Experimental Study of a Viscoelastic Surfactant-Based in Situ Self

Dec 6, 2013 - The existing models for worm-like channel development were developed by considering microprocesses at the individual pore level during c...
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Experimental Study of a Viscoelastic Surfactant-Based in Situ SelfDiverting Acid System: Results and Interpretation Guzel T. Bulgakova,*,† Rinat Ya. Kharisov,‡ Aleksey V. Pestrikov,‡ and Andrey R. Sharifullin‡ †

Ufa State Aviation Technical University, 12 Karl Marx Street, 450000 Ufa, Russia RN-UfaNIPIneft Limited, Rosneft Oil Company OJSC, 96/2 Revolutsionnaya Street, 450078 Ufa, Russia

‡

ABSTRACT: Recently, self-diverting acid systems based on viscoelastic surfactants (SDVAs) have been successfully used for carbonate reservoir treatment. Changes in the SDVA viscosity during the interaction with carbonate reservoirs are associated with the transformation of spherical surfactant micelles into worm-shaped micelles as the acid concentration decreases and the fluid salinity increases. The highly viscous fluid acts as a temporary barrier and diverts the acid fluid into the untreated lower permeability zones. After treatment, the SDVA barrier breaks down when it makes contact with newly formed hydrocarbons. The objectives of this study were to examine in detail the viscoelastic surfactant effect on the HCl and calcite reaction and to examine the effects of the surfactant and acid concentrations on the SDVA apparent viscosity. Rheological measurements were conducted using rotational viscometers at 25 °C. A proposed semi-empirical rheological model describes the relationships between the apparent viscosity, rate, and HCl concentration. The core flooding tests were conducted in a specialized laboratory unit to simulate the downhole conditions. The testing unit allows for control of the pressure, temperature, and injected volume of acid flowing across the core sample. The results showed that the pressure drop varies as a function of the pumped fluid pore volume. The steady increase in the pressure as the fluid entered the core was a good indication of the actual viscosity increase in the porous rock during acid spending. The experiments were conducted using various flow rates. The core flow tests indicated that the SDVA surfactant delayed the acid breakthrough in carbonate cores. The mathematical model describing the results, obtained from the acidification experiments, is presented.

1. INTRODUCTION Matrix is one of the most common methods for chemically treating the zones near oil wells to obtain enhanced oil recovery from carbonate reservoirs. The technology of enhanced oil recovery via stimulation of carbonate with hydrochloric acid has gained wide application in the past 15−20 years thanks to advanced laboratory research, development of new reagents, and mathematical modeling of the process. Studies by Daccord et al., Hung et al., Wang et al., Hoefner and Fogler, Fredd, Fredd and Fogler, Buijse, Glasbergen and Buijse,1−10 and others have shown that the efficiency of hydrochloric acid treatment is primarily controlled by the ability to form highly conductive porous channels, “wormholes”, where most of the acid is spent. The existing models for worm-like channel development were developed by considering microprocesses at the individual pore level during carbonate rock acidification and relate wormhole formation to the Damköhler number, which is defined as the ratio of the reaction rate to the rate of reagent convective transfer to a reaction surface. Despite extensive experience with acidification and the large amount of research work aimed at improving its efficiency, many treatment cases have not been successful. The improvement of acidification efficiency is an important task during oil production, especially in mature fields. Analysis of worldwide field experience and laboratory studies suggests the following main reasons for the limited effectiveness of acid treatment: (1) In the carbonate reservoir treatment process, acid flows primarily into the most permeable zones with the highest injectivity, and therefore, the rest of the © XXXX American Chemical Society

reservoir pores are poorly affected by the acid. (2) The rate of acid reaction with carbonate rocks in water-saturated zones is substantially higher than that with oil-saturated rocks, because of different wettability, and this results in higher acid activity in water zones and an increase in their injectivities. This suggests that a basic strategy for improving acidification efficiency would be to find methods for limiting fluid flow into high-permeability layers and for diverting acid into lowpermeability layers.10,11 After repeated acid treatments, a carbonate reservoir with vertical heterogeneity and a natural fractured network can become even more heterogeneous with regard to its permeability. The carbonate reservoir matrix containing residual oil may be affected to a greater extent in the surface layer when subjected to standard acid treatments; even in the best cases, cavities are formed. Consequently, one of the important parts of an acid treatment of a carbonate reservoir is to ultimately involve the matrix in the draining process. We consider flow divergence to be the most effective way of increasing the coverage of the acid exposure zone. Polymerbased viscoelastic liquids are the most commonly used flow diverters.11−14 However, some concerns have been expressed concerning the use of polymer-based liquids in matrix acidification.15−17 Special Issue: 14th International Conference on Petroleum Phase Behavior and Fouling Received: September 29, 2013 Revised: December 6, 2013

A

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Over the past 10 years, many studies on the properties of acid systems employing viscoelastic surfactants have been published.25−37 Laboratory studies of SDVA confirmed that using viscoelastic acid systems as acid diverters in the field is very efficient. It has been shown that SDVA ensures the highly efficient formation of highly conductive channels or “wormholes” in limestone and dolomite cores under a wide range of conditions. These formations significantly reduce channel leakage compared to conventional soldering and uncross-linked, thickened acids. It has been argued that SDVA can be effectively used to control fluid leakage during acid fracturing. Another paper25 shows that polyelectrolyte addition into a SDVA system increases the thermal stability of the fluids. The rheological fluid properties can be adjusted by adjusting the fluid pH, surfactant concentration, polyelectrolyte properties, and temperature. An example of the successful application of SDVA in oil wells is described. As presented in the literature,26 the high efficiency of viscoelastic acid systems as diverters in field applications has been proven by SDVA laboratory testing. It has been shown that SDVA provides a high efficiency for wormhole creation in limestone and dolomite cores under a wide range of conditions, significantly reduces the leakage in channels compared to conventional HCl, and does not cross-link gelled acids. It is stated that SDVA can be effectively used to control leakages during acid-fracturing jobs. Another paper27 describes the results of studies on the SDVA reaction with limestone using a rotating disc apparatus. The acid apparent viscosity was measured as a function of the surfactant concentration, temperature, and shear rate. The conclusions derived from these tests are as follows. The SDVA apparent viscosity is a complicated function of the shear rate, temperature, surfactant concentration, and corrosion inhibitor concentration. The rate of limestone dissolution by HCl does not depend upon the surfactant concentration if the concentration is above 4%. The effect of the temperature on the dissolution rate is similar to the surfactant viscosity− temperature relationship. The maximum viscosity for the system is observed at 50−60 °C. The chemical dissolution rate is controlled by H+ ion mass transfer to the limestone surface. The SDVA system reaction with calcite was studied28 using a rotating disk apparatus. It was shown that the viscoelastic surfactant (betaine type) reduces the rate of calcite dissolution by HCl. The surfactant reduces diffusion coefficient H+. The temperature influence on the diffusion coefficient does not follow the Arrhenius law as it does for conventional acids. The diffusion coefficient does not increase monotonically with the temperature growth. This is explained by the unique rheological properties of acid solutions with viscoelastic surfactants of this type. The optimization of fluid composition with cationic surfactants and low pH for acid diversion has been reviewed elsewhere.29 It has been demonstrated that the surfactant is stable and compatible with the majority of acid additives. The surfactant-based acids with the addition of corrosion inhibitor and anti-sludge agents lose their viscosity as the temperature increases. Other work30 shows the results of laboratory experiments on acidification of carbonate cores 20 cm long in comparison to previously described short cores that are not longer than 4 cm. The wormhole form studied using computer tomography for

The method currently used for regulating the reaction rate involves increasing the viscosity of the acid through the use of polymers, but this method is not efficient and has serious drawbacks. The viscosity of the initial acid solution is increased, thereby reducing the mass-transfer coefficient, which stimulates the formation of a high-conductivity channel. However, increasing the viscosity also decreases the injection rate because of increased hydraulic resistance. In addition, increasing the viscosity causes colmatation of the bottom-hole formation zone that remains after polymer-based acid treatment, and it allows only a limited range of operating temperatures (up to 90 °C). To overcome some of the problems associated with viscoelastic polymer-based fluids, acidic viscoelastic surfactantbased systems have been developed and used successfully in various fields.18−21 The technology of acid carbonate treatment using self-diverting acid systems based on viscoelastic surfactants (SDVAs) is commonly used by most service companies worldwide.22−24 Viscoelastic surfactant systems (VESs) use specialized surfactants to increase viscosity during the acid removal process. When a carbonate reservoir is stimulated with hydrochloric acid, increasing the pH (up to pH 2)21 forces the surfactant monomers to form rod-shaped micelles. The resulting chlorides further stabilize these structures, especially at high temperatures, leading to increased viscosity. A viscous fluid acts as a temporary barrier that diverts acid into untreated lowpermeability zones. After processing (with the pH reaching 3− 4) and in the presence of high-viscosity hydrocarbons, the highviscosity barrier collapses (Figure 1).21

Figure 1. Mechanism of VES self-diverting acid.

The literature21 describes the successful testing of new SDVA methods under laboratory and field conditions. The studies prove that SDVA systems stimulate oil recovery and, at the same time, divert fluids into low-permeability zones and work under a wide range of operating temperatures from ambient temperature to 150 °C. The VES reagent is a modified version of the OilSEEKER formula previously developed for the temporary isolation of water-saturated interlayers during acid treatment. It should be noted that the assertion of a SDVA viscosity increase at pH 2 and viscosity decrease at pH 3−422,23 is somewhat incorrect with respect to the treatment mechanism. The higher viscosity occurs because a pH value of 2 corresponds to a hydrochloric acid concentration of approximately 0.37%, which would be pH 3.5−4 or 0.0012−0.00037%. When the HCl concentration is 0.37%, it can be assumed that the acid is almost completely spent and dissolves carbonate rocks only slightly, which takes a considerable amount of time to reduce viscosity. This should have caused problems for the development of the field, but such problems were not observed. Along with describing this mechanism, the authors mentioned the volumes of acid spent in subsequent studies. B

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working agent in combination with other systems while acidifying carbonate reservoirs. When a SDVA design is developed, it is necessary to calculate reasonable estimates for the rate and volume of reagent injections and the number of the cycles required for working fluid injection to predict the well deliverability after treatment and to estimate the anticipated profits of the acid treatment. It is possible to make such forecasts based on mathematical modeling of the heterogeneous acid dissolution of carbonate reservoirs by SDVA. When a mathematical model of the acid composition filtration is created on the basis of viscoelastic surfactants, it is necessary to numerically describe the ongoing chemical reaction based on the experimental data and to describe the correlations based on the kinetics of the reaction and the rheology of the SDVA. In this case, the effectiveness of acid stimulation largely depends upon the mass flow rate and the kinetics of the surface reaction between the acid and carbonate rocks. Taking into account the need to improve the efficiency of acid treatment, it is very important to determine the rate constant and the reaction order of the limestone dissolution in SDVA. It is also important for conducting experimental studies on the kinetics of the chemical reactions of carbonates with hydrochloric acid in SDVA. It is necessary to rheologically test the SDVA solution to understand the relationship among the SDVA viscosity, waste acid concentration, and shear rate. For adequate mathematical modeling, it is essential to understand the unique chemical and physical properties of SDVA by conducting laboratory studies on the dissolution kinetics, rheology, and filtration in the core. The objectives of the present study are as follows: to study the rheological properties of surfactant-based acids, to study the kinetics of reaction with carbonate rocks, to measure the dissolution rate for surfactant-based acid and calcite, to analyze the propagation of surfactant-based acid in carbonate cores with pressure drop monitoring across the core sample, to identify key parameters affecting the characteristics of carbonate dissolution by acid, and to develop the averaged onedimensional (1D) model of SDVA propagation across cores based on the above-mentioned studies.

described studies reveals a great difference in dissolution feature shapes when using self-diverting or conventional acids. The work31 presents the results of laboratory experiments in relation to a three-dimensional reservoir model that includes three parallel cores with permeability ratios of 1:2:4. The SDVA system shows effective acid diversion in core samples of medium and low permeability. Field tests showed high diversion properties in SDVA systems. The results of core flow experiments for acids with a viscoelastic surfactant are described in another paper,32 where the surfactant concentration was measured in the injected fluid. The SDVA injection rates varied from 1.5 to 40 cm3/min. The results of these experiments showed that the amount of SDVA required for breakthrough depends upon the reagent injection rate. Another paper33 presents the results of core flow tests in relation to the three-dimensional reservoir model, including two cores with different permeabilities. The optimum injection rate was determined for achieving the maximum acid diversion efficiency. Other work34 presents the results of laboratory studies and field tests of VES applied to low-temperature wells (25 to 65 °C). The VES systems showed high acid diversion efficiency in low-temperature wells. The growth in the corrosion inhibitor concentration has a negative effect on the SDVA viscosity profile. SDVA does not tolerate ferric iron. After acidification in the SDVA application, the viscosity is substantially reduced when contacting hydrocarbons or during washing by mutual solvents. Another paper35 describes the results of applying a new SDVA system with effective viscosity support at temperatures of up to 150 °C. On the basis of the results, the conclusions are that the new surfactant system is compatible with HCl and a solvent, even in the case of a high iron contamination. It is determined that corrosion inhibitors may reduce the acid diversion properties of SDVA. Given the urgency of problems with oil recovery, domestic uses of SDVA technology have recently been implemented.36,37 Viscoelastic surfactant-based liquids were developed along with new reagents that avoid the negative features of polymeric liquids as acid diverters and, thus, avoid secondary plugging of treated reservoirs. The proposed reagent based on non-ionic surfactants has viscoelastic properties that are similar to those used in VES systems.23,24 In the descriptions below, this reagent is referred to as Surfogel. The new SDVA composition is already used by service companies for well acidification in the fields in the Ural−Volga region.37 Over the period of 2012− 2013, more than 15 wells were acidified. The average increase in oil production in treated wells amounted to more than 20 tons/day. Analysis of well acidification operations showed that the effectiveness of acid treatments with gelling agents is comparable to that of “basic” acid diversion technologies (viscous emulsion and gelled polymers) and even exceeds it in some wells. This technology does not require additional equipment (metering pumps) for chemical feeding of acid systems at the wellhead, because all operations are carried out by standard equipment used for acid treatment with diversion. The technology is applicable for carbonate reservoirs with permeability down to 5 mDa, a water fraction not higher than 70%, and a reservoir temperature not higher than 100 °C. Oil viscosity and formation water salinity do not have a critical impact on technology performance.37 The successful SDVA application based on Surfogel allows us to recommend it as a

2. SDVA RHEOLOGICAL TESTS This test was designed to quantify the changes in acid viscosities as the acid concentration decreases. For the measurements, a series of HCl and CaCl2 mixtures was prepared to simulate the composition of fluids with different percentages of spent HCl. During the laboratory tests, we studied the SDVA injection rate using 12% hydrochloric acid with a carbonate rock (limestone) and replacement of the spent acid by the appropriate amount of newly formed calcium chloride. To describe the relationship between SDVA viscosity and shear rate, we carried out rheological measurements at 22 °C for various acid concentrations in the range of 1−12%, using a rotary viscometer. The rheoviscosimetric tests for the studied fluids were performed on a VT-550 portable rheometer (Haake, Germany) with “cylinder− cylinder” sensing elements. This unit is a high-accuracy tool that generates flow curves for the tested objects based on the linear or steplike behavior of the rotation velocity or torque. This instrument also measures the yield stress based on the linear shear stress variation. The shear tests were performed for the studied fluids with shear rates in the range of 0.1−100 s−1. The shear test establishes the relationship between the shear stress (viscosity) and the shear rate. In the first stage, we determine the rheological type for one of the spent hydrochloric acid solutions, which contained 2.76% HCl, 9.15% CaCl2, 6% Surfogel, and 0.5% Akvatek-50-standard corrosion inhibitor. It was determined that the tested SDVA fluid is a non-Newtonian fluid C

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because the fluid viscosity decreases with increasing flow velocity or pressure. This means that SDVA is a thixotropic fluid because, after heavy deformations (300 s−1), the structure is not damaged. This result is supported by the absence of a hysteresis loop between the direct and return motion curves. The properties are completely restored after the load is removed, and the discharge line coincides with the load line. Figures 2 and 6 show the experimental viscosity

Figure 4. Apparent SDVA viscosity, inhibited with 0.4% Soling.

Figure 2. SDVA apparent viscosity versus shear rate for various hydrochloric acid residual concentrations in a reaction with limestone. curves (apparent viscosity versus shear rate) for the SDVA fluid, which was tested with various degrees of acid neutralization. The obtained function describing the viscosity versus shear rate is monotonic, which is typical for non-Newtonian fluids; the viscosity decreases as the shear rate increases. In contrast, the function describing the viscosity versus acid concentration is non-monotonic. At a shear rate of 1.2 s−1, the maximum viscosity of the Surfogel-based acid fluid corresponds to HCl exhaustion equal to 56.6% or a HCl residual concentration of 8.4% (when the initial concentration is 12%). In this test, the minimum residual HCl concentration was 13.7%. During the experiments, the effects of Ca2+ and a corrosion inhibitor on the SDVA apparent viscosity were investigated too. Figures 3−5

Figure 5. Effect of the corrosion inhibitor on the apparent SDVA viscosity with 6% hydrochloric acid and 21.29% CaCl2. The rheological properties of SDVA with Surfogel show that the agent has viscous−plastic properties. Acid fluid viscosity in a reservoir should increase with distance from the reaction site because the flow rate for flat-radial flow decreases proportionally to 1/r. The rheological parameters obtained from the experiments enable Ostwald’s model to be used for the reactive SDVA fluid while modeling carbonate rock dissolution under reservoir conditions.

3. RHEOLOGICAL MODEL FOR SDVA The rheological characteristics of the fluids were derived from mathematical processing of the flow curve by considering the rheological model of fluid behavior. The calculations indicated that the SDVA rheological behavior corresponds to Ostwald’s model and that the Surfogel-based fluid is characterized by pseudo-plastic properties.38,39 τ = Kγ ṅ

Figure 3. Apparent SDVA viscosity, inhibited with 0.5% Aquatech-50standard.

In the above equation, K represents fluid consistency (Pa s). A higher viscosity corresponds to a higher consistency. γ̇ is the shear rate (s−1). n is an indicator of non-Newtonian fluid behavior. The greater the difference between n and 1, the stronger the nonNewtonian properties of the fluid, when n < 1. τ is the shear stress. In the case of a power-law relationship between stress and shear rate, the apparent SDVA viscosity μap for a given hydrochloric acid concentration can be expressed as

show plots of the SDVA viscosity at different shear rates versus Ca2+ ion concentration under a constant 6% HCl concentration using inhibitors 0.5% Akvatek-50-standard and 0.4% Soling. The results show a non-monotonic relationship between the apparent SDVA viscosity and Ca2+ ion concentration for both inhibitors. The clearest non-monotonicity is observed at low shear rates. Analysis of the data shows that, at low shear rates (1.2 s−1), the SDVA viscosity with 0.5% Aquatech-50-standard is almost 5 times greater than the SDVA viscosity with 0.4% Soling. As the shear rate increases, the viscosity difference observed with different inhibitors decreases, and at a rate of 250 s−1, the difference becomes insignificant. SDVA inhibited with 0.5% Aquatech-50-standard is more sensitive to the shear rate.

μap = Kγ ṅ − 1

(1) −1

The shear rate γ̇ (s ) is a function of the fluid filtration rate v and rock properties D

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v ϕL

Article

(2)

where L is a characteristic length corresponding to the porous medium internal scale. When the porous medium is approximated as a capillary beam, L can be calculated as the average pore diameter40 L=2

8α*k φ

(3)

where α* is an experimentally defined shape factor describing the pore structure, v is the flow rate (Darcy’s law), ϕ is the porosity, and k is the permeability. As previously shown,41,42 the shape factor for carbonates varies from 1 to 15 and has different values for high- and low-permeability layers. For Newtonian fluids, n = 1 when K is equal to the fluid viscosity. The rheological tests show that the apparent SDVA viscosity is described by a complex function of the spent acid concentration and the shear rate. SDVA viscosity is significantly increased with spending acid in a chemical reaction, which leads to flow diversion and, consequently, to more uniform wormhole propagation in carbonates. However, there are no theoretical or empirical rheological models in the technical literature that describe the effect of the acid concentration on SDVA viscosity. The rheological properties, which are controlled by the acid concentration, remain the key factors in SDVA self-diversion. On the basis of the laboratory results, an empirical rheological model was developed. The function describing apparent viscosity based on the acid concentration at a given shear rate can be written as follows: ⎡ ⎛μ ⎞ ⎛ −α(c − c )2 ⎞⎤ max ⎟⎥ μap = ⎢1 + ⎜⎜ max − 1⎟⎟exp⎜ ⎢⎣ μ − c (1 c) ⎠⎥⎦ ⎝ 0 ⎠ ⎝

Figure 6. SDVA apparent viscosity versus spent HCl for the following shear rates: (1) 1.2 s−1, (2) 2.7 s−1, and (3) 112 s−1. () Calculation and (- - -) experiment.

The functional minimization was carried out sequentially for each parameter using the dichotomy technique. The functional minimum is achieved under the following model parameters: K = 0.011 Pa sn, n = 0.308, cmax = 0.405, and α = 3.5. Thus, all possible correlations between the apparent SDVA viscosity and acid concentration at different shear rates can be calculated using the four parameters of the empirical model (eq 5). The presented rheological model, describing the apparent SDVA viscosity behavior versus acid concentration and shear rate, can be applied in the design of well acidizing with the SDVA use in carbonate reservoirs.

4. SDVA REACTION KINETIC PARAMETERS The reaction kinetic parameters are important considerations for acid treatment optimization, and they can be derived only from experimental tests. Such tests can determine the kinetic parameters based on the rate of dissolution of rock samples in SDVA. In our case, this is represented by the rate of carbon dioxide release in a reaction with limestone carbonate rock. CaCO3 + 2HCl → CaCl 2 + H 2O + CO2 ↑

(4)

where μ0 is the basic viscosity (viscosity at acid mass concentration c = 0.12 or 0), μmax corresponds to the maximum viscosity, defined relative to the reference value because of gelation, cmax is the acid concentration at maximum viscosity, and α measures the acid concentration range for gel formation, which corresponds to the reciprocal variance (width) of the correlation between viscosity and acid concentration. With an α value increase, the range of gelation decreases. A combination of eqs 1 and 4 acts as a basis of the rheological model describing the apparent SDVA viscosity as a function of the acid concentration and shear rate. ⎡ ⎛μ ⎞ ⎛ −α(c − c )2 ⎞⎤ max ⎟⎥ μap = Kγ (̇ n − 1)⎢1 + ⎜⎜ max − 1⎟⎟exp⎜ ⎢⎣ ⎝ μ0 ⎠ ⎝ c(1 − c) ⎠⎥⎦

If the reaction occurs in a porous medium, these relationships are not valid because of capillary forces, heterogeneity of rock permeability, diffusion processes, etc. The reaction occurs primarily in high-permeability and water-saturated zones because oil-saturated rock decreases the rate of the reaction. This can be explained by the low relative permeability of water solutions at high residual oil saturation and the limited penetration of acid in a rock surface covered with oil. The derived kinetic curves can be described by the Avrami− Erofeev equation.44

(5)

Similarly, the variation of the viscosity of in situ cross-linked acids with the temperature, shear rate, and pH is considered by Ratnakar et al.43 Figure 6 shows the function of the apparent viscosity versus acid concentration at different shear rates (γ̇ = 1.2c−1, 2.7c−1, and 112c−1), calculated from eq 5, with μcalc as well as measured viscosity μexp derived from the experiments. For the values μmax = 1.33 Pa s and μ0 = 0.0087 Pa s, model parameters K, n, cmax, and α were determined from the functional minimization.

m

Vt = V0(1 − e−keff t )

(6)

This equation characterizes the kinetics of topochemical processes in a heterogeneous environment. Here, Vt is the volume of gas released at time t (mL); V0 is the maximum or total amount of released gas (mL); keff is the effective reaction constant (s−1); and t is time. In this equation, the exponent m is a measure of the anisotropy of the medium. If 0 < m < 1, then the kinetic curve does not show an induction period; if m > 1, the initial reaction rate is less than the maximum rate and is characterized by a gentle slope in the initial part of the curve.

∑ (μiexp − μicalc )2 → min i

E

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The initiation of induction in the oil-saturated porous medium can be explained by a low acid diffusion flux through the oil film. This result is confirmed by the value of m, which is approximately 2 (1.7−2.2) for the oil-saturated environment (the transient diffusion flux is proportional to t1/2). When m is close to 1, the Avrami−Erofeev equation can be reduced to an equation describing a first-order reaction. As the HCl concentration decreases, the reaction rate gradually declines. Existing publications about environmental effects on viscosity behavior generally describe the relationship between viscosity and the reacted acid. In our case, the surfactant molecules with higher pH (i.e., lower acidity) and a greater bivalent cation concentration (Ca2+ and Mg2+) form worm-like micelles. The methodology of the laboratory tests to determine the reaction kinetic parameters is described in detail in the literature.44 Thus, these details are omitted, but the findings can be presented. In the laboratory experiments, it was decided to use a disintegrated limestone sample. The rock sample was extracted with an alcohol−benzene mixture, dried at 105 °C, and saturated with oil or brine water. For this purpose, oil or brine water was filtered using a Buchner funnel through a 25−30 mm thick disintegrated rock layer. Surplus liquid was removed by filtration. The laboratory tests were performed at a temperature of 22 °C. For the experiments, we used degassed oil with a dynamic viscosity of 7.011 mPa s (at 20 °C) and a brine water model with two different salinities to analyze the effect of salinity on SDVA activity: 335 g/L (Table 1) and 15 g/L (80% NaCl and 20% CaCl2).

Figure 7. Rock dissolution kinetics in SDVA solutions.

The resulting kinetic parameters of the acid fluid reactions with rock are shown in Table 3. Table 3. Kinetic Parameters for the Dissolution of Waterand Oil-Saturated Rock by SDVA core saturation water-saturated 335 g/L

water-saturated 15 g/L

Table 1. Six-Component Composition of the Brine Water Model ions

oil-saturated

concentration (mg/dm3)

2+

75150.0 11370.0 32544.0 216000.0 110.0 0.0 335.0 1.2203 2.41

Ca Mg2+ Na+ + K+ Cl− SO42− HCO3− total salinity (g/dm3) density at 22 °C (g/cm3) viscosity at 22 °C (cP)

Table 2. Acid Fluids Used for Testing Acid Activity on Carbonate Rock acid fluid (AF)

AF-1 AF-2 AF-3

12% HCl, 6% Surfogel, and 0.5% Aquatech-50-standard 12% HCl and 0.5% Aquatech-50-standard 12% HCl, 6% Surfogel, and 0.4% Soling

m

AF1: 12% HCl, 6% Surfogel, 0.5% Aquatech-50-standard AF2: 12% HCl and 0.5% Aquatech-50-standard AF3: 12% HCl, 6% Surfogel, 0.4% Soling AF1: 12% HCl, 6% Surfogel, 0.5% Aquatech-50-standard AF2: 12% HCl and 0.5% Aquatech-50-standard AF3: 12% HCl, 6% Surfogel, 0.4% Soling AF1: 12% HCl, 6% Surfogel, 0.5% Aquatech-50-standard AF2: 12% HCl and 0.5% Aquatech-50-standard AF3: 12% HCl, 6% Surfogel, 0.4% Soling

0.00019

1

0.0078

1

and

0.00016

1

and

0.00012

1

0.0096

1

and

0.00017

1

and

0.0004

1

0.0054

1

0.0007

1

and

and

The results of the laboratory experiments revealed a significant effect for acid corrosion inhibitors on acid fluid activity. Furthermore, analysis of the rock dissolution rate using SDVA showed that the dissolution of carbonate rock substantially decreased in the initial stage as the diffusion process slowed because of an increase in SDVA viscosity (Figure 7). This allowed for the total length of the wormholes to increase. Water salinity has almost no effect on the kinetics of carbonate rock dissolution by acid fluid AF-3 (12% HCl, 6% Surfogel, and 0.4% Soling). The measured difference is within the range of measurement error. Thus, soling is less sensitive to water salinity. The rate of AF-1 (12% HCl, 6% Surfogel, and 0.5% Akvatek-50-standard) dissolution in the brine is slightly higher than that in low-salinity water. Under high-salinity conditions, the gel has a partial breakdown, which can lead to a slight increase in the dissolution rate. In low-salinity conditions, Aquatech gains strength more quickly; therefore, the reaction rate is slightly lower than that in the AF-3 case. In oil-saturated rocks, Surfogel increases the reaction rate by reducing the surface tension at the liquid−rock interface. In contrast, AF-2

The kinetic parameters were determined using the acid fluids described in Table 2. Figure 7 shows the kinetic curves of gas release in the SDVA reaction with a water-saturated rock. All of the results are displayed as a relationship between the carbon dioxide volume V and reaction time t.

number

keff (sek−1)

acid fluid (AF)

F

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Table 4. Reservoir Properties of Core Samples test number

sample code

air permeability (×10−3, μm2)

length (cm)

diameter (cm)

pore volume (cm3)

porosity (%)

1 2 3 4 5 6 7 8

5-13// 3-13// 11-13// 21-13// 10-13// 6-13// 1-13// 7-13//

1.22 1.41 1.65 2.01 2.40 1.30 1.40 2.43

8.18 8.17 8.04 8.16 8.20 8.03 8.02 8.15

2.94 2.95 2.94 2.94 2.94 2.93 2.90 2.94

7.21 7.37 7.47 7.82 6.95 7.24 8.10 8.26

14.28 13.26 13.70 14.11 12.48 13.38 15.29 14.94

The aim of the study was to determine the following filtration parameters: (1) water permeability of the model before and after acid treatment, (2) acid fluid injection volume before the breakthrough (the breakthrough time was determined from the point of sharp pressure differential decline) (Vpor fraction), (3) maximum pressure gradient for SDVA injection (dPmax/dL), and (4) relationship of the pressure gradient to injected volume. In the first stage, the model of reservoir water was injected through the prepared porous medium with 100% water saturation until the differential pressure stabilized and the water permeability of the medium was measured. Next, SDVA was injected into the core sample until the breakthrough and subsequent differential stabilization occurred. Then, the injected fluid volume was measured before the breakthrough and the maximum pressure gradient that occurred during acid injection. Then, formation water was injected again, and the water permeability of the porous medium was measured. All laboratory experiments were performed under the in situ pressure and temperature for the studied reservoir conditions. The agents were injected at different rates: 0.5, 1, 3, 6, and 12 cm3/min. As an example, we will describe the results of test 2, which used a constant flow rate of 1 cm3/min. The objective of the test was to determine the reservoir characteristics of core samples 3−13 flooded by agent AF-1. The pressure gradient profile for test 2 is shown in Figure 8.

decreases the reaction rate in oil-saturated cores, because acid is not completely neutralized as a result of the protection of surface pores by oil surfactant components, including asphaltenes, resins, and naphthenic acid.

5. PHYSICAL PROCESS MODELING OF SDVA INJECTION INTO CARBONATE ROCKS Core flooding experiments were performed to determine the key parameters controlling carbonate dissolution in SDVA and to enable the physical modeling of the process. The experiments involved monitoring the pressure drop in the core and measuring the amount of injected agent. In addition, the results of the laboratory tests can be used to adjust simulations of SDVA dissolution by accounting for acid fluid rheology. 5.1. Core Samples and Formation Fluid Preparation. In the laboratory experiments, we used rock samples (limestone) with a carbonate content of 98.9% from oil fields in the Volga−Ural petroleum basin. To carry out the acid fluid injection experiments on the available core material, we selected samples of porous and fractured porous rocks with similar linear and reservoir properties. Prior to the tests, the core samples were extracted with an alcohol−benzene mixture in a Soxhlet apparatus, washed with distilled water to remove salt, and dried in an oven. The gas permeability of the core samples was measured using an UltraPoroPermTM-500 permeameter−porosimeter. This unit allows for measurements of pore volume, specific gas permeability, Klinkenberg permeability, permeability (in the range from 0.01 mDa to 2 Da), and porosity (from 0 to 40%). The linear and reservoir properties of the samples used as porous medium models are shown in Table 4. The formation water model, used in laboratory experiments, has a salinity of 15 g/L (80% NaCl and 20% CaCl2), a density of 1.0095 g/cm3, and a viscosity of 1.05 mPa s at 25 °C. Acid fluids for core sample flooding were selected on the basis of the results of physical and chemical tests (Table 5). Table 5. Acid Fluids Selected for Core Flooding number

acid fluids

AF-1 AF-2

12% HCl + 6% Surfogel + 0.5% Aquatech-50-standard 12% HCl + 0.5% Aquatech-50-standard

Figure 8. Pressure gradient, test 2.

Acid fluid AF-2 was selected for a comparison. 5.2. Methodology of the Carbonate Rock Acid Dissolution Tests. The physical modeling of the dissolution of carbonate rocks in acids involves injecting SDVA into cores saturated by formation water. For the tests, we used the UIK-5 core laboratory unit (test 2).

Figure 9 shows photographs of core samples 5−13 after treatment with AF-1 in test 2. The figure demonstrates small cavities and a large wormhole at the inlet face. The wormhole (high-permeability channel) is also visible at the outlet face. The steady differential pressure growth during SDVA injection G

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6. SIMULATION OF CARBONATE ROCK DISSOLUTION BY SDVA IN A CORE SCALE In this section, the mathematical model describing the results, obtained from the acidizing experiments, is presented. 6.1. Main Assumptions of the Mathematical Model. To calibrate the adjusted mathematical model using the results of the SDVA injection tests, we propose applying the following scheme of fluid flow in water-saturated cores. The two-phase flow was initially neglected because we sought to review basic mechanisms affecting the process efficiency. The fluid flow includes the initial reactants and products of the chemical reactions (hydrochloric acid, water, carbon dioxide, soluble calcium salt, and carbonate rock). It was assumed that the flow zone was initially saturated with a fluid of viscosity μ0. Three zones of different mobilities with respect to the SDVA flow in a porous medium were identified (Figure 11).

Figure 9. End faces of core samples 5−13 after treatment by AF-1, test 2.

is a good indicator of the actual viscosity increase in the porous rock during acid spending. The results of the laboratory tests show that the breakthrough volume is a non-monotonic function of the injection rate (Figure 10). There is an optimal regime for carbonate rock

Figure 11. SDVA flow in a porous medium.

Descriptions of these zones are given as follows: (1) Zone 1 corresponds to the fluid originally saturating the formation and having initial permeability k0 and viscosity μ0. In this zone, the coordinate x varies from the resistance zone xr to the outlet core face. (2) Zone 2 is a resistance zone, in which a significant viscosity increase is observed. The fluid mobility in this zone is guided by the relationship of SDVA dynamic viscosity with the acid concentration at a constant injection rate and, hence, a constant shear rate. In this zone, the x coordinate varies from the dissolution zone length to xr. (3) Zone 3, in which the actual viscosity is broken down, is the wormhole zone with permeability kwh = k(ϕ). In this zone, the x coordinate varies from the inlet core face to dissolution boundary xwh. Let us introduce parameter θwh, the breakthrough wormhole volume, defined as the SDVA volume injected before the breakthrough and divided by the core pore volume. Let us also introduce parameter θr, the resistance zone breakthrough volume, defined as the SDVA volume injected before the resistance zone breakthrough and divided by the pore volume. The moment of resistance zone breakthrough can be defined on the basis of the experimental graph of the pressure gradient versus time at the beginning of the pressure gradient decline. Core flooding tests can provide flawed determinations of the fluid breakthrough volume, which are associated with the rock sample end effect. The error in the injected pore volume can reach 15%. To adjust the breakthrough volume, we introduced correction coefficients βwh and βr in the front propagation equation. The velocities of xwh and xr front propagation can be determined from the following equations:45

Figure 10. Injected AF-1 and AF-2 fluid volumes versus injection rate before breakthrough.

dissolution, characterized by the minimum volume injected before the breakthrough.1−7 In our case, the optimal SDVA treatment occurs at an injection rate of 1 cm3/min, and the minimum injected volume before breakthrough is 0.2 pore volume. When injecting conventional acid fluid (AF-2), the optimal treatment is obtained at a rate of 6 cm3/min, with a minimum injected volume before breakthrough of 0.23. These findings corroborate the dissolution kinetics studies. The chemical reaction constants for SDVA and conventional acid fluid (AF-1 and AF-2) are 0.00012 and 0.0096 s−1, respectively (Table 3). In the optimal treatment, a higher dissolution rate corresponds to a higher injection rate. When the injection rate is increased, we increase the pressure gradient by a corresponding amount. The SDVA injection pressure gradient exceeds the pressure gradient observed during conventional acid fluid injection, which confirms that the SDVA viscosity increases during flooding and acid depletion in cores. The experimental results demonstrate that the SDVA fluid flows through a more complex channel network than conventional hydrochloric acid because of its self-diverting properties. H

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dx wh v 1 = βwh θwh ϕ0 dt

(7)

dx r 1 v = βθ ϕ dt r r 0

(8)

are unchanged by the chemical reaction.47,48 It is also possible to apply Amagat’s law by assuming that the acid and reaction product concentrations are small and that the change in the fluid density is negligible. 6.2. Basic Equations. The basic equations describing the acid dissolution process in terms of Euler variables t and x for 1D linear flow have the following form for area 0 ≤ x ≤ xwh:

where ⟨v⟩ = Q/A is the average filtration rate (Darcy velocity), Q is the injection rate, ϕ0 is the initial core porosity, and A is the filtration zone cross-sectional area. Turning to practical problems with a variety of the conditions of the acid dissolution process, some assumptions and simplifications must be specified for the rock physical properties, which can be used for solving problems and do not have a considerable impact on the final result. In a large-scale approximation, the diffusion velocity of phase components can be neglected because of the small values of these parameters. The acid and salt fluid densities can be considered equal to the water density because of their low concentrations, and the equilibrium hypothesis can be accepted. Indeed, considering the order of the characteristic reaction time τ ≈ 102 s and the flow velocity ν ≈ 10−6 m/s, we can estimate the length of the chemical reaction zone as l = τν = 10−4 m. When this value is compared to the characteristic scale of the problem, L = 10−1 m (core scale) and L = 102 m (formation scale), we come to the concept of system equilibrium. Let us assume that the acidic solution has the same density as water. Because of the low concentration of dissolved substances, we accept the hypothesis of system equilibrium and consider the acid and salt solution densities to be equal to the water density ρ0l , i.e., ρa = ρs = ρ0l = constant and mineral density ρm = constant. While the carbonate matrix acidizing is simulated, the elastic rock properties are not usually taken into account. Generally, porosity changes are related to the carbonate rock dissolution. The solid medium compressibility is negligibly small. For example, the compressibility of limestone varies little with the pressure variation (0−25 MPa) and equals (25−27) × 10−6 MPa−1. For the simulation purposes, the porous media were considered as incompressible. The injected acid compressibility factor is approximately 10−4 MPa−1, which is negligible for the injection conditions given the small volume of injected agent. Therefore, we can assume fluid incompressibility. The reaction products are the highly soluble calcium chloride salt, CaCl2, and carbon dioxide, CO2. The solubility of the latter depends upon the reservoir temperature, pressure, formation water salinity, rock porosity, and acid concentration. To estimate the carbon dioxide concentration in the water phase within the acid plug zone, we can use the equilibrium reaction solution.46 The calculations presented in this work show that, when 20% hydrochloric acid is injected, the carbon dioxide concentration in the reaction front is 7%. Thus, the acid−rock reaction occurs in the presence of fluid injection and flow, in a porous medium, and most of the reaction products move in front of the acid plug. The concentrations of the products are less than they would be in a closed-volume reaction, and they do not significantly affect the acid dissolution process. This finding allows us to make an assumption about carbon dioxide solubility in the water phase. Generally, changes in phase density during a chemical reaction are small, and this is especially true for reactions in liquid solutions or gases. In these cases, Amagat’s law can be applied, meaning that both the mass and volume of the system

acid concentration equation ϕ

J ∂C ∂C +v =− k ∂t ∂x ρa

(9)

porosity variation equation J ∂ϕ = νm k ∂t ρm

(10)

pressure distribution equation ρ − ρa ∂ ⎛ k(ϕ) ∂p ⎞ ⎜⎜ − ⎟⎟ = νmJk m ∂x ⎝ μa ∂x ⎠ ρm ρa

(11)

Darcy’s law is given in the following form:

v=−

k(ϕ) ∂p μa ∂x

For the permeability calculation, we propose the following empirical relationship based on the Kozeny−Carman correlations: ⎛ ϕ ⎞n k = k 0⎜⎜ ⎟⎟ ⎝ ϕ0 ⎠

(12)

where k0 and ϕ0 are the initial permeability and porosity, respectively, and n is a constant derived from experimental data. In a previous study, a value of n > 6 was experimentally determined.49,50 In this zone, permeability can be calculated by eq 12 after calculating porosity from eq 10. The acid viscosity is constant and equal to the initial value μ = μa, νm is the stoichiometric reaction factor. Here, we note that formation pressures are high enough that liberated carbon dioxide stays in a liquid state and is completely dissolved in the water phase. The formulated system of equations must be completed by the kinetics law of the chemical reaction or the Jk function of reactant concentrations and reaction products. The reaction kinetics is described as follows: Jk = ρa keff C

(13)

In zone xwh ≤ x ≥ xr, the SDVA viscosity is a complex function of the acid concentration and shear rate (eq 5), μap = μ(C,γ̇). This relationship must be generated from the SDVA rheological test data. We assumed that the fluid flow was guided by the linear Darcy’s law v = −(k0/μ(γ̇,C))(∂p/∂x). In this zone, acid is spent and permeability is equal to its initial value. The pressure is described by the following equation: ρm − ρa k 0 ∂p ⎞ ∂ ⎛ ⎜− ⎟ = νmJk ∂x ⎝ μ(C , γ )̇ ∂x ⎠ ρm ρa

(14)

In zone xr ≤ x ≤ Lc, pressure can be calculated as I

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∂ ⎛ k 0 ∂p ⎞ ⎜⎜ ⎟⎟ = 0 ∂x ⎝ μ0 ∂x ⎠

It is necessary to evaluate the following parameters of the model: factors βwh, βr, Damköhler number Da, and exponent n in eq 12. The identification of model parameters is related to the solution of inverse problems that are ill-posed.51 Additional assumptions can turn ill-posed problems into well-posed problems. The use of supplementary (a priori) information of a qualitative nature yields the Ivanov method of quasisolution.52 The well-known published data are used as a priori information.6,7,45,49,50 In this case, as the inverse problem solution, the parameter set (βwh, βr, Da, and n) gives a minimum of the residual function (eq 19) that can be considered as a correct solution for the described problem. Essentially, the obtained set (βwh, βr, Da, and n) acts as a quasisolution for the ill-posed problem.52 The model was adjusted against the actual and calculated pressure drop dynamics (Δpexp and Δpcalc i i , respectively) by minimizing the corresponding function.

(15)

The initial and boundary conditions are specified as C|t = 0 = 0,

Cx = 0 = C0

(16)

ϕ|t = 0 = ϕ0

(17)

k(ϕ) ∂p μa ∂x

p|t = 0 = p0 ,

=− x=0

Q , A

p|x = Lc = p0 (18)

The specified conditions mean that, at the inlet core face of area A, the acid is injected at a constant rate Q. The filtration unit maintains conditions similar to the reservoir conditions. At the onset time, pressure equals the reservoir pressure p0, and porosity has an initial average value of ϕ0 throughout the entire core length. The concentration at the inlet core face was maintained at its maximum level for the duration of the test. When calculating pressure, permeability and viscosity are determined separately for each zone. This system of equations is supplemented by the frontal advance eqs 7 and 8. The considered mathematical model of carbonate core dissolution using SDVA based on a viscoelastic surfactant is a nonlinear system of differential equations, for which only numerical solution is possible. For the numerical simulation, the differential equation system (eqs 9−18) was non-dimensionalized. The calculations involve the following dimensionless variables: x̅ =

x , L

p̅ = t0 =

t̅ = p , p0 ϕ0ALc Q

t , t0

k̅ = ,

C̅ = k , k0

t̅ =

C , C0

μ̅ = tQ , ϕ0ALc

ϕ =

J(βwh , βr , Da , n) =

∑ (Δpiexp − Δpicalc )2 → min i

(19)

An iterative algorithm for minimization of the residual functional was developed. Figure 12 compares the simulated and experimental data (test 2): pressure gradient on the core sample versus SDVA pore volumes injected.

ϕ , ϕ0

μ , μ0 Da =

keff ALc Q

where Lc is the core length, C0 is the initial mass concentration of an agent, v0 is the linear velocity of reagent injection, and p0 and t0 are the characteristic pressure and time, respectively. The variable t ̅ represents dimensionless time, which is equal to the ratio of injected fluid to pore volume, i.e., the number of pore volumes. Previous works mention the Damkohler number,6,7 which describes the relationship of chemical reaction rates and reagent supply. The non-dimensionalized system of equations provides a similar criterion with the identical meaning.

Figure 12. Pressure gradient versus injected pore volumes (test 2, with a shear rate of 244 s−1).

The minimum value (eq 19) for optimal core treatment (test 2, Figure 8) is achieved when Da = 0.51, n = 7, βr =1.32, and βwh = 1.12. The results of the mathematical modeling are consistent with the experimental observations. Thus, the numerical simulation proves the correctness of the mathematical model of carbonate dissolution by SDVA in a core scale. The modeling and laboratory data are in a good agreement. This validates the model describing reactive dissolution with SDVA. 6.3. Analysis of Results. The numerical simulation of acid core treatment using SDVA allowed for the analysis of the experimental results. When self-diverting acid was injected, the pressure drop along the core sample initially increased, but after

k L k AL Da = eff c = eff c v0 Q

The results of the calculations for the flooding tests were simulated using the empirical rheological model (eq 5), chemical dissolution constants k eff (Table 3), and a mathematical model of carbonate acid reaction in SDVA in a core scale (eqs 9−18). For the numerical solution, we developed a software package. All physical parameters have the values accepted for the experiments. Densities of aqueous solutions and minerals are ρ0l = 1023 kg/m3 and ρm = 2160 kg/m3. J

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Other modifiers, including corrosion inhibitors, also have a considerable effect on viscosity. The thixotropic properties of Surfogel-based SDVA show that the viscosity increases with the distance from the bottom hole, where the shear rate falls. This effect leads to the diversion of the less viscous acid that causes a less active reaction. The laboratory experiments made it possible to determine the physical−chemical parameters needed for adjusting the mathematical model of the layered heterogeneous carbonate reservoir treatment with SDVA fluids. This model is used to predict the final interlayer distribution of injected acid volumes. The injection tests on core samples revealed the optimal SDVA injection regimes, which must be considered when optimizing the SDVA treatment design for well stimulation. The developed empirical model describes the dependence of surfactant-based acid viscosity upon the acid concentration during chemical reactions. The model is consistent with the experimental data and describes the key features of SDVA rheology: maximum viscosity increases from the reference value, and the behavior is controlled by the acid injection rate. The laboratory experiments on physical process modeling confirmed the correctness of the proposed carbonate dissolution model under SDVA treatment. On the basis of the results, we adjusted the model of acidizing water-saturated carbonate cores using self-diverting acid fluids. The mathematical model of carbonate rock dissolution by SDVA allowed for the simulation of acid treatment in the core scale. The simulation results are very consistent with the laboratory observations of pressure gradient behavior in core samples. The constructed mathematical model of carbonate treatment with self-diverting acid based on a viscoelastic surfactant is based on the laws of physical and acid−carbonate interactions and laboratory observations. It can be used for the design and optimization of well-acidizing reactions for a wide range of reservoirs and reservoir fluids.

gel breakthrough, it began to decline. The process of SDVA acidizing occurs in three phases. In the first phase, a gelation process occurs, forming a resistance zone. At this time, we observe a minor permeability variation, but because of the gel formation, there is a considerable increase in fluid viscosity (Figure 8). Therefore, the effective mobility decreases, and the pressure drop through the core increases. Next, there is a phase of gel propagation: the resistance zone reaches its maximum width, extending at a constant velocity. In this phase, viscosity is almost constant, but permeability and effective mobility start to grow slowly and the overall pressure drop starts to decrease. In the third and final phase, the gel begins to flow out of the core; thus, the fluid viscosity decreases. As a result, the effective mobility increases very quickly and the pressure drop starts declining sharply. This is the moment of wormhole breakthrough. The rate of mobility growth is very high. The pressure gradient drops to almost zero. The plot of pressure drop versus dimensionless time shows two important points. The gel breakthrough time corresponds to the pressure gradient inflection point (the beginning of the decline section), and the wormhole breakthrough time corresponds to the time when the pressure drastically drops to almost zero. In dimensionless variables, these points correspond to the number of pore volumes injected before resistance zone breakthrough and the number of injected pore volumes before the wormhole breakthrough, respectively. Figure 13 shows the extension of the resistance zone along the core sample in test 2. The SDVA viscosity growth in the

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS

■

REFERENCES

This study was supported by the Russian Foundation for Basic Research (Project 11_01_97015_r_povolzh’e.a).

Figure 13. Acid viscosity along the core sample at different time points.

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resistance zone points to a gelation process. With time and spent acid, the width of this zone increases slightly. The maximum viscosity depends upon the concentration and shear rate.

7. CONCLUSION The laboratory experiments showed that the hydrochloric acid injection rate and calcium ion concentration have a significant effect on SDVA properties during carbonate rock treatment. There is no significant effect for pH. The maximum acid viscosity occurs at pH 2−4, when hydrochloric acid is almost spent. K

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