Modeling of reactive transport with particle tracking and kernel density estimators
The defense will take place:
Friday, February 23rd 2018, 12:00
UPC, Campus Nord
Building C1. Classroom: 002
C/Jordi Girona, 1-3
Random walk particle tracking methods are a computationally efficient family of methods to solve reactive transport problems. While the number of particles in most realistic applications is in the order of 106-109, the number of reactive molecules even in diluted systems might be in the order of fractions of the Avogadro number. Thus, each particle actually represents a group of potentially reactive molecules. The use of a low number of particles may result not only in loss of accuracy, but also may lead to an improper reproduction of the mixing process, limited by diffusion. Recent works have used this effect as a proxy to model incomplete mixing in porous media. The main contribution of this thesis is to propose a reactive transport model using a Kernel Density Estimation (KDE) of the concentrations that allows getting the expected results for a well-mixed solution with a limited number of particles. The idea consists of treating each particle as a sample drawn from the pool of molecules that it represents; this way, the actual location of a tracked particle is seen as a sample drawn from the density function of the location of molecules represented by that given particle, rigorously represented by a kernel density function. The probability of reaction can be obtained by combining the kernels associated with two potentially reactive particles. We demonstrate that the observed deviation in the reaction vs time curves in numerical experiments reported in the literature could be attributed to the statistical method used to reconstruct concentrations (fixed particle support) from discrete particle distributions, and not to the occurrence of true incomplete mixing. We further explore the evolution of the kernel size with time, linking it to the diffusion process. Our results show that KDEs are powerful tools to improve computational efficiency and robustness in reactive transport simulations, and indicates that incomplete mixing in diluted systems should be modeled based on alternative mechanistic models and not on a limited number of particles.
Motivated by this potential, we extend the KDE model to simulate nonlinear adsorption which is a relevant process in many fields, such as product manufacturing or pollution remediation in porous materials. We show that the proposed model is able to reproduce the results of the Langmuir and Freundlich isotherms and to combine the features of these two classical adsorption models. In the Langmuir model, it is enough to add a finite number of sorption sites of homogeneous sorption properties, and to set the process as the combination of the forward and the backward reactions, each one of them with a pre-specified reaction rate. To model the Freundlich isotherm instead, typical of low to intermediate range of solute concentrations, there is a need to assign a different equilibrium constant to each specific sorption site, provided they are all drawn from a truncated power-law distribution. Both nonlinear models can be combined in a single framework to obtain a typical observed behavior for a wide range of concentration values. This approach opens up a new way to predict and control an adsorption-based process using a particle-based method with a finite number of particles.
Finally, by classifying the particles to mobile and immobile states and employing transition probabilities between these two states, we take into account the porosity of the diluted system in the KDE model. The state of a particle is an attribute that defines the domain at which the particle is present at a given time within the porous medium. The transition probabilities are controlled by two parameters which implicitly determine the porosity. Simulations results show a good agreement with the analytical solutions of complete and incomplete mixing solutions, independent of the number of particles. A transition between the complete and incomplete mixing solutions is also obtained, showing a good match with a transition probability function. These results show the potential of our proposed model to simulate reactive transport problems in porous media.