Simulation of Dynamic Foam-Acid Diversion Processes
Authors: Cheng L. , Kam S.I. , Delshad M. , Rossen W.R. 
Published: December 2007

Abstract
A new simulator for foam-acid diversion is described. The simulator explicitly accounts for the first time for the effects of gas trapping on gas mobility in foam and in liquid injected after foam, and for the effects of pressure gradient on gas trapping. The foam model fits steady-state foam behavior in both high- and low-quality flow regimes and steady-state liquid mobility after foam. Previously, laboratory experiments suggested that a relatively slow transition between steady states during foam and acid injection may control the diversion process in the field. The simulator fits this transition period in laboratory corefloods qualitatively with no additional adjustable parameters. A procedure for fitting the simulator parameters to laboratory data is described.
The dynamics in the transition period are complex. For instance, simulations indicate that most of the core experiences a period of drier flow at the start of liquid injection, due to expansion of gas already in the core. Simulations and new laboratory results suggest that a dead volume present upstream of the core in previous studies strongly affects the transition period seen in those experiments. Simulations and data agree that the transition is faster at higher pressure (with lower gas compressibility) and that response to a shut-in period depends on how much gas escapes during the shut-in - i.e., on how long the shut-in lasts.
Extended to radial flow, the simulator suggests that the transition period may not be so crucial to field application as at first appeared from laboratory corefloods. In the cases examined, injection-well pressure approaches its steady-state value within about 15 minutes or less of the start of liquid after foam.

Introduction
In foam/acid well stimulation, foam can help divert acid to reservoir layers most in need of stimulation.1–8 Foam acts by partially blocking undamaged or higher-permeability layers and allowing acid to enter other layers in greater need of stimulation. Foam may be injected along with acid or in alternating slugs with acid. This paper focuses on foam injected in alternating slugs with acid, although the simulator described could be applied to either process.
Numerous studies report that foam does not directly alter the relative-permeability function krw(Sw) or viscosity of the aqueous phase.9–14 Foam does enormously reduce the mobility of gas, in part by trapping a large fraction of gas in place.14,15 Foam reduces the mobility of acid by reducing the mobility of gas in flowing foam, which drives up the gas saturation, and then trapping this gas in place during subsequent injection of acid.3,16,17 The success of foam then depends on both the mobility of flowing foam and the trapped-gas saturation during acid injection.
Several studies8,13,18–21 show that foam exists in porous media in two flow regimes, depending on foam quality fg (gas volume fraction in injected foam), as illustrated in Fig. 1. In this figure pressure gradient is plotted as a function of gas and liquid superficial velocities ug and uw. Foam quality increases as one moves from lower-right to upper-left in this diagram. At high foam qualities (upper-left portion of plot) pressure gradient is nearly independent of gas flow rate, while at low foam qualities (lower-right portion of plot) flow rate is nearly independent of liquid flow rate. In the high-quality regime, sometimes called the "coalescence regime,"22 foam is controlled by foam stability and capillary pressure.23–25 In the low-quality regime, behavior is controlled by gas trapping, and bubble size appears to be largely independent of flow rates.20,21 The foam quality at the transition between regimes, fg*, depends on surfactant formulation, permeability, and probably other compositional and petrophysical properties.21