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 p. 385
A new method for calculating acid penetration distance in fractures bas been developed and tested experimentally. The method combines spending-time data from rotating-disk reaction pots with mass-transfer data obtained from laboratory fractures, thus allowing for both the effects of surface reaction kinetics and actual mixing patterns in the fracture. It is shown that the new method successfully predicts the acid spending obtained in laboratory fractures in both turbulent and laminar flow, using a reaction-rate constant obtained with a rotating-disk apparatus, This appears to be the first method that is easily applicable to small core samples and it allows properly for acid mixing in the fracture. Introduction Recently, there has been considerable interest in and research toward developing a more accurate method for calculating the acid penetration distance in a reservoir fracture. The acid penetration distance, defined as the distance the acid will travel before spending to some predetermined degree, is essential for estimating the production improvement obtainable by fracture acidizing. The first and probably most widely used method for calculating the penetration distance was based on the static-reaction test, in which a small core sample and a known quantity of acid were allowed to react for a given time in a small pot. By equating the spending time of acid in the pot to the residence time of acid flowing down the fracture (t = L/vi), a penetration distance was calculated. it has become penetration distance was calculated. it has become apparent that, because of the extremely fast surface reaction occurring in many acid-rock systems, the over-all acid spending rate to a large degree depends on the extent of fluid mixing at the rock surface. Since fluid-mixing patterns in small reaction pots may not be necessarily the same as those occuring in fractures, several experiments have been performed using actual laboratory model fractures. performed using actual laboratory model fractures. Recent investigations of this type have shown that, because of variable fluid properties, the mixing patterns in real fractures are very complex. To patterns in real fractures are very complex. To allow for this mixing and thereby to calculate more accurately the penetration distance in real fractures, design methods based on experiments in laboratory model fractures have been developed. Therefore, there appear to be two basic approaches to calculating the acid penetration distance, one using data obtained in small reaction pots, and the other using data gathered from model laboratory fractures. Both methods have some advantages. The former is quick, simple to operate, and applicable to the small core samples usually available for tests, while the latter method is more costly, more time consuming, requires special equipment, and is not applicable for use with small core samples. However, for reasons noted above, the latter method is probably more representative of mixing in actual reservoir fractures. In this paper we present a new method for calculating acid penetration distance that combines the advantages of both the above methods without incurring the disadvantages. The new method combines data from both reaction-pot experiments and laboratory-model fracture tests in a manner such that both the reaction rate of the actual rock (obtained conveniently from a small core sample) and the mixing occurring in an actual fracture are allowed for. Reaction-rate constants are obtained using a small batch reaction pot containing a rotating-disk core sample. These rate constants are then used with mass-transfer coefficients obtained from laboratory fractures to predict the acid penetration distance. penetration distance.The combined mass-transfer coefficient/ rate-constant method proposed here has several advantages over existing methods for predicting penetration distance. Since general correlations penetration distance. Since general correlations can be developed for mass-transfer coefficients (in fact, many applicable correlations already exist, most notably in the related field of heat transfer), A is not necessary, nor is it usually possible, to perform experiments in laboratory fractures for each perform experiments in laboratory fractures for each new field core sample obtained. SPEJ P. 277
In a fracture acidizing treatment the acid reacts with the fracture faces. This acid/rock reaction generates heat that causes the acid temperature itself to increase. To predict accurately the temperature profile and acid spending rate of acid traveling down a hydraulically created fracture, this heat must be considered.Since the heat generated by reaction depends on the reaction rate, the thermal energy equation must be coupled with the acid spending equation. A model has been developed that, for the first time, examines the effect of the heat of reaction on fluid temperature and acid penetration in a fracture. Some sample calculations have also been made to illustrate the effects of the most important parameters on acid penetration in a fracture. Introduction Acid hydraulic fracturing is a common method of stimulating a reservoir. Acid selectively reacts with, and dissolves, portions of the fracture wall so that a finite fluid conductivity remains when the well is returned to production. An important aim in designing such fracturing treatments is determining the distance that live acid will penetrate down the hydraulically induced fracture. This distance is usually called the acid penetration distance and is essential to estimate the production improvement from a given treatment.Because of its importance in predicting stimulation ratio, acid penetration in fractures has been studied by numerous investigators. They assumed the temperature in the fracture was uniform. In real fractures, however, the temperature will vary from the wellbore to the tip of the fracture. Therefore, the assumption of constant temperature seems to be an oversimplification.Whitsitt and Dysart were among the first to study the temperature distribution in a fracture. They constructed a model but it could be applied only to a nonreacting fluid flowing in a fracture because the heat generated by an acid/rock reaction was not considered. In a fracture acidizing treatment, the acid is reacting with the rock walls. This acid/rock reaction generates heat, which causes the acid temperature itself to increase. To predict accurately the temperature profile along the fracture, this heat also must be considered. A model has been developed that, for the first time, examines the effect of the heat of reaction on fluid temperature and acid penetration distance. Mathematical Development The mathematical model is a modification of that introduced by Whitsitt and Dysart to allow for the heat of reaction in the energy-balance equation. Since the heat generated by the acid reaction also depends on the reaction rate, the thermal-energy equation is coupled with the mass-balance equation. These two equations must be solved simultaneously .The model for acid spending in a fractures is illustrated in Fig. 1. The fluid leakoff velocity Vw is assumed constant over the fracture length. Assuming steady-state flow in a vertical fracture and constant fluid properties, the mass-balance equation for acid flowing in a fracture is ................(1) SPEJ P. 501^
Experiments show that after certain formations have been fractured, paraffin may precipitate and damage the permeability if a reservoir crude paraffin may precipitate and damage the permeability if a reservoir crude oil is cooled below its cloud point. The extent of such damage depends on formation permeability, the amount of paraffin precipitated, and the bottom-hole temperature. Introduction Crude oil in an untapped reservoir exists in a state of chemical and physical equilibrium. As the oil is produced through the formation, this equilibrium no produced through the formation, this equilibrium no longer exists, and the fluid undergoes physical and chemical changes. The volatile liquid constituents are continuously lost from the crude oil after it enters the fracture and the wellbore, since the pressure there is less than the pressure driving the fluid through the reservoir. Also, because of the pressure differentials that exist in the well, the crude oil begins to cool below formation temperature. A loss of light ends and a decrease in temperature combine to cause the solution to become saturated with paraffin. Then the paraffin begins to precipitate, and often it collects paraffin begins to precipitate, and often it collects on the tubing, in flow lines, etc. The equilibrium balance of the crude oil can also be disrupted within the formation by the injection of cold fluids such as those used in fracturing. This is particularly true when the surface fluid temperature particularly true when the surface fluid temperature is much colder than the formation temperature. If the fluid in the formation is cooled to a temperature below the cloud point, paraffin precipitates and may deposit in the formation pores, partially blocking or plugging the fluid flow channels and thus restricting plugging the fluid flow channels and thus restricting the flow. The precipitation of paraffin is almost irreversible in that the wax, once removed from solution, is very difficult to redissolve in the same fluid, even after original formation temperatures are restored. Obviously, formations having temperatures higher than the melting point of the precipitated paraffin would not be affected. Paraffin precipitation in the pores of the formation often causes serious problems and is difficult to detect. The damage resulting from such wax accumulation near the fracture faces may manifest itself in decreased production, slow cleanup of wells after fracturing, or failure to attain predicted production increases. Because such symptoms are often thought to result from poor fracturing treatments, very little attention has been given to the idea of paraffin precipitation in a formation. Our investigations deal with theoretical considerations of reservoir cooling during fracturing and with formation damage that can occur if the crude oil is cooled below the cloud point. From well and reservoir fluid information, the effect on production can be calculated. Experimental results of paraffin damage in cores cooled below the crude-oil cloud point will be discussed. Method of Calculation Wellbore and Fracture Temperature Hydraulic fracturing is commonly used to increase oil or gas production in formations with low permeability. The process involves applying enough permeability. The process involves applying enough hydraulic pressure to overcome the stresses in the formation. When this hydraulic pressure is applied, the formation ruptures perpendicular to the least principal stress and the created fracture will continue principal stress and the created fracture will continue to propagate until the end of the treatment if the pressure is maintained. The least principal stress is pressure is maintained. The least principal stress is generally accepted to be in the horizontal plane; therefore, most fractures would be vertical. JPT P. 997
This work proposes to investigate the reaction rates of hydrofluoric- hydrochloric acids on silica and various clays and mixtures thereof, and then mathematically to determine the amounts of both sand and clay removed from formations at various depths. Also calculated are theoretical increases in the productivity of undamaged and mud-damaged formations and of formations naturally damaged by clay. Introduction Treatment of sandstone formations by mixed hydrofluoric-hydrochloric acids has been used as a means of removing damage caused by the presence of days. The removal of such damage presence of days. The removal of such damage results from the dissolution of clay by reaction with the hydrofluoric acid: 36 HF + Al2Si4O10(OH)2->4H2SiF6 + 12 H2O+ 2H3A1F6. The acid will also react with sand and other siliceous minerals: 6 HF + SiO2 ->H2SiF6 + 2 H2O. These reactions, though they appear simple, are really very complex. Further reactions may take place that produce insoluble reaction products. It is for this reason that excess hydrochloric acid should be maintained in the mixture and dilute hydrochloric acid solutions are used as preflushes ahead of the mixed acids. A discussion of these reactions and their effect on productivity has been presented by Smith and Hendrickson. An awareness of these possible problems allows us to approach such treatments with problems allows us to approach such treatments with proper design to circumvent undesirable results. proper design to circumvent undesirable results. Another equally important aspect of successful acid application is that of adequately describing the effect of hydrofluoric acid reaction on depth of acid penetration. It is this aspect that ultimately penetration. It is this aspect that ultimately determines the extent of damage removal and subsequent productivity increase. productivity increase. Relatively little has been done in the area of investigating the reaction rates of hydrofluoric acid, or the mixed acids, on silica and silicates. It has been generally state that the reaction is faster on clays than it is on sand, but without quantitative work the desired calculations have not been possible. It has long been assumed by some that these acid systems could be injected into a sandstone formation to dissolve clay at almost any depth from the wellbore, and that the reaction on sand is so slow that little reaction takes place. It has been our purpose to investigate the reaction rates of hydrofluoric-hydrochloric acids on silica, various clays and mixtures thereof, and then to determine mathematically the amounts of both sand and clay removed from formations at various depths, as a function of acid reaction. Also, theoretical increases in productivity have been calculated, on the basis of the penetration determinations, for the cases of undamaged and mud-damaged formations and for formations naturally damaged by clay. Reaction Rate Studies Experimental The determination of reactant concentrations has been a problem in following the HF-silica reaction and probably accounts for the meager data available in the literature. Development of a specific ion electrode for fluoride ions by the Orion Corp. has enabled us to obtain much information accurately and rapidly. The analytical procedure and equipment have been previously described by Gatewood. previously described by Gatewood. JPT P. 693
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