A method of calculating acid spending of foams in a fracture has been developed. Experimental data are presented for foamed acid flow in limestone and dolomite laboratory-prepared fracture systems 60 in. [152.4 cm] long at 70 deg. F at [21.1 deg. C]. The surface reaction in an HCl-limestone system is fast compared to the mass transfer to the rock surface. The overall acid spending rate is to a large degree dependent on the extent of fluid mixing in the fracture. This is called a mass-transport- or diffusion-controlled reaction. The surface reaction rate in an HCl/dolomite system is finite compared with the rate of mass transfer to the rock surface. This is called a "kinetic-controlled" reaction. Earlier work showed that whether the acid was gelled or weighted, diffusion control still would be dominant for the HCl/limestone system and kinetic control would dominate the HCl/dolomite system. A method of predicting the penetration of foamed acid is presented based on combining the mass transfer coefficient of the acid in the fracture with the surface reaction that occurs at the fracture surface. Example calculations are presented to show the effect of various variables,
The use of foamed acid in fracture acidizing treatments has gained widespread acceptance in the past few years. Various properties of foamed acid have been investigated and its successful use in stimulation treatments has been reported. Surface reaction kinetics in fracture acidizing have been studied for nonfoamed systems by several authors. This paper describes work where acid spending of foams has been measured directly and the effect of foam on surface kinetics in fracture acidizing has been studied. Laboratory data are presented that describe the effects of foam quality, foam flow rate, and foam stabilizers on the surface reaction kinetics of foamed acid on limestone and dolomite.
A mathematical model describing the spending of foamed acid was developed similar to that presented earlier by Guin and Roberts. The model for foamed acid spending is illustrated in Fig. 1. The foam leakoff velocity, V, is assumed constant over the fracture length. Assuming the foam quality is constant and there is steady-state flow in a vertical fracture, the mass balance equation for foam in a fracture is
where u and V represent the foam velocity components in the x and y direction, respectively. Gamma represents the foam quality (volume fraction of N in foam) and C the concentration of acid in the foam. (D+D ) represents the effective diffusivity. Eq. 1 must be solved subject to the following boundary conditions.
The velocity components u and V are functions of x and y, satisfying the continuity equation:
These partial differential equations can be transformed into ordinary differential equations (Appendix A) to give
Eqs. 4 through 6 were derived by following the procedures proposed by Roberts and Guin for acid penetration for plain acid fracture treatments.