Research

Direct Numerical Simulation in Pore-space

Due to the advent of X-Ray tomography technology, and computational architecture, we can now utilize pore-scale models based on a realistic description of the pore space and a detailed analysis of displacement mechanisms to predict multiphase flow properties based on available data from thin section analysis, capillary pressure and wettability indices.

Direct flow simulation in 3-D pore-space images of rocks allows a deeper understanding of complex flow structures within rock pores which is unattainable by field experiments. An example of such approach is illustrated by the investigation of the impact of porous media heterogeneity on the non-Darcy flow behaviour. In the figures below, the flow streamlines in Estaillades carbonate are given for three different Reynolds numbers corresponding to flows at the Darcy, transition, and the Forchheimer regime. The steady eddies emerging at elevated flow velocities act as the precursor of the inertial regime.

This explains a much earlier onset of non-Darcy flow regime in Estaillades carbonate when compared to more homogeneous Bentheimer sandstone and a beadpack. As predicted by our computations, the onset of these samples are given in the table below. Note that the onset of non-Darcy flow for Estaillades is much earlier than predictions of many empirically derived correlations. Indeed this demonstrates the utility of our approach - to simulate flow directly in the system of interest.

References:

1. B. P. Muljadi, M. J. Blunt, A. Q. Raeini, B. Bijeljic,  Advances in Water Resources, "The Impact of Porous Media Heterogeneity on Non-Darcy Flow Behaviour from Pore-Scale Simulation", doi:10.1016/j.advwatres.2015.05.019 (in-press)  

Multiscale Finite Element Method

figure: Contours of magnitude of velocity |u| along with their streamlines in a domain depicted in figure 3(b), with U according to equation (17), computed using Crouzeix-Raviart MsFEM on (a) 64 × 32 coarse elements; compared with (b) the reference solution.

The MsFEM method relies on the expansion of the solution on special basis functions which are precalculated by means of local simulations on a fine mesh which represent a model of the microstructure of the flow. These basis functions do not depend on any analytical models. 

figure: The solutions of advection-diffusion problems with 3600 non-periodic fine perforations and with circulating velocity field on (a)8X8, (b)16X16, (c)32X32, (d)64X64, (e)128X128 cells compared with (f)Reference sol. solved with Q1-Q1 FEM on 1024X1024 cells. Computational speed up of up to four orders of magnitude is achieved for a comparable accuracy with reference solution.

It is known that the approximation of boundary condition on coarse element edges when computing the multiscale basis functions critically influences the eventual accuracy of any MsFEM approaches. To meet the challenge of providing the most flexible and efficient platform to solve multiscale problems in perforated media, we adapt the Crouzeix-Raviart finite element space in the context of MsFEM. The weakly enforced continuity of Crouzeix - Raviart function space across element edges leads to natural boundary conditions for the multiscale basis functions on element edges, thus relaxes the sensitivity of our method to complex patterns of perforations. Another ingredient to our method is the application of bubble functions which is shown to be instrumental in maintaining high accuracy amid dense perforations. The extension of the Crouzeix-Raviart MsFEM onto perforated domains is straightforward and convenient through the means of penalization method inside the perforations which allows extensive use of simple Cartesian meshes. 

In nature perforations are mostly, if not always, non-periodic, this method shall give a flexible and efficient approach to solve problems in complicated geometries without resorting to complex unstructured meshes. 

References:

1. P. Degond, A. Lozinski, B. P. Muljadi, J. Narski, Communication in Computational Physics (17)4 887-907, 2015, "Crouzeix-Raviart MsFEM with Bubble Functions for Diffusion and Advection-Diffusion in Perforated Media" 

2. B. P. Muljadi, J. Narski, A. Lozinski, P. Degond, SIAM: Multiscale Modeling and Simulation (4)13 1146-1172, 2015, "Non-Conforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media. Part I: Methodologies and Numerical Experiments" 

3. B. P. Muljadi,  "Multiscale method for Oseen problem in porous media with non-periodic grain patterns", preprint.  

Rarefied (Semiclassical) Gas Dynamics

figure: shock diffraction problem of Bose-Einstein gas near continuum limit. Ms=2.

My PhD works focused on the development of theoretical, and computational solvers for semiclassical Boltzmann-BGK. Discrete velocity method is implemented to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source terms. My work hinges on the inclusion of Fermi-Dirac and Bose-Einstein's statistics alongside their classical counterpart—Maxwell-Boltzmann; hence providing a unified treatment for different types of carriers.

Several high order methods e.g., Space-Time CE/SE method, TVD and WENO have been incorporated for solving discrete semiclassical Boltzmann-BGK equations resulting in an accurate solver for a wide range of Knudsen regimes. 

figure: shock diffraction problems for Fermi-Dirac gas from near-continuum to transition regimes.

figure: steady state problem of semiclassical gas flow pass a square cylinder.

References: 

1. B. P. Muljadi, J. Y. Yang, Proceedings of Royal Society A., (468) 651-670, 2012, "Simulation of shock wave diffraction by a square cylinder in gases of arbitrary statistics using a semiclassical Boltzmann Bhatnagar Gross Krook equation solver" 

2. J. Y. Yang, B. P. Muljadi, S. Y. Chen, Z. H. Li, Computers and Fluids, 2013, "Kinetic numerical methods for solving the semiclassical Boltzmann-BGK equation" 

3. B. P. Muljadi, J. Y. Yang, H. X. Zhang, Z. H. Li, Communication in Computational Physics, (14) 242-264, 2013,"A Direct Solver for Initial Value Problems of Rarefied Gas Flows of Arbitrary Statistics" 

4. J. Y. Yang, B. P. Muljadi, Journal of Statistical Physics, (145) 1674-1688, 2011, "Simulation of Shock Wave Diffraction over 90° Sharp Corner in Gases of Arbitrary Statistics" 

5. B. P. Muljadi, J. Y. Yang, Computers and Fluids, (63) 184-188, 2012, "Space time CE/SE and discrete ordinate method for solving gas flows of arbitrary statistics"