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Mathematical Inhibitor Model for Barium Sulfate Scale Control Shiliang He,* Amy T. Kan, and Mason B. Tomson Department of Environmental Science and Engineering, Rice University, P.O. Box 1892, Houston, Texas 77251 Received October 16, 1995. In Final Form: January 9, 1996X A semiempirical mathematical model has been developed to predict inhibitor efficiency for barium sulfate scale control in industrial processes. This model can be used for selecting effective inhibitors and determining the minimal effective concentration needed for a given system. The model incorporates experimental data of the nucleation and inhibition kinetics. Specifically, the induction period in the presence and absence of scale inhibitors has been measured experimentally and inputted into the model: Cinh ) (1/b) log[tinh/t0], where Cinh is the inhibitor concentration, tinh is the inhibition time (e.g., 20 min), t0 is the nucleation induction period of the scaling mineral crystal, and b is the inhibitor efficiency. The inhibition kinetics of barium sulfate nucleation with bis(hexamethylene)triaminepenta(methylenephosphonic acid) (BHMTPMP) and several other polyphosphonate and polyacrylate inhibitors have been measured. Many factors which are important to nucleation and inhibition kinetics, such as the degree of supersaturation, temperature, and solution pH, have been included in the inhibitor model. The model prediction for barium sulfate scale control was in good agreement with laboratory observations and field experience.
Introduction
Inhibitor Treatment Model
Formation of mineral scale deposits is undesirable in various industrial processes where water and water treatment are involved, such as cooling systems, boilers, heat exchangers, filtration, mineral processing, oil and gas production, and geothermal systems.1 Common waterformed scales include calcium carbonate, calcium sulfate, and barium sulfate. Control and treatment of barium sulfate deposits containing naturally-occurring radioactive materials (radium in particular) (NORMs) is the latest task, particularly in the petroleum industry.2 NORM scales have been found in both offshore and onshore wells in production facilities in the Gulf Coast of Louisiana, Texas, the Michigan Basin, the North Sea, and many other areas.3-5 Governmental regulations concerning NORMs require an effective approach to eliminate their formation. The most practical and efficient method to control scale deposits is to prevent their formation by chemical inhibition using scale inhibitors1,2,6 such as polyphosphates, phosphate esters, phosphonates, polyacrylates, and polyelectrolytes. Currently, there is no theory or model available to predict the appropriate inhibitor to use and its effective dosage. Static and dynamic inhibitor testing under specific conditions in the laboratory is often required. This paper describes a semiempirical mathematical model for inhibitor treatment which can be applied to barium sulfate under a wide range of conditions. The model is based on the nucleation theory and experimental data of induction periods in the presence of scale inhibitors and can be used to estimate the effective inhibitor and its dosage under a given condition, on the basis of physical chemical parameters commonly measured.
It is commonly observed in laboratory experiments of both spontaneous precipitation and seeded crystal growth that the presence of an inhibitor additive, even in the threshold range (typically between 0.1 and 10 mg/L) in a supersaturated solution, will delay the nucleation and precipitation of sparingly soluble minerals and thus prolong the induction period.6-14 These minerals include calcium carbonate, calcium sulfate, strontium sulfate, and barium sulfate. The prolongation of induction period depends on the nature of the inhibitor and its effective concentration at a given condition. For a given inhibitor, generally the higher the inhibitor concentration, the longer the induction period. The only exception found is wherein the inhibitor precipitates with divalent cations (e.g., Ca2+, Mg2+, Sr2+, Ba2+, and Fe2+) when the solution is supersaturated with the metal inhibitor salts.14 The induction period of barium sulfate nucleation in synthetic brines with chemical composition of a typical oil and gas well in Texas (the Guerra well) in the presence of inhibitor additives was measured by the turbidity method (for details, see refs 14 and 15). The brine composition is shown in Table 1. Four types of inhibitors were used in this study, including polyphosphonates, polyacrylates, phosphinopolycarboxylates, and sulfonated polyacrylates (Table 2). The induction period of BaSO4 in the presence of various inhibitor concentrations at SI ) 2.7 and 70 °C is listed in Table 3. A dimensionless parameter called the “nucleation inhibition index (NII)” can be defined as the logarithm of the induction period in the presence of inhibitor divided by the induction period in the absence of inhibitor. As
* To whom correspondence should be addressed. E-mail:
[email protected]. X Abstract published in Advance ACS Abstracts, March 15, 1996. (1) Cowan, J. C.; Weintritt, D. J. Water-Formed Scale Deposits; Gulf Publishing Co.: Houston, TX, 1976. (2) Oddo, J. E.; Tomson, M. B. Oil Gas J. 1994, 1/3/94, 33. (3) Smith, A. L. J. Petrol. Technol. 1987, 39, 697. (4) Anderson, B. Ocean Industry 1990, 90, 33. (5) Gray, P. R. J. Petrol. Technol. 1993, 45, 12. (6) Veter, O. J. J. Petrol. Technol. 1972, 24, 997.
(7) Naono, H.; Miura, M. J. Chem. Soc. Jpn. 1965, 38, 80. (8) Naono, H. J. Chem. Soc. Jpn. 1967, 40, 1104. (9) Sarig, S.; Raphael, M. J. Cryst. Growth 1972, 16, 203. (10) Liu, S.-T.; Nancollas, G. H. J. Colloid Interface Sci. 1975, 52, 582. (11) Rizkalla, E. N. J. Chem. Soc., Faraday Trans. 1 1983, 79, 1857. (12) Van der Leeden, M. C.; Kashchiev, D.; van Rosmalen, G. M. J. Colloid Interface Sci. 1992, 152, 338. (13) Bromley, L. A.; Cottier, D.; Davey, R. J.; Dobbs, B.; Smith, S.; Heywood, B. R. Langmuir 1993, 9, 3594. (14) He, S. L.; Oddo, J. E.; Tomson, M. B. Appl. Geochem. 1994, 9, 561. (15) He, S. L.; Oddo, J. E.; Tomson, M. B. J. Colloid Interface Sci. 1995, 174, 319.
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Table 1. Chemical Composition of the Synthetic Guerra Brine concentration species Na+ Mg2+ Ca2+ Sr2+ Ba2+ ClSO42IS (M) pH
mg/L
mM
19871.8 54.0 6500.0 700.0 550.0 43000.0 48-384
864.4 2.2 162.2 8.0 4.0 1228.6 0.5-4.0 1.42 6.50
Table 2. List of Inhibitors and Their Molecular Weights Used in This Study inhibitor name
abbreviation
molecular weight
Bis(hexamethylene)triaminepenta(methylenephosphonic acid) Bis(hexamethylene)triaminetetra(methylenephosphonic acid) Diproplethylenetetraaminehexa(methylenephosphonic acid) polyacrylic acid phosphinopolycarboxylic acid phosphinopolycarboxylic acid sulfonated polyacrylic acid
BHMTPMP
685
BHTMP
591
DETHMP
738
PAA PAA PPPC SPA
3500 3500 3800 3000
Table 3. Induction Period of Barium Sulfate Nucleation in the Presence of Inhibitors at 70 °C (SI ) 2.7, pH ) 6.5) in the Synthetic Guerra Brine tind (s) Cinh (mg/L) BHMTPMP BHTMP DETHMP PAA PPPC SPA 0.0 1.0 2.5 5.0 7.5 10.0 15.0 20.0
5 7 34 196 2391 24132
5 6 23 131
5 6 12 48 577 2180
5 6 11 21 87 405
5 5 12 19 28 58 25 9
5 5 7 10 16 24 77 258
will be shown below, the NII is found to be directly proportional to the concentration of the inhibitor used, as in eq 1 (see Figure 1).
NII ) log(tinh/t0) ) bCinh
(1)
where t0 and tinh are the induction period of barium sulfate nucleation in the absence and presence of the inhibitor, Cinh is the effective inhibitor concentration in mg/L or mmol/L, and b is the slope called the inhibition efficiency in L/mg or L/mmol and is assumed to be a constant at a given condition, such as supersaturation, temperature, and pH (see Table 4). Generally the greater the value of b, the greater the inhibition efficiency. Therefore, the appropriate inhibitor to be used under a given condition can be predicted on the basis of the inhibition efficiency, that is, the value of b, for different inhibitors. For example, BHMTPMP has the highest b value and is thus predicted to be the most effective inhibitor among the various scale inhibitors we tested. The inhibitor dosage (mg/L) can be estimated on the basis of eq 2, rearranging from eq 1.
Cinh ) b-1[log(tinh) - log(t0)]
(2)
The inhibition time (tinh) in eq 2 is equal to the time period during which the system is required to be protected from scaling. The value of tinh may be chosen on the basis of the needed detention time in a process or chosen
Figure 1. Nucleation inhibition index (log[tinh/t0]) as a function of the inhibitor concentration for several inhibitors. All experiments were done at 70 °C, an SI of 2.7, and a pH of 6.5. Solid and open circles are for BHMTPMP and PAA, respectively. Solid and open diamonds are for DETHMP and PPPC, respectively. Crosses are for SPA. The curved-down shape of inhibitor PPPC is because of precipitation of calcium PPPC salts as the PPPC concentration increases over the saturation limit (for details, see ref 16). Straight lines are the curve fitting for each inhibitor. Table 4. Inhibition Efficiency (b) of Several Inhibitors for Barium Sulfate Scale at 70 °C (SI ) 2.7, pH ) 6.5) inhibition efficiency (b) inhibitor
L/mg
L/mmol of polymer
BHMTPMP BHTMP DETHMP PAA PPPC SPA
0.38 0.30 0.27 0.19 0.11 0.09
260.3 177.3 199.3 665.0 418.0 270.0
arbitrarily, e.g., 20 min. In the case of the oil/gas production system, it will be the residence or detention time of produced water in the system prior to treatment or disposal, which includes the flowing time from the bottom of the hole to the surface and the detection time of the brine in the surface equipment (ts), as in eq 3:
tinh )
πDR2fW + ts QW
(3)
where D is the depth of the well, R is the inner radius of the pipe, fW is the fraction of the brine in the total fluid, and QW is the brine production rate (e.g., m3/s). The typical inhibition time in oil or gas wells ranges from minutes to hours. The nucleation induction time (t0), in eq 1 or 2, can be calculated from experimental data of barium sulfate nucleation rates in the absence of inhibitors, which is a function of saturation index and temperature on the basis of classic nucleation theory.15
log t0 ) a1 + a2 SI + a3/T
(4)
where a1 is a constant, a2 is the supersaturation coefficient, a3 is the temperature coefficient, SI is the saturation index of the brine solution with respect to barium sulfate, which is the logarithm of the saturation state (Ω) defined in eq 5, and T is the absolute temperature in Kelvin.
Ω)
aBa2+aSO42Ksp
(5)
where aBa2+ and aSO42- are the activities of barium and
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Table 5. Summary of the Inhibitor Efficiency (b) of BHMTPMP under Various Conditions for Barium Sulfate Inhibition T (°C)
pH
SI
b (L/mg)
25 30 40 50 60 70 80 90 70 70 70 70 70 70 70
6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 3.19 4.00 5.17 7.45 6.50 6.50 6.50
2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.70 2.70 1.70 2.00 3.00
1.71 1.37 0.97 0.62 0.46 0.38 0.15 0.06 0.07 0.12 0.17 1.40 7.10 2.70 0.09
sulfate and Ksp is the thermodynamic solubility product of barium sulfate. Model Parameters All the model parameters can be obtained from experimental data. Parameters for the nucleation induction period have been fitted from the experimental data of barium sulfate nucleation in NaCl solutions from 0 to 6 m at temperatures from 25 to 90 °C.15 They are
a1 ) -0.595, a2 ) -1.125, and a3 ) 1520 A negative supersaturation coefficient (a2) and a positive temperature coefficient (a3), coupled with T being in the denominator, are consistent with classic nucleation theory,15,16 which implies that higher SI and T both increase nucleation rate and lower the induction period. The inhibition efficiency (b) is an inhibitor specific constant at a given condition. An example is given as follows for barium sulfate under the conditions of SI of 2.7, T of 70 °C, and pH of 6.5, as in Table 4. The value of b varies from 0.38 L/mg (BHMTPMP) to 0.09 L/mg (SPA) on the weight base or from 665 L/mmol (PAA) to 177 L/mmol (BHTMP) on the molar base. The inhibition efficiency for a given inhibitor is a function of saturation index, temperature, and pH (see Table 5), as in eq 6:
log b ) a4 + a5 SI + a6 pH + a7/T
(6)
where a4 is a constant, a5 is the supersaturation coefficient, a6 is the pH coefficient, and a7 is the temperature coefficient, which can be fitted from experimental data (Figure 2). The functional form was based on nucleation theory and the experimental observation of the influence of inhibitor additives on the nucleation rates as a function of supersaturation, temperature, and pH.14 A pH term was added here because it is commonly observed that the inhibition of nucleation and crystal growth depends upon the solution pH due to dissociation of the inhibitor acid.14,17 These constants or coefficients (a4-a7) are inhibitor specific. An example is given for the inhibitor BHMTPMP as follows:
a4 ) -5.6, a5 ) -1.4, a6 ) 0.275, and a7 ) 2377 A negative a5 means that as SI increases the inhibition efficiency decreases, which is supported by classic nucle(16) Mullin, J. W. Crystallization, 3rd ed.; Butterworth Heirmann: New York, 1993. (17) Austin, A. E.; Miller, J. F.; Vaughan, D. A.; Kircher, J. F. Desalination 1975, 16, 345.
Figure 2. Dependence of the inhibitor efficiency (b, L/mg) on saturation index (SI), pH, and temperature (T): A, SI dependence of log(b - SI) at 70 °C and a pH of 6.5; B, pH dependence of log(b - pH/(b - 6.5)) at 70 °C and an SI of 2.7; C, T dependence of log(b - T/(b - 70)) at an SI of 2.7 and a pH of 6.5.
ation theory16 and often observed experimentally.6,12,14 Generally the inhibition efficiency increases as the solution pH increases, probably because of the advancing dissociation of the inhibitor acid.17 Also, the inhibition efficiency decreases as the temperature increases, suggested by the 1/T term and the positive coefficient (a7). In order to check the validity of the inhibitor model, dynamic inhibitor testing has been performed to obtain the minimal inhibitor dosage using a dynamic flowthrough apparatus.18 The apparatus simulates the oil/ gas production system in the laboratory in a wide range (18) Oddo, J. E.; Tomson, M. B. J. Petrol. Technol. 1982, 34, 2409.
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Figure 3. Comparison of the model prediction with experimental data of the minimal effective inhibitor dosage. The solid line is the prediction from the inhibitor treatment model, and the solid circles are the experimental data. Table 6. Data of the Minimal Inhibitor Concentrations to Inhibit Barium Sulfate Scaling in the Flow-Through System under Different Degrees of Supersaturation at 70 °C in the Synthetic Guerra Brine SI (barite)
Cmin (BHMTPMP) (mg/L)
1.70 1.82 1.89 1.95 2.00 2.05 2.35 2.52 2.65 2.74
0.19 0.25 0.31 0.38 0.46 0.65 1.8 4.7 8.2 13.5
of temperatures (-18 to 150 °C), pressures (1-300 bars), and flow rates (0.1-20 mL/min). The dynamic inhibitor test was undertaken at 70 °C with a synthetic brine with a composition similar to that of the Guerra well in south Texas, which has a barium sulfate scale problem. Two streams of solutions (one with barium and another with sulfate only, but both with the same composition of other components) were preheated to the desired temperature and pumped into a loop of poly(ethere ether ketone) (PEEK) tubing. The inhibitor was added into the solution containing the sulfate ion. The residence time of the synthetic brine after mixing in the testing system was about 20 min. The sulfate concentration of the effluents was monitored, and the measurable difference between the input and output in sulfate concentrations was used as an indicator of scaling, in addition to measuring the pressure differentials.18 Initially, high concentrations of inhibitor were used and no scale was formed. Then, the inhibitor concentration was lowered gradually until scaling was encountered and the inhibitor concentration just prior to the occurrence of scaling was determined as the minimal inhibitor dosage. The barium sulfate saturation index of the Guerra brines was varied from 1.7 to 2.7 by changing the sodium sulfate concentration from 0.5 to 4.0 mM. The minimal inhibitor (BHMTPMP) concentrations needed to prevent scaling in the system under various SI conditions are listed in Table 6. A comparison was made between the model prediction and the minimal effective inhibitor dosage over a wide range of saturation indices obtained from the flow-through inhibitor test (Figure 3). It can be seen that the model prediction and experimental data were in good agreement.
Figure 4. Model prediction of the minimal effective inhibitor dosage of BHMTMP for a typical oil and gas well under different conditions. The inhibition time is about 21 min: A, At a pH of 6.5 (T of 100, 85, 70, and 50 °C); B, At 85 °C (pH of 4.5, 5.5, and 6.5).
Application of the Model to the Oil and Gas Production Systems The present inhibitor treatment model is a novel method to predict scale control and treatment. When the model parameters have been obtained from experimental data for a series of scale inhibitors, the model can be applied to choose the appropriate inhibitor to use and the effective concentration needed for the production system to be protected. Once the brine composition and production parameters are obtained, the present model can be coupled with a saturation index (SI) program (e.g., EQPITZER19 or Oddo-Tomson SI20) to design effective scale control and treatment. In the case of downhole scale control, the model can also be very useful to determine the time to resqueeze after the inhibitor concentration drops below the predicted effective range. An example is given in Figure 4 to illustrate the use of the model in a typical gas well (e.g., D ) 10 000 ft, R ) 1.25 in., QW ) 2000 bbl/d, fW ) 0.5, ts ) 0, and tinh ) 20 min). The effective inhibitor (BHMTPMP) concentration is calculated to be from 0.6 to 4.2 mg/L for saturation indices from 1.75 to 2.25 for a temperature of 85 °C and a pH of 6.5. In the case of the Guerra well, the measured sulfate concentration is about 5-10 mg/L and that of barium is 550 mg/L. The estimated SI of barite is about 0.8-1.1 at 70 °C. The calculated effective inhibitor dosage is from (19) He, S. L.; Morse, J. W. Comput. Geosci. 1993, 23, 1. (20) Oddo, J. E.; Tomson, M. B. SPE Prod. Facil. 1994, 94, 47.
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0.02 to 0.05 mg/L BHMTPMP and from 0.03 to 0.06 mg/L BHTMP, assuming the same SI, T, and pH coefficients. The assumption and estimation may be reasonable because these two inhibitors (BHMTPMP and BHTMP) belong to the same class and have similar structures. This well was treated with an inhibitor (BHTMP) squeeze in December 1993. The inhibitor concentration had dropped below the detection limit (0.3 mg/L in the Guerra brine) after 20 months, and no scaling of barium sulfate had been observed by the operator. These preliminary field observations seem to be consistent with our model predictions. In addition, the mathematical inhibitor model presented provides a practical and systematic approach for correlating laboratory and field data for scale inhibitors. The model can therefore be used as a reference for comparing and ranking the efficiency of scale inhibitors under various conditions. Parametrization for other inhibitors and other types of scales (calcite in particular) is currently being studied and will be reported in future communications. Conclusions A novel semiempirical model has been presented to choose an inhibitor and predict the effective concentration
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which can be applied to many industrial processes including the oil and gas production system. The model has been based on nucleation theory and laboratory experimental data of the nucleation induction period of barium sulfate in brines over a wide range of conditions. The model predictions are in good agreement with experimental data of the minimal inhibitor dosage obtained from the dynamic flow-through inhibitor testing and preliminary field observations. The inhibitor model also provides a practical method for correlating laboratory and field data for scale inhibitors. Among many commercial inhibitors tested, BHMTPMP is an effective scale inhibitor for inhibition of barium sulfate precipitation over a wide range of saturation indices, temperatures, pH’s, and brine compositions. Acknowledgment. This work was supported by the Gas Research Institute, but in no way does this constitute an endorsement by GRI of any products or views contained herein. In addition, the work was also supported by a consortium of companies including Texaco, Inc.; Conoco, Inc.; Champion Technologies, Inc.; FMC Corporations Product Additives Division; and Zapata, Inc. LA950876X
