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EXERCISES 1. Consider the calcium carbonate precipitation reaction given by: Ca²+ + HCO3 CaCO3(s) + CO2(g) + H₂O a. Present an expression for the equilibrium constant for this reaction. b. Obtain the values of the equilibrium constant at 1 atm and 50°C conditions from a data reference book. c. Determine the saturation ration value for a solution containing 0.1 M (mole/L) Ca²+ and 0.5 M HCO3 ions at 1 atm and 50°C conditions. What can be said as to the saturation condition of this solution? (7.70) 2. Based on the data given in Figure 7.7, estimate the calcium carbonate induction time for a solution containing a ratio of 2.0 magnesium-to-calcium ions concentrations and a F, = 10.0 saturation ratio. 3. The values of the rate constants of the calcium carbonate crystal growth at 25°C and 50°C temperature conditions are 1.4 and 10.3 µm/s, respectively. Determine the values of the activation energy and the high-temperature limit of the rate coefficient. 4. Assuming a set of representative values for p,, k', o, and F.: a. Derive an analytical solution for Eqs. (7.55) and (7.56). b. Prepare a plot of porosity variation by scale deposition as a function of time.
1. Using Einstein's equation, Eq. (8.6), estimate the Brownian diffusivity of a 1.0 um diameter fine particle in water at 20°C temperature. (Answer: D = 4.3 x 10 cm²/s; McDowell-Boyer et al., 1986). 2. Estimate the Stokes' motion velocity of a particle using Eq. (8.3) in a fluid. The particle density is 2.5 g/cm³, the mean spherical volume equivalent diameter is 0.005 mm, and the fluid density and viscosity are 1.0 g/cm³ and 1.0 centi-poise, respectively. 3. Consider a suspension of clay particles in brine. The volume flux of the clay suspension is given as 1.5 X 104 mL/cm²-min. The clay particle volume fraction in the clay suspension is 0.10. The porosity of the rock sample is 25%. The particle deposition rate constant is 8.3 X 107 cm. The stationary deposition factor value is 5.2 x 10-5 mL/cm²-min. Estimate the rate of deposition of the clay particles over the pore surface. 222 CHAPTER 8 PARTICULATE PROCESSES IN POROUS MEDIA 4. Consider the data given in Figure 8.7. The particle concentration of a drilling mud is 10 lb/gallon. Answer the following questions: a. What is the critical pore throat-to-mud particle diameter ratio for bridging to occur at the sand face? b. Would pore-throat bridging occur if the pore throat-to-mud particle diameter ratio had been equal to 4.0? 5. Based on Eq. (8.63), estimate the wall shear stress for a non-Newtonian gel having k' = 5.0 and n' = 2.0 and flowing at a 0.05 cm/min interstitial velocity through a 0.0001 cm diameter capillary flow path in porous rock. Hi
6. Derive an analytical solution for the set of ordinary differential equations given by Eqs. (8.72)-(8.74) for the special case of aA = BA= 1, subject to the condition given by Eq. (8.75) similar to Gupta and Civan (1994a). Prepare the plots of these analytical solutions for typical parameter values. 7. Consider a suspension of clay particles in brine. The volume flux of the clay suspension is given as 1.2 x 10 ml/cm2-min. The clay particle volume fraction in the clay suspension is -4 0.15. The porosity of the rock sample is 20%. The particle deposition rate constant is 8.0 X 10 cm. The stationary deposition factor value is 5.0 x 105 mL/cm²-min. Estimate the rate of deposition of the clay particles over the pore surface. 8. Consider the pore-throat plugging criterion given by the exponential equation Eq. (8.42). The parameter values are A = 4.6, B = 0.153, and C= 1.52. Particle concentration of a drilling mud is c = 0.15 g/L. u = 0.1 cm/min. D, = 25 µm. = 0.20.= 1 cp. T = √2. Note that 1 μm = 106 m. Answer the following questions: a. What is the critical pore throat-to-mud particle diameter ratio for bridging to occur at the sand face? b. Would pore-throat bridging occur if the pore throat-to-mud particle diameter ratio had been equal to 3.0? 9. Explain the salinity, velocity, and temperature shock phenomena, and their impact on formation damage. Discuss the differences owing to single-phase versus multiphase pore-fluid systems. 10. Develop a theoretically meaningful correlation of the experimental data of Ver vs DID of Maroudas et al. (1965) given in Table 8.1. Table 8.1 Experimental Data of Maroudas et al. (1965) DJDp Ver (cm/s) 16 36 22.2 34 30.8 25 40 15 100 2