Modelling viscous-dominated fluid transport using modified invasion percolation techniques

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Abstract

Invasion percolation is a computing technique used for solving the transport problem for flow systems dominated by capillary and gravity forces. This work introduces a new technique that attempts to extend the percolation algorithms to flow systems dominated by viscous forces, while retaining the ability to honour capillary and gravity effects.

Introduction

Traditional invasion percolation (IP) algorithms (e.g. Wilkinson and Willemsen, 1983) can be used to solve the transport problem for capillary dominated systems or systems dominated by both capillary and gravity forces (e.g. Carruthers, 2000). However, techniques for extending the rapidly solvable IP algorithms to a system that contains capillary, gravity and viscous forces, have been elusive.

This work demonstrates some early results of our attempt to unify the three force systems into a single IP-based transport algorithm. The motivation for this work is computing efficiency: multi-million gridcell IP-based models can be solved in a matter of seconds or minutes, whereas the timescales for comparable Darcy-based simulators are measured in days or weeks.

Wilkinson (1984) introduced some ideas about how the viscous case might be handled using invasion percolation. He rightly states that this “is a fundamentally more complicated problem than the [hydrostatic] buoyancy case, because the pressure variations in the system are dynamically determined by the spatial fluid configurations rather than being purely hydrostatic”. To extend the percolation techniques to viscous scenarios, Wilkinson (1984) suggests the adoption of a mean field description of the fluid flow in which each interface moves under the average pressure field produced by the motion of the other fluids. In order to estimate these viscous pressure fields, he suggests using the multiphase Darcy equations (1):Pwx=μwυwkkrwPox=μoυokkroPo−Pw=Pcwhere w and o are water and oil phases, P is the phase pressure, μ the viscosity, υ the Darcy velocity, k the absolute permeability, and kr the relative permeability

This equation system assumes that at any given saturation, each fluid flows in its own set of flow channels and exerts negligible shear stress on the other fluid. On the other hand, from the microscopic point of view, the only effect of the bulk flow is to produce a local capillary pressure between the phases, which evolves in time. The interface motions respond to the pressure difference in the same way as they would in the quasistatic limit. Under these assumptions, the sequence of configurations that a given region passes through is independent of the bulk capillary number (only the rate at which these changes take place is altered). Therefore, the relative permeabilities and capillary pressure are unique functions of saturation, independent of flow rate (Wilkinson, 1984).

As Wilkinson (1984) points out, the fluid configurations at a given saturation are dependent on the flow rate which created the configuration and it would seem implausible that the relative permeabilities and capillary pressures would be only functions of saturation.

To model these highly complex systems within reasonable timescales, simplifications must be made, but made in such a way that the force balances are always honoured. The goal is to couple Eq. (1) with IP algorithms that honour both capillary and gravity forces (e.g. Carruthers, 2000), in such a way as to not negate the computational benefits of the IP techniques. What is needed is a rule-based system, which is founded on Eq. (1) but which avoids the necessary pressure solving required to arrive at a solution for the differential equations. The procedure needs to mimic the behaviour of the flow system, without actually implementing any of the mathematical overheads.

As a first attempt at such a system, the following technique is proposed.

Section snippets

A new algorithm

The first step is to discretise the flow field into absolute permeabilities and capillary entry pressures (an example is given in Table 1). It is known that the fluids will not flow unless the invading phase pressure is greater than the gridcell threshold pressure (since the invading phase relative permeability is zero below this state). To determine which gridcells in the model are “flowable”, we need to have an estimate of the pressure gradient across the flow field. This can be calculated in

An example

The images in Fig. 1give an example of the new algorithm being applied to a complex fabric. The model contained approximately 100,000 gridcells and took a few minutes to execute on a modest workstation (SGI O2). The greyscales in Fig. 1(a) represent the capillary threshold pressures used in the simulation. Permeability values were based on an inverse square relationship with threshold pressures (although they could have been entered separately). The system is water wet, and an invading oil

Discussion

This work has demonstrated that it may be possible to extend IP algorithms to viscous-dominated flow regimes. These new techniques are extremely computationally efficient, with the potential for the transport problem being solved for very high resolution scenarios, in reasonable timescales. Although the processing power of computers is forever increasing, fluid flow models which rely on the full pressure solving of Darcy's equations are still limited to tens of thousands of gridcells on

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