## Two-phase flow in porous media under steady-state conditions

Two-phase flow in porous media is a process found in many natural and technological processes. Examples are found in fuel cells, paper pulp drying, the food industry, oil recovery and bio-medical technology [1].

Two fluids can be classified, respectively, as wetting and non-wetting for a given solid by the contact angle. If the contact angle in a fluid is smaller than $$\pi/2$$ it is wetting, otherwise it is non-wetting.

If the non-wetting fluid displaces the wetting fluid, the flow is referred to as imbibition and, conversely, if the wetting fluid displaces the non-wetting fluid the flow is referred to as drainage.

### Network model

One way to describe two-phase flow in porous media is through network models. These models represent a compromise in complexity and computational effort between coarse models, like the generalized Darcy equations, and fully resolved models that involves solving the PDEs governing the flow.

We consider incompressible flow in a network of $$N$$ nodes, connected with links. The volume flow from node $$j$$ to node $$i$$ is denoted $$q_{ij}$$. Conservation of mass then gives $\sum_j q_{ij} = 0,$ at each node $$i \in \left[0,\dots,N-1\right]$$. For flow in small pores, where viscosity dominates and inertial effects are negligible, the time-dependent volume flow can be approximated by $q_{ij} = - g_{ij} \left( p_i - p_j - c_{ij}\right).$ Herein, $$p_i$$ is the pressure at node $$i$$, $$g_{ij}$$ is the link mobility, which depends on the link geometry and fluid viscosity, and $$c_{ij}$$ is the capillary pressure in the link, which depends on the distribution of fluids in the link, the link geometry and the interfacial tensions.

Inserting the approximation of $$q_{ij}$$ into the mass conservation equation, we obtain a linear system of equations for the node pressures, $\sum_j A_{ij} p_j = b_i, \quad \forall i \in \left[0,\dots,N-1\right].$ By solving this system, we also know the instantaneous flow distribution in the network.

### References

 [1] V. Joekar-Niasar and S. M. Hassanizadeh. Analysis of fundamentals of two -phase flow in porous media using dynamic pore-network models: A review. Critical Reviews in Environmental Science and Technology 42 (2012), 1895-1976. doi:10.1080/10643389.2011.574101.