Research Spotlight Application of pore-network models in two-phase flow in porous media

Two-phase flow is the major physical process in many natural and industrial porous media. In the last two decades, with significant improvement of computational and imaging facilities, fundamentals of two-phase flow as well as new theories have been extensively investigated.

Compared to other pore-scale methods for simulating two-phase flow in porous media, pore-network modeling has been used more extensively, as it is relatively simple and computationally less demanding. Pore-network modeling has also its own cons and pros. While pre-processing is required to idealize the pore space geometry and topology into simplified inter-connected geometries, the significant benefit is that larger physical domains can be simulated. Other pore-scale models such as lattice Boltzmann (LB), smoothed particle hydrodynamics (SPH), and level set (LS) methods are resolution based, while pore-network model is resolution-free. When simulating flow in a domain that is physically larger than the REV¹, other methods are impractical for analysis of continuum-scale theories.

Pore-network modeling has four steps:

• Network generation: A pore network is usually made of large pores (pore bodies) connected to each other through narrow long pores (pore throats). The topological and geometrical information can be obtained from imaging techniques to construct either a statistically-equivalent network or an exact network based on the imaged sample.
• Deriving analytical solutions: Analytical relations to define the pore-scale mechanisms for given pore geometry should be derived. For instance, for a two-phase flow problem, pore-scale flow equation, entry capillary pressure relations, and snap-off capillary pressures relation should be provided.
• Solving balance equations over the whole network: Systems of equations should be composed to solve the pressure field and calculate phase flows over the whole network.
• Averaging from pore-scale to Darcy scale: Pore-scale quantities will be averaged to obtain a Darcy-scale quantity. In our research, we focused on analysis of capillarity theories developed by Hassanizadeh and Gray, [1990]. These theories propose new physical insights that are absent in classical Darcy’s law. They proposed specific interfacial area as a new state variable that can eliminate some complexities related to the hysteresis in capillary pressure-saturation relation. Furthermore, they propose that Darcy’s law can be generalized by inclusion of gradient free energy that is related to fluid phases and interfacial area.

The following objective was analyzed in our research:

a) to investigate whether there exists a unique relation among capillary pressure, saturation and specific interfacial area under all drainage and imbibition conditions. If validated, hysteretic behavior of Pc-S curves can be removed by including the specific interfacial area, and

b) to analyze the functional dependences of non-equilibrium effects on phase pressure dif-ference and specific interfacial area. Consequently, we have developed quasi-static and dynamic pore-network models. The descriptions are given below:

Quasi-static pore-network models: These models provide equilibrium saturation profiles for given boundary pressures. Three models were developed for hypothetical, 2-D micro-model, and 3-D glass bead samples to analyze the uniqueness of P^c-S-a^nw. The validation of the pore-network models against experiments showed the reliable applicability of these models to investigate the theory. These models cover different levels of complexity. While the hypothetical model is quite simple (having regular lattice with circular cross sections), the glass bead pore-network model is unstructured irregular with mixed hyperbolic cross sections. Thus, the relationships for entry capillary pressure, specific interfacial area, etc. had to be derived.

Moreover, the wide range of analysis illustrated how Pc-S-anw surface can change depending on pore geometry. A medium with high porosity was also simulated using quasi-static pore-network models. A medial axis transform was used to extract the topology and geometry of the porous medium and as shown in Figure 1, the drainage and imbibi-tion cycles were successfully simulated using this model.

Figure 1: Simulated and observed saturation profiles un-der drainage and imbibition. A quasi-static pore-network was developed using the medial axis transform. Right: light grey: non-wetting phase, dark grey: wetting phase, black: solid phase. Left: blue: non-wetting phase, light grey: wetting phase

Dynamic pore-network models: The 2^nd type of pore-network models are dynamic and were initiated by Koplik and Lasseter [1985]. These models provide information about change fluids topology with time. But, because of two major technical issues (listed below), they were not used as extensive as the quasi-static ones.

1) In the literature it has been reported that dynamic pore-network models are numerical instable especial-ly at very small flow rates. Thus, ad hoc assumptions were made to stabilize the algorithm. However, this problem was resolved using a two-pressure algorithm and a semi-implicit saturation update. Therefore, we developed the DYPOSIT² model [Joekar-Niasar et al., 2010; Joekar-Niasar and Hassanizadeh, 2010]. The DYPOSIT model includes major pore-scale mechanisms, such as counter-current flow within the pores, capillary pinning, snap-off, pinch-off, piston-like movement, variable corner flow, variable capillary pressure, and mobilization of blobs by viscous forces. It was tested for a wide range of capillary number (10-8-10-2) and a wide range of viscosity ratios (0.01-10). Moreover, the consistency between the quasi-static and DY-POSIT model results was validated for gradual stepwise change of pressure at boundaries.

2) The second issue is that only few of the previous pore-network models could simulate mobilization of a trapped phase. Those ones that succeeded were based on exhausting and time-demanding search algo-rithms. In the DYPOSIT model, as the non-wetting phase pressure gradient inside the disconnected blob is solved numerically and the curvature of a capillary interface is dynamically changing with the pressure field, we can calculate the location where the non-wetting fluid will accumulate. In this location, the capillary pressure will build up until it can overcome the entry capillary pressure of a neighboring pore throat. This opening will facilitate mobilization of the disconnected blob or fragmentize it into smaller blobs. For illus-tration, Fig. 2 shows the change of invasion pattern for different capillary numbers and different viscosity ratios.

Employing versatile formulations in the DYPOSIT model allow it to be used for different two-phase flow scenarios. For instance, the DYPOSIT model has been successfully used to analyze the fate of the fluid-fluid interfacial area under drainage and imbibition under different dynamic conditions. It has been also used to investigate the non-equilibrium effects on phase pressure differences, effects of dynamic conditions on trapping and its consequences for fluid-fluid interfacial area, dynamics of capillary rise in porous media. A snapshot of the capillary rise problem simulated by the DYPOSIT model has been shown in Fig. 3 that illustrates the trapped air. ‘The Immiscbles’ (see Joekar-Niasar [2010]) not only provides new technical achievements for pore-network modeling, but also new insights into the physics of two-phase flow in porous media.