Temperature and Common Ion Effects on Effective Acid Penetration in a Fracture

Abstract

Summary The effective acid penetration distance, an important parameter in acid-fracturing-treatment design, has a great influence on the effectiveness of the treatment. The temperature in a fracture and the acid/rock reaction products simultaneously affect the acid penetration greatly. But until now, no information concerning this aspect has been published. To satisfy practical calculation requirements in the treatment design, this paper develops a new mathematical model for the reaction of acid flowing in the fracture by considering the principal factors impacting the effective penetration of the acid. This model was combined with the model of temperature distribution in a fracture, the model of velocity field, and the model of acid/rock surface reaction kinetics to give a simultaneous solution, and a corresponding computer program was developed to perform the calculation. By analyzing and correlating figures and tables of data on the influences of common ion effect, acid concentration, formation temperature, and heat generated by the acid/rock reaction on effective acid penetration distance have been obtained. The computer program and conclusions in this paper can be used in practical acid-fracturing-treatment design. Introduction The effective penetration distance, Le, of acid, which reacts with rock along fracture walls, is an important factor affecting acid-fracturing treatment and is a main parameter in acid-fracturing design. The study of acid's Le began in the 1960's on the basis of physical simulation. Researchers at Dowell Co. measured the spent time of the activity of acid under various flowing reaction conditions and then calculated the acid penetration distance when flowing acid reacted with rocks along the fracture. Workers at Halliburton Co. and Esso Petroleum Development Co. studied Le by the mathematical/physical simulation method. None of these studies considered the effects of common ion effect and temperature in the fracture. Lee and Roberts studied the effects of formation temperature and reaction heat on Le. Ren et al. studied the influence of the common ion effect on Le. So far, two techniques are available for solving the flowing acid/rock-reaction mathematical model: the difference Method and the lumping method. The difference method solves the mathematical model directly; the lumping method uses finite difference. A computer transforms the partial-differential equations into ordinary differential equations by the lumping method and solves these equations numerically. On the basis of the above studies, this paper considers the comprehensive influence of the common ion effect and temperature ithe fracture on Le and on the calculation of Le in a prepad fluid acid-fracturing treatment and discusses the effects of the common effect, acid concentration, formation temperature, heat generated during reaction, and variable fracture width. After analysis and correlation, it is shown that the influences of the common ion effect, heat generated by reaction, and temperature distribution in the fracture should be considered, and that the variable fracture width should be applied in the calculation of Le in a prepad acid-fracturing design. Mathematical Modeling Flowing Acid/Rock Reaction Model. During acid fracturing, acid reacts with rocks along the vertical fracture (see Fig. 1). Four assumptions are made.Acid flow in the fracture is a steady-state, linear flow.Acid is an incompressible liquid.The acid density is uniform, and the effects of the natural convection on mass transfer are negligible.Length and height of the fracture are constant. Forced convection of acid does not occur, and there is no acid-concentration difference along the direction of the fracture height. In the past, the effective mass-transfer coefficient of hydrogen ion, KH, was known as a constant. Later Roberts and Guin studied the effect of temperature distribution on acid-concentration distribution in the fracture. Our research shows that KH relates not only to the temperature but also to the acid concentration; i.e., the common ion effect occurs in the acid/rock reaction and greatly affects the acid-concentration distribution. According to the mass conservation law, the mathematical model of flow describing that acid/rock reaction that considers both temperature and common ion effect can be formulated as u () + v() = ()[KH()],...................(1) where x=O, c=co, y=O, =0, and y= b/2, -KH()=Rs (1-). According to the theory of Arrhenius, KH = [KH(cT)] and the reaction-rate constant Rs =RsT can be derived as follows: KH = KHO exp[E2(T-To)/RTTo], and Rs =RTO exp[E2(T-TO)/RTTo], where the coefficients a' and b' are determined by experiments. The velocity terms u and v can be determined by Bemun's flow function: u=(2/b)[(b/2) - V1x]f (), v = V1f1 (), f1() = 3/2 - 1/2 3, and =y/(b/2). Eq. 1 can simulate the flowing acid/rock reaction in a fracture during acid-fracturing treatment, but the acid concentration in the direction of the fracture height is constant, the fracture width is unchangeable, and acid flow is steady state. Temperature Model in the Fracture. Model 1. This model uses u()+v()=()(),.................(2) where × = 0, T= Tbh, Y = 0, (a T/4) = 0, y= b/2, qh (t)=)], and = --------- ------------ . Model 2. With Whitsitt and Dysart's temperature model in the fracture as a basis, when the leakoff rate of the wall is constant, a temperature model in the fracture can be derived as follows: (T- Tbh)/(Ti - Tbh) = 1 -[1-(2e, xV1/Qo)], ............(3) where 1() = [e-)]. SPEPE P. 221^