ABSTRACT
CONTENTS Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 12.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
12.2.1 Basic Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 12.2.2 Modeling for Rate of Deposition Function . . . . . . . . . . . . . . . . . . . . . . 492
12.2.2.1 Present model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 12.2.2.2 The Model by Herzig et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 12.2.2.3 Discussion of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
12.2.3 Permeability Impairment, Injectivity Ratio, and Impedance Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 12.2.3.1 One-dimensional rectilinear case . . . . . . . . . . . . . . . . . . . . . . 498 12.2.3.2 Radial case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
12.3 One-Dimensional Rectilinear Problem with Constant Injection Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 12.3.1 Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 12.3.2 Constant-Rate Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
12.3.2.1 Nondimensional variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 12.3.2.2 Characteristic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
12.3.3 Variable-Rate Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 12.3.3.1 Present model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 12.3.3.2 Solution for the model by Herzig et al. . . . . . . . . . . . . . . . 512 12.3.3.3 Comparison and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 12.3.3.4 Average permeability and impedance index . . . . . . . . . 518
12.4 One-Dimensional Rectilinear Problem with Time-Dependent Injection Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
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12.5 Radial Problem with Constant Injection Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 12.5.1 Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 12.5.2 Constant-Rate Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
12.5.2.1 Nondimensional variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 12.5.2.2 Characteristic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
12.5.3 Variable-Rate Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 12.5.3.1 Solution for present model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 12.5.3.2 Partial solution for model by Herzig et al. . . . . . . . . . . . 532
12.6 Radial Problem with Time-Dependent Injection Rate . . . . . . . . . . . . . . . . . 533 12.7 Applications and Validation of Analytic Solutions . . . . . . . . . . . . . . . . . . . . . 533
12.7.1 One-Dimensional Rectilinear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 12.7.1.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 12.7.1.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 12.7.1.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
12.7.2 Radial Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 12.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
One-dimensional rectilinear and radial macroscopic phenomenological models along with analytical solutions and applications for impairment of porousmedia bymigration and deposition of fine particles, and effects on the injectivity decline during flow of particle-fluid suspensions, are presented in this chapter. The mechanism and kinetics of the fine particle deposition in porous medium for two different models are described and compared. The present approach considers the rate of deposition at a given location to be proportional to the particle flux, with the proportionality factor being a function of the cumulative particles passing by the location per unit volume. The popular model by Herzig et al. [Herzig, J.P., Leclerc, D.M., and Le Goff, P., Flow of suspensions through porous media — application to deep filtration, Industrial Eng. Chem., 62(5), 8-35, 1970.] stems from the assumption that the proportionality factor, called the filtration coefficient, is a variable depending on the deposition function itself. The present new system of equations has a similar appearance to that developed by Herzig et al., but the equivalent constitutive relations are subtly different. The formulation and analytic solution for the constant and time-dependent
injection-rate cases are carried out. A methodology for determination of the parameters of the deep-bed filtration process is provided. Typical scenarios
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are simulated, illustrating the parametric sensitivity and application of the present analytical solutions.
