3D Simulation of Packed Particle Bed and Transport Properties Prediction for Product Optimization through Virtual Experiments

Porous media are heterogeneous systems. The microstructures of the pore spaces influence their transport properties. A quantitative geometrical characterization of the pore space is crucial for accurate prediction of porous media transport. Thus, a 3D simulation of porous media was developed based on randomly packed glass beads. Unconsolidated porous media are reconstructed through Monte Carlo gravitational particle packing simulation. A mathematical morphology based three-dimensional image processing algorithm is developed to characterize the pore space in the simulated porous media. This algorithm calculates the bulk porosity, average particle contact numbers, and specific surface area of the porous media. It also generates the pore-throat network with details of pore size distribution, location, and throat tortuosity distribution. The simulation results are validated by statistical comparison with the bulk porosity and pore size distribution obtained from x-ray micro-tomographic images of randomly packed glass beads. Simulation of porous media, given a specific size distribution of constituent particles, followed by pore-space characterization provides a powerful tool for predicting transport processes. Introduction Transport in porous media is encountered in numerous physical systems from water resources management to industrial processes of varying length scales. It is a complex phenomenon. Many efforts have been made to develop continuum mechanics based models to describe porous media transport. These mathematical models often tried to correlate the transport behavior of a specific porous medium with their bulk porosity. Sometimes, in these models, effective permeability of the porous medium is represented as a function of porosity as well as an average tortuosity, in order to match the experimental data. These descriptions of porous media flow are not purely mechanistic models, but based on semi-empirical relationships among capillary pressure, saturation level, and relative permeability, obtained through controlled experiments. These models do not account for the topography of the microstructure in the porous medium. But, the transport of fluids and solutes in porous media is a function of its geometry and topological characteristics. The microscale phenomena happening in the pore-scale level translates very well into the macro-scale behavior of a porous structure. Hence, pore-network based modeling of various properties of porous media is a powerful method to generate realistic prediction of macroscopic behavior. Early attempts at pore network modeling describe porous media transport based on capillary tubes. These models attempted to explain the permeability without accounting for the interconnectivity among the pore channels. More recently the concept of 2D and 3D networks to describe the pore-space topography has been introduced. 2D pore networks have limited application since most porous media involve 3D flow and hence the the connectivity between pores can not usually be adequately defined in 2D. The existing 3D network models in the literature are not realistic, those are generated based on certain assumptions regarding the pore and throat size distribution. Thus, there is a need to develop a realistic 3D pore network model, which shows more promise to a realistic prediction of porous media transport. Pore network models can be generated using indirect or direct methods. In indirect method, an equivalent network is produced based on distributions of major pore-space structures, pore body and pore throat, and their positional correlations. In contrast, the direct method, extracts the pore-throat network from the pore space directly using 3D image. The direct method requires no assumption related to the topological positions or dimensions. In order to directly map the pore network, a 3D data representing the pore-space, with enough resolution, is essential. Many attempts have been made to generate this dataset using high resolution non-invasive threedimensional imaging techniques like laser scanning confocal microscopy, x-ray microtomography, etc. There are three major limitations of this approach. First, one is restricted to investigating only those porous media, of which one already has samples of. Thus, one can not use this approach for virtual product design. Second, the pore-space data set resolution is limited by the instrument capability that is used to acquire the 3D data. The third limitation is the contrast in the dataset. The edges of the pore-space, i.e. the grain boundary in the instrument-generated images will not be absolute. This sharpness of the edges will depend on how the sample material interferes with the instrument signal. The imperfection in obtaining the correct edges can introduce serious errors in the pore network extracted. All these limitations are avoided in this work by extracting the pore network from a digitally generated consolidated porous media through computer simulation and mathematical morphology based image processing. A random packing of particles represents an unconsolidated porous media. Various methods have been used by different researchers to generate a random packing of particles. Some of them are: sequential addition, ballistic drop with mechanical interaction calculation, region growing, and mechanical contraction. Simple rules of sequential addition, where particles are added at random positions as long as they do not share spaces with each other, can generate random particle packs, but the packing tend to be fairly loose. The reason for this is while placing a new particle it stops moving as soon as it hits another and do not slide along other particles boundary, which is more realistic. Ballistic drop method of random particle packing simulation utilizes calculations of mechanical forces each time a falling particle encounters another on its path. Although this method follows the mechanics of the process realistically and very accurately, many detailed calculations in each step of the movement for a huge number of particles become extremely demanding on the computing power. Thus, this method of simulation tends to be much slower, and to generate a random packing with fairly large numbers of particles, a very high amount of computing time is required. Region growing method of random particle packing is used in simulating aggregated powder particles. It starts with one central particle and attaches particles of random size at random available locations. This method is not representative of randomly droppping particles into a container randomly to generate an unconsolidated porous media. The mechanical contraction method of random particle packing is motivated by simulation of amorphous packing and is based on the idea of density quenching a system, which undergoes no thermal fluctuations. Three-dimensional Monte Carlo simulation under gravity is a method of random particle packing, which closely resembles random gravitational dropping of particles and at the same time computationally less demanding. Hence this method was selected to generate the porous media model in this work. Since, the porous media models are constructed through computer simulations, the resolution of the pore-space was limited only by the available memory (RAM) and processor speed of the computer used. This model enabled perfect binarization of 3D pore-space images without any error from edge or boundary detection. A 3D image processing algorithm was developed for this work to map the pore-throat network. Skeletonization of the 3D pore-space was the first step in mapping of the pore-throat network. This conversion of the 3D image into the pore-throat network has a wide variety of applications and has been an active area of research for many decades. Many researchers have tried different sequential, parallel, as well as non-iterative thinning techniques towards the same goal; Hamilton-Jacobi, local flux driven extraction, three-dimensional template based exclusion rule, medial axis extraction, are to name a few. These algorithms, in spite of being effective in finding the skeleton in specific kind of structural features, fail to correct centerline in other types of shape features. Especially when the structure is as complex as pore-space it becomes very challenging to come up with a single algorithm that will converge to a perfect skeleton in an unsupervised manner. A morphological thinning algorithm is developed in this work which thins the pore-space with same flux vector from all directions while preserving the continuity. This algorithm also generated the bulk porosity, co-ordination number, specific surface area. Using the distance-transform principle; pore size distribution, and average tortuosity of the pore-space are also calculated in an automated manner by this algorithm. All these values directly calculated from the three-dimensional morphology of the pore-space are more realistic input parameters for pore network based modeling. Thus porous media based virtual product design, aiming at a specific transport behavior, becomes much faster and complete.