REE1: Lattice Boltzmann methods for multiphase flow in porous media with applications to CO2 sequestration

Project completed December 31, 2011

Robust and efficient algorithms for predicting multiphase flow (for example, flow involving substances in both a liquid and a gas phase) are becoming essential for engineering and environmental applications, perhaps most notably in the oil industry. An important research challenge is to properly model the interfaces that exist between phases, as these are prone to small scale instabilities (see Figure 1).

The lattice Boltzmann equation (LBE) is a promising alternative to the traditional methods of computational fluid dynamics (CFD). In fields such as multiphase fluid mechanics where intensive, time-consuming computations are commonplace, the LBE is extremely attractive because it allows one to exploit massively parallel modern computer architectures, including graphics processing units (GPUs), leading to very fast computations.

Techniques and Challenges

Accurately representing the behaviour of fluid interfaces remains a challenge to CFD, as when these systems are simulated numerically there are often undesirable and unrealistic effects. To address these issues, researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) have developed a formulation for simulating multiphase flow using the LBE by examining interface sharpening methods to mitigate numerical effects such as pinning and facetting.

Physically, the interface between the phases is a boundary with zero thickness. However, when simulated numerically, the interface is a region with non-zero thickness. This leads to modelling problems, as state-of-the-art models are only able to make predictions that are theoretically sound when the flow parameters are such that the interface region is many times thicker than is realistic in practical applications.

It is possible to tune model parameters to sharpen the interfaces, but this leads to unphysical behaviour, such as pinning, where the interface does not move, and facetting, where the interface consists of planar segments.

Results

To help understand pinning and facetting, the researchers introduced an order parameter. The order parameter undergoes a rapid but smooth variation in the region of the interface between the phases. For example, in regions where there is only one phase (e.g. oil), the order parameter might be 0, and in regions where there is only the other phase (e.g. water), the order parameter might be 1, and within the interface region, the order parameter changes smoothly from 0 to 1.

In principle, the interface is simply advected or transported by the fluid velocity. however, due to the discrete nature of the system, numerical diffusion is inevitable, meaning that the interface increases thickness artificially which is undesirable. It is common to add a ‘sharpening term’ to counteract the numerical diffusion and project the order parameter in the diffused interface region back to either 0 or 1 to control the width of the interface. The direction of this projection can be determined by whether the post-advection order parameter is above or below a constant critical value. The width of the interface is then controlled by a balance between diffusion and the sharpening term. However, the system becomes numerically stiff and difficult to solve when attempting to minimise the width of the interface, which causes boundaries to become fixed or pinned to the grid and facetted.

As a promising alternative to the constant critical value, the researchers found that using a random threshold given by a sample from the van der Corput sequence predicts the correct average propagation speeds, even for very sharp boundaries (see Figure 2).

The Future

The stochastic sharpening model developed shows much promise and can be incorporated into existing multiphase lattice Boltzmann algorithms and used to simulate pore-scale flows with surface tension.

Further work will focus on increasing the permitted density ratio between the phases and the simulation of real-world applications, such as the unstable displacements that occur in CO₂ sequestration.

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