Abstract
A mathematical model is developed that yields the distance to which live aid may penetrate into a fracture under conditions in which the over-all reaction kinetics. The model is solved by an explicit finite-difference method, and the results are presented in graphical form. An example design presented in graphical form. An example design calculation is given for HC1 reaction in a dolomite fracture. Experimental data are presented for acid flow in limestone and dolomite laboratory - prepared fracture systems 4.1 t 9.7 ft long, at 71, 190, and 290F. From these experiments was determined a parameter appearing in the mathematical model-termed the effective mixing coefficient. The mixing coefficient has a minimum in the low Reynolds number region, indicating that rectilinear laminar flow is approached more closely just before the flow becomes turbulent. The mixing coefficient also appears to be dependent upon temperature in the laminar flow region. The mathematical solutions given in this paper are applicable to situations in which the over-all rate of acid reaction is not determined solely by mass transfer.
Introduction
Acids are widely used in the hydraulic fracturing of reservoirs to stimulate wells. Roughly speaking, the purpose of the acid is to selectively react with and dissolve portions of the fracture wall so that a finite fluid conductivity remains when the well is returned to production. One important variable that must be known in designing these acid fracturing treatments is the distance to which acid will penetrate the fracture before completely reacting penetrate the fracture before completely reacting and becoming spent. This distance is usually termed the acid penetration length and is an essential part of the information needed for predicting productivity after acidizing. Other important design variables include the dynamic fracture geometry and the residual fracture conductivity. Because of its importance in predicting stimulation ratios, acid penetration into a fracture has been studied by several investigators. Both static tests and dynamic tests have been used to predict acid reaction rates in fractures. It seems predict acid reaction rates in fractures. It seems reasonable that a dynamic acid reactor test will be useful for predicting acid spending rates, since the mass transfer rate in an actual fracture may be approached in this type of test. One experimental apparatus used for acid flow tests in parallel plate system such as that used by Barron et al. plate system such as that used by Barron et al. and by Williams and Nierode. In these tests, acid is pumped at a known flow rate through a fracture of known geometry and the inlet and outlet acid composition is measured. From the resulting information it is possible to predict acid penetration in a real fracture with the aid of a mathematical model having experimentally determined parameters. We present here the results of an investigation of the use of mathematical model for predicting acid spending a fracture. Using Williams and Nierode's approach to calculating acid penetration, we have extended their method to allow for the fact that the surface reaction rates of several acid-rock systems (e.g., HC1-dolomite) may be finite compared with the rate of mass transfer to the surface. Experimental data are presented for determining the parameters appearing in the mathematical model and a sample calculation illustrates its use.
MATHEMATICAL MODEL FOR ACID FRACTURING
The mathematical model presented here is a modification of that introduced by Williams and Nierode to allow for the occurrence of finite reaction rates. This modification makes it possible to calculate theoretical penetration distances for acid featuring when reaction kinetics are important as in the case of the HC1-dolomite reaction. Since an analytical solution of the model is not possible, a finite-difference method was developed and is presented in Appendix A. presented in Appendix A. The model for acid formula is fracturing is presented in Fig. 1. Here the acid leakoff velocity, presented in Fig. 1. Here the acid leakoff velocity, is assumed constant over the fracture length.
SPEJ
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Interpolation of Boundary Condition at Time-Interval of Unknown Lenghth for the Problem of Convective Diffusion in a Three-Layered Water Filter
