Mean-field population balance equations for ganglia dynamics in porous media models

We study experimentally the dynamics of non-wetting ganglia flowing simultaneously with a wetting fluid in a quasi-two-dimensional porous medium consisting of random obstacles. The ganglia continuously merge, forming larger ones (coalescence), and break up into smaller ones (fragmentation), leading to an overall dynamic equilibrium between the two processes over longer time scales. We develop a clustering algorithm for the identification of fragmentation and coalescence events that records the size of ganglia prior to and immediately after each event from high-resolution videos of the immiscible flow experiments. The results provide significant insight into the main physical features of these two processes, such as ganglia size distributions, breakup and coalescence frequency as a function of total flow rate, and the size distributions of the ganglia formed by either the fragmentation or coalescence of other ones. One of the salient features of the fragmentation process in our study is that ganglia that are smaller than the typical pore size exhibit a higher probability of producing two almost identical children (in size), whereas larger ganglia break up into two children of different sizes. In the latter case, one of the children is found to have a dimension that is practically equal to the typical pore size. Our experimental results are also interpreted in the framework of a mean-field approach, where the dynamics of the ganglia sizes is expressed through an integro-differential population balance equation that comprises terms for the description of the rates of size gains and losses by either fragmentation or coalescence. We recover appropriate expressions for the relevant coalescence and fragmentation kernels, as functions of the ganglia sizes that participate in each event. A rather surprising result is that for a given ganglion size population, we obtain an equilibrium between the gains by fragmentation and the losses by coalescence. Furthermore, the opposite is also true, as the population gains by coalescence are found to be equal to the losses by fragmentation.