We present a new 2D analysis based on the recently developed stochastic bubble population foam model, focusing on the effect of the core heterogeneity. In the frame of the model presented in a parent paper in the conference, we assume that the bubble generation kinetics is dependent on layer permeability. We present experiments consisting of co-injection of N2 gas and surfactant solution in layered cores, with layering parallel and to the flow directions. The cores are obtained by combining two porous media chosen from Benteimer and Berea sandstone and sintered glass, with large permeability contrast. X-ray computed tomography (CT) scans are used to visualize and quantify local fluid distributions and differentiate foam propagation in the different layers. From both the model and the experiments we conclude that foam is primarily generated in the high-permeability layers, where it propagates at a much higher speed than in the low permeability layer. The propagation of foam in the low permeability layer requires that the pressure gradient is higher than the capillary entry pressure for the layer. The new stochastic population balance foam model reproduces rather well the main features of foam motion in heterogeneous cores containing a surfactant. Core floods combined with CT scan imaging provide valuable specific information about the effect of heterogeneity for better design of acid diversion operations.
Foam is a highly efficient acid diversion agent for matrix stimulation operations. It is inherently non-damaging and low cost, allowing easily recursive treatments in case of an unsuccessful operation. Foam has also been widely used as a mobility control agent in Enhanced oil Recovery (EOR) as a profile correction agent [1–6]. The description of foam behavior in porous media relies on macroscopic modeling. Although many works have been devoted to foam, the study of the effect of core media heterogeneity has been minimal.
A variety of methods have been proposed for modeling of foam flow and displacement in porous media. These models include fractional flow modeling, population balance model and percolation models. The foam fractional flow model was first introduced by Rossen et al. [2,7–8]. The implicit assumption of foam incompressibility makes this model unsuitable when pressure changes are relatively high in comparison with the backpressure. Moreover, fractional flow models do not account for the evolution of bubble population and therefore might not be accurate when describing transient motion . Percolation models which take into account the pore level mechanisms for the foam seem unlikely to be useful in transient displacement in large scales, because of large amount of calculations .
The population balance approach introduced by Patzek  and further elaborated by Kovsec et al. [10–11] and Falls et al.  originates from the principle that foam mobility depends on the bubble density (number of bubble per unit gas volume). The population balance model splits gas saturation into flowing and trapped fractions. The introduction of parameters that may be difficult to measure experimentally is disadvantageous and it is preferred to use parameters that are measurable experimentally.
Recently Zitha  developed an alternative population balance theory for foam motion in porous media. The principals of this model are as follows: Foam is a complex fluid, characterized by a yield stress and above the yield stress, by power law behavior. Its rheology is described using the Herchel-Bulkley model.
Foam rheology depends essentially on the bubble density. Since on a large scale foam generation mechanisms happen in a large number of randomly interconnected pores, the bubble generation can be treated as a stochastic process. The kinetic of the foam generation is described by a simple exponential growth function involving two parameters. This makes it easier to determine those parameters from the experimental results.