Method obtains alkali formate packer fluid thermal conductivity

Dec. 23, 2002
A study determined alkali formate, a packer, by using existing correlations. An ion characteristic coefficient for formate (σHCOO–) is –0.007535.

A study determined alkali formate, a packer, by using existing correlations. An ion characteristic coefficient for formate (σHCOO–) is –0.007535.

The oil and gas industry uses aqueous salt solutions as completion fluids (packer fluids) in producing wells. These packer fluids range in density from 1.02 g/ml for a KCl solution to 2.52 g/ml for a ZnBr2 solution.

Operators select packer fluids based on bottomhole pressure, corrosion, carbon dioxide content in the flowing fluid, and crystallization temperature.

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The insulating property of the packer fluid has become recently an issue as the industry produces from deeper water. The cooling effect of the water can cause the producing fluids to precipitate substances that interfere with the well's production capabilities.

Thermal conductivity of a packer fluid and salt water is an important variable in calculating heat loss from the well to the surrounding formation and predicting the surface temperature of flowing gas and oil.

Several references1 2 provide some thermal conductivity data for sodium chloride and calcium chloride solutions; however, the thermal conductivity for the alkali formates has not been available.

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To find these values for alkali formates, the study used existing correlations and the ion characteristic coefficient to calculate the thermal conductivity. Measured values confirmed the accuracy of the thermal conductivity for the cases studied.

Riedel equation

The Riedel equation (Equation 1 in the Equations box) correlates thermal conductivity of electrolyte solutions for various salt concentrations.2 The concentration of solution, Ci, is given in molarity. The terms inside the bracket calculate the thermal conductivity of the solution at a reference temperature T0. The multiplier, the ratio of water thermal conductivities, is a temperature correction factor.

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Riedel3 and Reid2 provide the characteristic coefficients, si, for various cations and anions at 20° C. Table 1 lists the coefficients for salts commonly found in the oil field.

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Miller presents the thermal conductivity of water at 20° C. and other temperatures.4

A more recent study by Uzbek and Phillips reports the thermal conductivity of water at 20° C. as 0.603 w/m-K.5 For elevated temperatures, Equation 2 calculates the thermal conductivity of water. In Equation 2, T is in °C.

Potassium, sodium formates

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Horton provides the density and thermal conductivity of potassium formate at 25 and 67° C. for various weight percent (Table 2).6 These data are used to determine the characteristic coefficient of the formate anion.7

Equation 1 yields Equation 3 after rearrangement of the terms and differentiation. Equation 3 can be used to least-square fit the data.7

In Equation 3, (λ(T); exp)κ,exp)i, is the i-th measured thermal conductivity of formate solution at 20° C., and Ci is the molarity of the i-th data set. The characteristic coefficient of the potassium cation, σcation, is –0.00756 (Table 2).

The characteristic coefficient, σHCOO, obtained from the least-square fit is –0.007535 (Table 1). One can use this value to calculate the thermal conductivity of potassium and sodium formate solutions.

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Table 3 compares the calculated thermal conductivities of potassium formate at 25° and 67° C. with the measured values. The deviation is less than 10% in most cases.

Bridgman equation

According to Bird, the Bridgman equation was first developed in 1923 to predict the thermal conductivity of monatomic liquids. Equation 4 shows the original form of Bridgman equation.8

In the equation, k is the Boltzmann constant, N is Avogadro's constant, V is molar volume, and r is the density of the aqueous solution. The equation also predicts the thermal conductivity for pure polyatomic liquids.8

The equation is similar to the equation developed by Power, Roseveare, and Eyring. Recently, Curtiss and Bird extended the application of the equation to polymeric liquids.9

The Bridgman equation, after being rearranged, can be written in terms of isothermal compressibility of solution. Equation 5 gives the isothermal compressibility of solution, kT.

Combining Equations 4 and 5 results in Equation 6, a convenient form of the Bridgman equation.

One can assume that the isothermal compressibility of a solution is equal to that of pure water. Values of it can be found in the CRC Handbook for various temperatures,10 or it also can be calculated from the pressure-volume-temperature (PVT) data in steam tables.

The International Critical Tables and Perry's Handbook present the density of many aqueous electrolyte solutions at various temperatures and concentrations.11 12 Values for potassium formate solutions (Table 2) were obtained from an unpublished work by Horton.6

The basic thermodynamic relations 13 show that Equation 7 can obtain the heat-capacity ratio. The coefficient of thermal expansion, a, is available in the CRC Handbook10 for water.

For water and many electrolyte solutions, the square-root of heat-capacity ratio is nearly unity.14

The temperature correction factor is needed to calculate thermal conductivity at elevated temperatures. It is obtained from the least-square fit of thermal conductivity of water at various temperatures.

Equation 8 is the modified Bridgman equation with the temperature correction factor. The equation calculates the thermal conductivities of potassium formate at temperatures 25° C. and 67° C., with the assumption that the isothermal compressibility of aqueous electrolyte solutions is almost equal to that of pure water. Table 3 lists the results of these calculations.

These results have a reasonable agreement with the measured values. The deviations are within 12% for the cases studied.

Figs. 1 and 2 compare the thermal conductivities of potassium formate calculated by Riedel and Brigman equations with the measured values.

Sample calculation

Using the Bridgman equation, find the thermal conductivity of aqueous potassium formate solution at 21.1° C. The concentration of the solute is 20.00 wt %.

From the CRC Handbook of Chemistry and Physics,10 the isothermal compressibility of water is 45.76 x 10–6 bar–1, or 4.576 x 10–11 sec2 cm/g.

The density of the solution at the given temperature and concentration is 1.139 g/ml, according to Horton.6 Then, 1/ρ kT is equal to 1.962 x 1010 sq cm/sec2.

From the weight percent given, the mole fraction of the solute is found to be 0.0508. The mole-fraction average molecular weight is: 84.01 x 0.05083 + 18.02 x 0.9245 = 21.38 g/g-mole. The molar volume is: 21.38/1.114 = 19.18 cc/g-mole. Then, the calculated thermal conductivity is 0.5534 w/m-K.

The measured thermal conductivity of the 20 wt % solution at 25° C. is 0.5588 w/m-K. or 0.323 btu/hr ft °F.

Acknowledgments

The authors thank John H. Hallman from Clearwater Inc. for providing the thermal conductivity data for the acetates and formate solutions. We also thank the management of BJ Services Inc. for permission to publish this work.

References

  1. Assael, M.J., Charitodou, E., Stassis, J.C., and Wakeham, W.A., "Absolute Measurements of the Thermal Conductivity of Some Aqueous Chloride Salt Solutions," Phys. Chem., Vol. 93, 1989, pp. 887-92.
  2. Reid, R.C., Prausnitz, J.M., and Poling, B.E., The Properties of Gases and Liquids, 4th Edition, New York: McGraw-Hill, 1987.
  3. Riedel, L., "Neue Warmeleitfahigkeitsmessungen an organischen Flussigkeiten," Chem. Ing. Technik., Vol. 23, No. 59, 1951, pp. 321 and 465.
  4. Miller, J.W. Jr., McGinley, J.J., and Yaws, C.L., "Correlation Constants for Liquids-Thermal conductivity of Liquids," Chemical Engineering, Vol. 83, No. 23, 1976, p. 133.
  5. Ozbek, H. and Phillips, S.L., "Thermal Conductivity of Aqueous Sodium Chloride Solutions from 20 to 330° C.," J. Chem. Eng. Data, Vol. 25, No. 3, 1980, pp. 263-67.
  6. Horton, R.L., Mixing and Blending Tables for Potassium Formate Brines, OSCA Internal Report, February 1995.
  7. Vollmer, D.P., and Fang, C.S., "Thermal Conductivities of Aqueous Electrolyte Solutions Containing Chlorides, Formates, and Acetates," AIChE Spring Meeting, Paper No. 144i, Mar. 10-14, 2002, New Orleans.
  8. Bird, R.B., Stewart, W.E., and Lightfoot, E.L. Transport Phenomena, New York: Wiley, 2002.
  9. Curtiss, C.F. and Bird, R.B., "Thermal conductivity of dilute solutions of chainlike polymers," J. Chem. Phys., Vol. 107, No. 13, 1997, pp. 5254-67.
  10. Lide, D.R., Handbook of Chemistry and Physics, 71st Edition, Boston: The Chemical Rubber Publishing Co., 1990.
  11. Washburn, E.W., International Critical Tables of Numerical Data, Physics, Chemistry and Technology, Vol. III, New York: McGraw-Hill, 1928.
  12. Perry, R.H., and Green, D.W., and Maloney, J.O., Chemical Engineers' Handbook, 6th Edition, New York: McGraw-Hill, 1984.
  13. Sandler, S. I., Chemical and Engineering Thermodynamics, 3rd Edition, New York: John Wiley, 1999.
  14. Garvin, J., "Use the Correct Constant-Volume," Chem. Eng. Prog., July 2002, pp.64-65.

The authors

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Daniel P. Vollmer is region engineer of BJ Services Co. Lafayette, La. He holds a BS and MS in chemical engineering from the University of Louisiana at Lafayette.

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Cheng-Shen Fang is professor of chemical engineering at the University of Louisiana at Lafayette. He holds a BS from the National Taiwan University, Taipei, and MS and PhD degrees from the University of Houston.

Correction

Graeme Walker
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The article "Disappearing plug eliminates riser for subsea completion" by Craig Stair, Joe Stuckey, and Graeme Walker (OGJ Dec. 2, 2002, p. 51) included an incorrect photo of Graeme Walker. Here's the correct one.