	mole fraction of component $\kappa$ in phase $\alpha$	$\mathit{g}$	gravitational acceleration
$X_{\alpha }^{\kappa }$	mass fraction of component $\kappa$ in phase $\alpha$	$q_{\alpha }^{\kappa }$	volume source term of $\kappa$ in $\alpha$
${\varrho }_{\text{mol},\alpha }$	molar density of phase $\alpha$	$u_{\alpha }$	specific internal energy
${\varrho }_{\alpha }$	mass density of phase $\alpha$	$h_{\alpha }$	specific enthalpy
$M$	molar mass of a phase or component	$c_{\text{s}}$	specific heat enthalpy
$k_{\text{r}\alpha }$	relative permeability	${\lambda }_{\text{pm}}$	heat conductivity
${\mu }_{\alpha }$	phase viscosity	$q^h$	heat source term
$D_{\alpha }^{\kappa }$	diffusivity of component $\kappa$ in phase $\alpha$	${\mathit{v}}_{\alpha }$	velocity

Porous medium properties

The porosity $\varphi$ is defined as the fraction of the volume occupied by fluids in an REV $V_{\mathrm{fluid}}$ divided by the total volume of the REV $V_{\mathrm{total}}$.

$$\varphi =\frac{V_{\mathrm{fluid}}}{V_{\mathrm{total}}}=1-\frac{V_{\mathrm{solid}}}{V_{\mathrm{total}}}.$$

The intrinsic permeability is a measure on the REV scale of the ease of fluid flow through porous media. It relates the potential gradient and the resulting flow velocity in the Darcy equation. As the porous medium may have a structure leading to preferential flow in certain directions, intrinsic permeability is in general a tensorial quantity $\mathbf{K}$. For isotropic porous media, it can be reduced to a scalar quantity $K$.

Mixture properties

The composition of a phase is described by mass or mole fractions of the components. The mole fraction $x_{\alpha }^{\kappa }$ of component $\kappa$ in phase $\alpha$ is defined as:

$$x_{\alpha }^{\kappa }=\frac{n_{\alpha }^{\kappa }}{\sum _i{n}_{\alpha }^i},$$

where $n_{\alpha }^{\kappa }$ is the number of moles of component $\kappa$ in phase $\alpha$. The mass fraction $X_{\alpha }^{\kappa }$ is defined similarly using the mass of component $\kappa$ in phase $\alpha$ instead of $n_{\alpha }^{\kappa }$, $X_{\alpha }^{\kappa }=\frac{{\mathrm{mass}}_{\alpha }^{\kappa }}{{\mathrm{mass}}_{\alpha }^{total}}$.

The molar mass $M^{\kappa }$ of the component $\kappa$ relates the mass fraction to the mole fraction and vice versa.

Fluid properties

The most important fluid properties to describe fluid flow on the REV scale are density and viscosity.

The density ${\rho }_{\alpha }$ of a fluid phase $\alpha$ is defined as the ratio of its mass to its volume $({\rho }_{\alpha }=\frac{{\mathrm{mass}}_{\alpha }}{{\mathrm{volume}}_{\alpha }})$ while the molar density ${\rho }_{\mathrm{mol},\alpha }$ is defined as the ratio of the number of moles per volume $({\rho }_{\mathrm{mol},\alpha }=\frac{{\mathrm{moles}}_{\alpha }}{{\mathrm{volume}}_{\alpha }})$.

The dynamic viscosity ${\mu }_{\alpha }$ characterizes the resistance of a fluid to flow. As density, it is a fluid phase property. For Newtonian fluids, the dynamic viscosity relates the shear stress ${\tau }_{\mathrm{s}}$ to the velocity gradient:

$${\tau }_{\mathrm{s}}={\mu }_{\alpha }\frac{d{v}_{\alpha ,x}}{dy}.$$

The kinematic viscosity ${\nu }_{\alpha }:=\frac{{\mu }_{\alpha }}{{\rho }_{\alpha }}$ is more often used in the description of incompressible systems.

Both density and viscosity generally depend on pressure, temperature, and phase composition.

Fluid-fluid and Fluid-solid interactions

If more than a single fluid is present in the porous medium, the fluids interact with each other and the solids, which leads to additional properties for multi-phase systems.

The saturation $S_{\alpha }$ of a phase $\alpha$ is defined as the ratio of the volume occupied by that phase to the total pore volume within an REV. As all pores are filled with some fluid, the sum of the saturations of all present phases is equal to one.

Capillary pressure. Immiscible fluids form a sharp interface as a result of differences in their intermolecular forces translating into different adhesive and cohesive forces at the fluid-fluid and fluid-fluid-solid interfaces creating interfacial tension on the microscale. From the mechanical equilibrium which has also to be satisfied at the interface, a difference between the pressures of the fluid phases results defined as the capillary pressure (phase difference pressure) $p_{\mathrm{c}}$:

$$p_{\mathrm{c}}=p_{\mathrm{n}}-p_{\mathrm{w}}.$$

On the microscale, $p_{\mathrm{c}}$ can be calculated from the surface tension according to the Laplace equation.

On the REV scale, however, capillary pressure needs to be defined by quantities of that scale. Several empirical relations provide expressions to link $p_{\mathrm{c}}$ to the wetting-phase saturation $S_{\mathrm{w}}$. An example is the relation given by Brooks and Corey [18] to determine $p_{\mathrm{c}}$ based on $S_{\mathrm{e}}$, which is the effective wetting-phase saturation, the entry pressure $p_{\mathrm{d}}$, and the parameter $\lambda$ describing the pore-size distribution:

$$p_{\mathrm{c}}=p_{\mathrm{d}}{S}_{\mathrm{e}}^{-\frac{1}{\lambda }},S_{\mathrm{e}}=\frac{S_{\mathrm{w}}-S_{\mathrm{w},\mathrm{r}}}{1-S_{\mathrm{w},\mathrm{r}}},$$

where $S_{\mathrm{w},\mathrm{r}}$ is the residual wetting phase saturation which cannot be displaced by another fluid phase and remains in the porous medium.

Relative permeability. The presence of two fluid phases in the porous medium reduces the space available for flow for each of the fluid phases. This increases the resistance to flow of the phases, which is accounted for by the means of the relative permeability $k_{\mathrm{r},\alpha }\le 0$, which scales the intrinsic permeability. The relative permeability strongly and nonlinearly depends on the saturation. The relations describing the relative permeabilities of the wetting and nonwetting phase are different as the wetting phase predominantly occupies small pores and the edges of larger pores while the nonwetting phases occupies large pores. The relative permeabilities for the wetting phase $k_{\mathrm{r},\mathrm{w}}$ and the nonwetting phase in a two-fluid-phase system can be modeled, for instance following Brooks and Corey [18], by

$$k_{\mathrm{r},\mathrm{w}}=S_{\mathrm{e}}^{\frac{2+3\lambda }{\lambda }},k_{\mathrm{r},\mathrm{n}}={(1-S_{\mathrm{e}})}^2(1-S_{\mathrm{e}}^{\frac{2+\lambda }{\lambda }}).$$

Also see Dumux::FluidMatrix::BrooksCorey for where these constitutive relations are implemented in DuMux.

Transport processes in porous media

On the macro-scale, mass transport can be characterized by the driving force of the transport process. Pressure gradients result in the advective transport of a fluid phase and all the components constituting the phase, while concentration gradients result in the diffusion of a component within a phase.

Advective transport is determined by the flow field. On the macro-scale, the Darcy or filter velocity $\mathbf{v}$ is calculated using the Darcy equation depending on the potential gradient $(\mathrm{\nabla }{p}_{\alpha }-{\rho }_{\alpha }\mathbf{g})$, accounting for both pressure difference and gravitation, the intrinsic permeability of the porous medium, and the viscosity $\mu$ of the fluid phase:

$$\mathbf{v}=-\frac{\mathbf{K}}{\mu }(\mathrm{\nabla }p-\rho \mathbf{g}).$$

$\mathbf{v}$ is proportional to $(\mathrm{\nabla }p-\rho \mathbf{g})$ with the proportional factor $\frac{\mathbf{K}}{\mu }$. This equation can be extended to multi-phase flow by considering phase velocities ${\mathbf{v}}_{\alpha }$ of phase $\alpha$ and modeling phase interaction through the relative permeability $k_{\mathrm{r},\alpha }$,

$${\mathbf{v}}_{\alpha }=-\frac{k_{\mathrm{r},\alpha }\mathbf{K}}{{\mu }_{\alpha }}(\mathrm{\nabla }{p}_{\alpha }-{\rho }_{\alpha }\mathbf{g})$$

Molecular diffusion is a process determined by the concentration gradient. It is commonly modeled as Fickian diffusion following Fick's first law:

$${\mathbf{j}}_{\mathbf{d}}=-{\rho }_{\alpha }{D}_{\alpha }^{\kappa }\mathrm{\nabla }{X}_{\alpha }^{\kappa },$$

where $D_{\alpha }^{\kappa }$ is the molecular diffusion coefficient of component $\kappa$ in phase $\alpha$. In a porous medium, the actual path lines are tortuous due to the impact of the solid matrix. This tortuosity and the impact of the presence of multiple fluid phases is accounted for by using an effective diffusion coefficient $D_{\mathrm{pm},\alpha }^{\kappa }$:

$$D_{\mathrm{pm},\alpha }^{\kappa }=\varphi {\tau }_{\alpha }{S}_{\alpha }{D}_{\alpha }^{\kappa },$$

where ${\tau }_{\alpha }$ is the tortuosity of phase $\alpha$ (see Dumux::DiffusivityConstantTortuosity).

Gas mixing laws

Prediction of the $p$- $\varrho$- $T$ behavior of gas mixtures is typically based on one of two concepts: Dalton's law or Amagat's law. Both laws make the same predictions for ideal gases but differ for non-ideal gas mixtures. In the following the two concepts will be explained in more detail.

Dalton's law assumes that the gases in the mixture are non-interacting (with each other) and each gas independently applies its own pressure (partial pressure), the sum of which is the total pressure:

$$p=\sum _i^{}{p}_i.$$

Here $p_i$ refers to the partial pressure of component i. As an example, if two equal volumes of gas A and gas B are mixed, the volume of the mixture stays the same but the pressures add up (see figure)

The density of the mixture, $\varrho$, can be calculated as follows:

$$\varrho =\frac{m}{V}=\frac{m_{\mathrm{A}}+m_{\mathrm{B}}}{V}=\frac{{\varrho }_{\mathrm{A}}V+{\varrho }_{\mathrm{B}}V}{V}={\varrho }_{\mathrm{A}}+{\varrho }_{\mathrm{B}},$$

or for an arbitrary number of gases:

$$\varrho =\sum _i^{}{\varrho }_i,{\varrho }_m=\sum _i^{}{\varrho }_{m,i}.$$

Amagat's law assumes that the volumes of the component gases are additive. The interactions of the different gases are the same as the average interactions of the components. This is known as Amagat's law:

$$V=\sum _i^{}{V}_i.$$

As an example, if two volumes of gas A and B at equal pressure are mixed, the pressure of the mixture stays the same, but the volumes add up.

The density of the mixture, $\varrho$, can be calculated as follows:

$$\varrho =\frac{m}{V}=\frac{m}{V_{\mathrm{A}}+V_{\mathrm{B}}}=\frac{m}{\frac{m_{\mathrm{A}}}{{\varrho }_{\mathrm{A}}}+\frac{m_{\mathrm{B}}}{{\varrho }_{\mathrm{B}}}}=\frac{m}{\frac{X_{\mathrm{A}}m}{{\varrho }_{\mathrm{A}}}+\frac{X_{\mathrm{B}}m}{{\varrho }_{\mathrm{B}}}}=\frac{1}{\frac{X_{\mathrm{A}}}{{\varrho }_{\mathrm{A}}}+\frac{X_{\mathrm{B}}}{{\varrho }_{\mathrm{B}}}},$$

or for an arbitrary number of gases:

$$\varrho =\frac{1}{\sum _i^{}\frac{X_i}{{\varrho }_i}},{\varrho }_m=\frac{1}{\sum _i^{}\frac{x_i}{{\varrho }_{m,i}}}.$$

Ideal gases. An ideal gas is defined as a gas whose molecules are spaced so far apart that the behavior of a molecule is not influenced by the presence of other molecules. This assumption is usually valid at low pressures and high temperatures. The ideal gas law states that, for one gas:

$$p=\varrho \frac{RT}{M}={\varrho }_{m}RT.$$

Using the assumption of ideal gases and Dalton's law (or equivalently Amagat's law) leads to the following expression for the (mass) density and molar density of the mixture:

$$\varrho =\frac{p}{RT}\sum _i^{}{M}_i{x}_i,{\varrho }_m=\frac{p}{RT}.$$

Also see Dumux::IdealGas for where this is implemented in DuMux.

Porous medium flow models

A list of porous medium flow models implemented in DuMux can be found in Porous medium flow (Darcy). The module description found under the link features a description of the governing equations describing each mathematical model.

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