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Abstract
The movement of low viscosity fluid through a porous medium containing extremely viscous fluid is an important phenomenon in several petroleum engineering applications. These include the recovery of heavy oil by solvent injection, the preferential reduction of water flow using polymer gels, and the enhancement of acid fracturing treatments. The displacement of one fluid from a porous medium by a second immiscible fluid has been extensively studied in two cases: when capillary forces are dominant, and when viscous forces are comparable to capillary forces.

This study examines a third case: when viscous forces are dominant. The viscosity of the fluid initially present in the porous medium is several orders of magnitude greater than the viscosity of the displacing fluid. Consequently, the displacement through an individual pore will be dictated by the hydrodynamic forces required to move the high viscosity fluid. This study develops a mathematical model of the viscosity-dominated displacement in a network of conduits.

Key practical insights from the model are the nature of the displacement and effects of geometry and hydraulic conductivities on the sweep efficiency.

Introduction
The interesting physics related to the displacement of a viscous fluid by a less viscous one in a porous material has been the subject of past and recent years’ research, particularly due to its close connections with hydrology and oil recovery7,12. Network models are the most commonly used structure for this type of simulation. Network models typically simulate multiphase flow through an idealized representation of the pore space to calculate average properties, such as relative permeability, capillary pressure, and oil recovery5,3,10,11. The models can only make direct predictions of multiphase properties if both the geometry of the porous medium and the displacement process are precisely known8.

Modeling Approach
Numerical simulators use a three dimensional confined regular lattice structure, which dictates a certain number for connectivity. This number is constant for all the nodes in that structure. It can only be less if we apply zero conductivity to some of the bonds, but it cannot be higher than the degree of the lattice. For example, a node in a 3D cubic lattice cannot have a degree of connectivity higher than six.
In this research, we are developing a three dimensional simulator with the ability to accept up to seventeen connections between nodes. This development creates the ability to generate connections, even between the nodes that are not physically located near each other (e.g., non-adjacent pores) and have one or more nodes distance in between, even in different directions with different degrees of connectivity.

A real network sample or model is complicated in terms of size, connectivity, hydraulic conductivity, geometry, and etc. Figure 1 shows an example of this development. As an example in Figure 1, nodes number 8 and 12 are connected, although there are two nodes (numbers 9 and 10) that physically created a distance between them.