Continuity statement for the wetting liquid phase in a porous medium

The constitutive behavior for pore fluid flow is governed either by Darcy's law or by Forchheimer's law. Darcy's law is generally applicable to low fluid flow velocities, whereas Forchheimer's law is commonly used for situations involving higher flow velocities. Darcy's law can be thought of as a linearized version of Forchheimer's law. Darcy's law states that, under uniform conditions, the volumetric flow rate of the wetting liquid through a unit area of the medium, $sn{\mathbf{v}}_w$, is proportional to the negative of the gradient of the piezometric head (Bear, 1972):

where $\widehat{\mathbf{k}}$ is the permeability of the medium and $\varphi$ is the piezometric head, defined as

where z is the elevation above some datum and g is the magnitude of the gravitational acceleration, which acts in the direction opposite to z. On the other hand, Forchheimer's law states that the negative of the gradient of the piezometric head is related to a quadratic function of the volumetric flow rate of the wetting liquid through a unit area of the medium (Desai, 1975):

where $\beta \left(\mathbf{x},e\right)$ is a “velocity coefficient” (Tariq, 1987). This nonlinear permeability can be defined to be dependent on the void ratio of the material. We see that, as the fluid velocity tends to zero, Forchheimer's law approaches Darcy's law. Also, if $\beta =0$, the two flow laws are identical.

$\widehat{\mathbf{k}}$ can be anisotropic and is a function of the saturation and void ratio of the material. $\widehat{\mathbf{k}}$ has units of velocity (length/time). [Some authors refer to $\widehat{\mathbf{k}}$ as the hydraulic conductivity (Bear, 1972) and define the permeability as

where $\nu$ is the kinematic viscosity of the fluid (the ratio of the fluid's dynamic viscosity to its density).]

We assume that g is constant in magnitude and direction, so

where $\mathbf{g}\stackrel{\mathrm{def}}{=}-g\partial z/\partial \mathbf{x}$ is the gravitational acceleration (we assume that ${\rho }_w$ varies slowly with position).

The permeability of a particular fluid in a multiphase flow system depends on the saturation of the phase being considered and on the porosity of the medium. We assume these dependencies are separable, so

where $k_s\left(s\right)$ provides the dependency on saturation, with $k_s\left(1\right)=1.0$ and $\mathbf{k}\left(\mathbf{x},e\right)$ is the permeability of the fully saturated medium.

The function $k_s$ can be defined by the user. Nguyen and Durso (1983) observe that, in steady flow through a partially saturated medium, the permeability varies with $s^3$. We, therefore, take $k_s=s^3$ by default.

Introducing the flow constitutive law allows the mass continuity equation (Equation 1) to be written

The bulk behavior of the grains was discussed in Constitutive behavior in a porous medium. From Equation 2,

Combining this with Equation 4 and neglecting all but first-order terms in small quantities, we obtain

Using Equation 1 and again neglecting second-order terms in small quantities, we obtain

Combining this result with Equation 4 and again approximating to first-order in small quantities,

Because $u_w$ measures pressure in the wetting liquid and we neglect the pressure in the other fluid phase in the medium (see Effective stress principle for porous media), the medium is fully saturated for $u_w>0$. Negative values of $u_w$ represent capillary effects in the medium. For $u_w<0$ it is known (see, for example, Nguyen and Durso, 1983) that, at a given value of capillary pressure, $-u_w$, the saturation lies within certain limits. Typical forms of these limits are shown in Figure 1.

Figure 1. Typical liquid absorption and exsorption behavior.

We write these limits as $s^a\le s\le s^e$, where $s^a\left(u_w\right)$ is the limit at which absorption will occur (so that $\dot{s}>0$), and $s^e\left(u_w\right)$ is the limit at which exsorption will occur, and thus $\dot{s}<0$. We assume that these relationships are uniquely invertible and can, thus, also be written as $u_w^a\left(s\right)$ during absorption and $u_w^e\left(s\right)$ during exsorption. We also assume that some wetting liquid will always be present in the medium: $s>0$.

Bear (1972) suggests that the transition between absorption and exsorption and vice versa takes place along “scanning” curves. We approximate these with a straight line, as shown in Figure 1.

Saturation is treated as a state variable that may have to change if the wetting liquid pressure is outside the range for which its value is admissible according to that actual data corresponding to Figure 1. The evolution of saturation as a state variable is defined as follows. Assume that the saturation at time t, ${s|}_t$, is known. It must satisfy the constraints

We solve the continuity equation for ${u_w|}_{t+\mathrm{\Delta }t}$, initially assuming ${s|}_{t+\mathrm{\Delta }t}={s|}_t$. We then obtain ${s|}_{t+\mathrm{\Delta }t}$ by the following rules:

where ${\left(ds/d{u}_w\right)|}_s$ is the slope of the scanning line. These choices are shown in Figure 2.

Figure 2. Evolution of s in unsaturated cases.

The Jacobian contribution from the continuity equation is obtained from the variation of Equation 2 with respect to $\mathbf{x}$ and $u_w$ at time $t+\mathrm{\Delta }t$.

Consider first the surface integral. The surface divides into that part across which the liquid mass flow rate, ${\rho }_{w}sn\mathbf{n}\cdot {\mathbf{v}}_w$, is prescribed and that part where the wetting liquid pressure, $u_w$, is prescribed. Thus, the only contribution of this term to the Jacobian is the variation in the integral caused by change in surface area in that part where the mass flow is prescribed. We neglect this contribution.

The remaining part of the variation of Equation 2 is

Using Equation 3 we have

and, thus, neglecting small terms compared to unity,

Equation 5 shows that $J\left({\rho }_w/{\rho }_w^0\right)n_t\approx h_t$, which is defined by the evolution equation given in Constitutive behavior in a porous medium and so makes no contribution to the Jacobian.

Finally, the Jacobian contribution from the permeability term is rather complex in the general case of the nonlinear Forchheimer flow law. Although we include it in the software, here we only write the linearized flow version reflecting Darcy's law ($\beta =0$):

Using these results provides the Jacobian of the continuity equation as