﻿ The Mathematical Model for Particle Suspension Flow through Porous Medium Geomaterials, 2012, 2, 57-62 http://dx.doi.org/10.4236/gm.2012.23009 Published Online July 2012 (http://www.SciRP.org/journal/gm) The Mathematical Model for Particle Suspension Flow through Porous Medium Hooman Fallah, Hosein Barzegar Fathi, Hamed Mohammadi Department of Petroleum Engineering, Firoozabad Branch, Islamic Azad University, Firoozabad, Iran Email: hooman.fallah2@gmail.com Received May 12, 2012; revised June 14, 2012; accepted June 24, 2012 ABSTRACT Transport of suspensions and emulsions in porous media occurs in numerous processes of environmental, chemical, petroleum and civil engineering. In this work, a mass balance particle transport equation which includes filtration has been developed. The steady-state transport equation is presented and the solution to the complete advective-dispersion equation for particulate suspension flow has been derived for the case of a constant filter coefficient. This model in-cludes transport parameters which are particle advective velocity, particle longitudinal dispersion coefficient and filter coefficient. This work recommends to be investigated by particle longitudinal dispersion calculation from experimental data, directly. Besides, the numerical model needs to be developed for general case of a transition filter coefficient. Keywords: Particulate Suspension Flow; Filtration Theory; Filtration Coefficient; Transport; Porous Medium; Mathematical Model 1. Introduction The transport of particulate suspensions in porous me-dium occurs in a variety of industrial and natural process such as wastewater treatment, propagation of pollutants in subsurface environment, fouling of membranes and seawater injection in oil reservoirs. Flow of solid and liquid particles with particle capture by the rock takes place during injection of seawater and produced water in oil fields. Particle transport and retention are mostly important for the environmental processes, where the particle con-centration must not exceed a safety value, while the per-meability changes is important for petroleum production due to its effect on well productivity and injectivity . The study of particle retention in porous medium can be dated back to the work of Iwasaki  who proposed the following basic empirical for deep bed hydrosol fil-tration. cut (1) where:  is the specific volume of deposited material (particle retained concentration), is the fluid superfi-cial velocity, is the particle suspended concentration and uc is called the filtration coefficient. The must used approach for evaluating colloid migra-tion, retention and detachment is solute transport mass balance equation with th e sink term for particle retention [3,4]. 22occcUDtxx  (2) detcU kt (3) The term  in Equation (3) is called filtration coeffi-cient. Equations (1) and (2) together with the formula for coefficient  are called the classical filtration theory in above refere n ces. In the current work, a theoretical investigation of par-ticulate suspension flow through porous media has been done. The steady-state transport equation is presented and the solution has been derived. The filtration coeffi-cient  will be discussed as a dominant parameter in the particle transport and retention through porous me-dium. 2. Particulate Suspension Flow through Porous Medium 2.1. Filtration Theory When fluid containing particles reaches a porous medium, the liquid and the solid phase in the suspension can be separated, either by depositing in the pore or accumulat-ing in front of the surface. This is similar to what hap-pens in the filtration process. Then, the retention process of particles when flowing through a porous media is called Deep Bed Filtration . Th e deep filtration occurs because of several mechanisms: the contacting of parti-Copyright © 2012 SciRes. GM H. FALLAH ET AL. 58 cles with the retention site, the fixing of particle sites and the breaking away of previously retained particles . 2.2. Mechanisms of Particles Retention The particle capture in porous media can be caused by different mechanisms . • Size exclusion (large particles are captured in small pores and pass through large p ores ); • Electrical forces (London-Van der Waals, double electrical layer, etc.); • Gravity segregation; • Multi particle bridging. 3. Basic Formulation of Particles Transport through Porous Medium The model includes transport parameters which are parti-cle advective velocity, particle longitudinal dispersion coefficient and filter coefficient. These parameters have been defined by dimensional analysis using the pertinent variables of the porous medium system. 3.1. Advective Velocity The averaged particle velocity in the porous media, has been found to be either the same or slightly higher than that of the carrier fluid. This deviation is caused by the particle’s size. The expected difference can be deter-mined by analyzing the velocity profiles of both the fluid and the particles in a pore. The model has been formu-lated for a capillary tube which has a constant rate with the following assumption: No interactions between the particles and the wall, suspension is well-mixed with a constant concentration across the cross section .There is no tran sverse flow. 20(0 )rr1SOorUU r 0(0)221pPOopoarUU rr  rr a SV (4) As the particle travels through a tube, Brownian mo-tion and shear action will cause the particle to travel across the entire cross-section of the tube except that the center-line of the particle will be excluded from the im-mediate region of the wall due to its radial dimension. After the particle has traveled far enough longitudi-nally through the tube, the particle will have spent equal amounts of time in all radial position across the capillary tube. Integration of the velocity profile of Equation (4) over the range of possible radii shown in Equation (4) for both the particle and fluid yields the higher average ve-locity for the particle than that for the carrier flu id. The av erage flu i d velocity, in a capillary tube is: 2OSUV (5) The average velocity of a particle, PV in a capillary tube is: 221112ppPOooaaVU rr (6) By inspecting Equation (5) and (6), the particles are expected to have a larger average velocity than the car-rier fluid velocity. This enhanced velocity of the particle can be expressed as a fractional difference between the two average velocities: 2723ppPSSooaaVVVVrr V (7) This equation shows that as the radius of the particle increase, the difference between the average particle ve-locity and the average fluid interstitial velocity also in-crease. This increase is not unbounded but reaches a maximum value for 37poar; as 37poar the velocity difference decrease. In physical sense, the pore radius can be estimated to approximately equal to one-fifth of media grain diameter (15ogrd); therefore the largest possible particle to be able to fit through the porous bed has a radius to this pore radius (poar). For a particle with poar, the particles have been shown to collect on the bed surface in a cake . These references show that the onset of deep bed filtration oc-curs for a particle radius pa.less than one-twentieth of the media grain diameter (120pgad5). Particles with the radii larger than this will not transport into the bed, but will collect on the surface. By lettingc godr, the largest particle which will transport has a radius equal t o one-fourth of the pore radius (14poar). 3.2. Longitudinal Dispersion Coefficient An important element of any dispersion model is the re-presentation of the geometry of the porous medium. Houseworth  has thoroughly reviewed such longitudi-nal dispersion model for solute tracers. Instead of mod-eling the internal structure of a porous medium, dimen-sional analysis is used to analyze the problem. In this study, the effect of mechanisms is expected to scale with the pertinent transport variables. The pertinent variables for solute dispersion are: DL = longitudinal dispersion coefficient (L·T–1) D = free fluid molecular diffusion coefficient of Copyright © 2012 SciRes. GM H. FALLAH ET AL. 59solute (L2·T–1) Vs = fluid interstitial velocity (L·T–1) and dg = media grain diameter (L) From the Buckingham pi theorem, the following pairs of groups are formed: ,oSgLSgVdDFVd Dralternatively SgVdFDDLD (8) where Pec let nuPe mber SgVdD Experimental data for solu te longitudinal dispersion in uniform media show good correlation with these dimen-sionless groups . When the Peclet number is grater than 1, the two groups can be re duced to one: ConstantSgDLVd (9) where P Dedynamic Peclet numbeSgLVdrD An order to magnitude approximation for the longitu-dinal dispersion coefficient for solutes can be made with: gLSDdV (10) Particle longitudinal disp ersion is expected to be simi-lar to that of solutes. Currently, no particle breakthro ughs have been performed by others form which particle lon-gitudinal dispersion coefficient be determined. 3.3. Filter Coefficient Two approaches exist for analyzing the filter coefficient. These are the macroscopic mass balance approach and the microscopic trajectory analysis approach. Deep bed filtration studies have been conducted to analyze both th e system variables and the underlying mechanisms in-volved in the processes of capturing and retaining parti-cles in porous media . 3.3.1. Macros c opi c Approach The macroscopic process of filtration or change in sus-pension concentration over depth is first-order decay with distance in steady flow. The filter coefficient may be expressed as a function of single collector efficiency and the single collector efficiency may be related to mi-croscopic filtration mechanism. Filtration results for steady state flow, neglecting lon-gitudinal dispersion, gives: CCx (11) OC C expx The filter coefficient theoretically may be expr essed as the single collector efficiency as follows: 132ecTgd (12) For T, the particles under consideration are those contained in a cylinder of diameter gd which is coinci-dent with the vertical axis through the media grain col-lector.  can be found either experimentally or from the individual collector efficiency, T and system vari-ables using Equation (12). The classical filtration theory assumes simultaneous particle capture and dislodging. On the contrary, the proposed model assumes that the particle capture takes place only if the total of torques for electrostatic and gravity forces prevails over the drag and lifting forces, so the resulting torque presses the particle towards the ma-trix or the internal cake. 3.3.2. Microscopic App roach Microscopic study of particle motion near a collector is defined as trajectory analysis. In porous media, particle path far from media grains follow fluid streamline. As a particle approach a media grain, the motion deviate from the streamline because of various forces and torques act-ing on the particles. These forces are presented by trans-port and attachment mechanisms. The transport mechanisms are: gravity settling, inter-ception, Brownian diffusion and advection. The attach-ment mechanisms are considered to be gravity, Lon-don-van der Waals att racti on, do u bl e lay er fo rces. From a combination of the trajectory analysis, the three major contribution of filtration can be formulated. These are collection due to Brownian diffusion, intercep-tion and settling. Here, we assumed double layer repul-sive forces and hydrodynamic retardation (slow drainage of fluid from between two closely interacting particle which occurs before contact of particles) are negligible. The equations for these collection efficiencies are given in the following . Collection due t o Br ow ni an di ff usi o n, 12334DSpAPe (13) 134SA is constant. Also, either the particle Peclet number can be allowed to reach a minimum value of approxi-mately 1, or as 1pPe the efficiency reaches an as-ymptotic value of 1. Collection due to interception, 221.5ISRAN (14) 21.5 is constant here. Also the relative size group, RN, can be allowed to reach a minimum value of ap-Copyright © 2012 SciRes. GM H. FALLAH ET AL. 60 proximately 1 in order for the efficiency to remain less than or equal to 1. As we mentioned before, the size of the largest parti-cles which are able to penetrate and transport through a porous bed is one twentieth of media grain diameter (120pgda) or one-fourt h of the p ore radius (14poar). Collection due to settling, 292pfpsssGgaVVTDIGw (15) The limit for this collection efficiency is the best pos-sible collection which occurs for the settling velocity equaling the interstitial velocity. As the settling velocity becomes grater than the interstitial velocity, the effi-ciency remains at a value of 1. The equation for to tal collection efficiency is: T (16) In this case,  has an asymptotic maximum value of 1. For Brownian particles (p), there is good agreement between experimental and theoretical result but for advective particles (p), it is not, because of neglecting hydrodynamic retardation and London-van der Waals attraction. 1.0dmicron1.0dmicronSo, the exact analytical solution is : 12 11533 88411240.722.4 3TSp SLORSGR0APeANNEAN N  (17) The first term represents filtration due to Brownian diffusion. The second and third terms represent the com-bined effects of interception and gravity when the retar-dation and London-van der Waals attraction are incl uded. The effects of surface double layer forces are ignored (attractive surface double layer is controlled by transport processes and not depend on surface chemistry). 4. Modeling of Particle Transport and Filtration In this part, the steady-state transport equation is pre- sented and the solution to the complete advective-dis- persion equation for particulate suspension flow has been derived for the case of a constant filter coefficient. This model includes transport parameters which are particle advestive velocity and particle longitudinal dispersion coefficient. These parameters have been defined by di-mensional analysis using the pertinent variables of the porous media system. 4.1. Particle Advective Velocity The result of the size exclusion for particles flowing in capillary tube can be written as Equation (6) where 23. By using the Equations (5), (6) and (7) we will have: 1VV (18) *PSVVV (19) As the particle size increase, the difference between particle velocity and fluid velocity increase. 4.2. Particle Longitudinal Dispersion Coefficient In modeling particle dispersion, the following variable substitutions are used: PDD PVV LLPDD Particle size variable can be removed by using the particle properties as shown, provided 1pgdd also the effect of particle size is included in the enhanced advec-tive velocity for the particles. This analysis shows that particle and solute longitudi-nal dispersion are similar. When the particle Peclet number (pPe =PgpPVdPe D is grater than 10, the two groups can be reduced to one : constantLPPgDVd  (20) An order of magnitude approximation for the longitu-dinal dispersion coefficient for particles can be made with: LDPPgVd1pPe 10Pe  (21) As mentioned before, the dimensional argument for defining the longitudinal dispersion coefficient is only valid when . For uniform media, this restriction is seen to be p. Flow conditions are simultane-ously limited to the linear, laminar regime for which the Reynolds number must be less than 10. 4.3. Steady-State Transport Equation and Solution Particle removal or filtration occurs as a particle suspen-sion flows through a porous medium due to the interac-tion of the advecting particles and grains of the medium. Iwasaki  is credited with being the first to express fil-tration a first-order decay of particle concentration with distance: Copyright © 2012 SciRes. GM H. FALLAH ET AL. Copyright © 2012 SciRes. GM 615. Conclusion CCx00Cx C (22) In this work, a mass balance particle transport equation which includes filtration has been d eveloped. This model includes transport parameters which are particle advec- tive velocity, particle longitudinal dispersion coefficient and filter coefficient. The steady-state transport equation is presented and the solution to the complete advective- dispersion equation for particulate suspension flow has been derived for the case of a constant filter coefficient. (23) and a solution in dimensional terms: 0expCC x***expCx (24) or in dimensionless terms:  (25) This work recommends to be investigated by particle longitudinal dispersion calculation from experimental data. Besides, the numerical model needs to be devel-oped for general case of a transition filter coefficient. where *0CCC 00CCx* 6. Acknowledgements d gThe authors thank Professor P. Bedrikovetsky, Chair in Petroleum Engineering, Australian School of Petroleum, University of Adelaide. *gxxd A complete equation of steady-state filtration can be formulated by using the general steady-state advection- dispersion equation of transport for particle concentration with a sink term to describe particle removal due to fil-tration: REFERENCES  S. Pang and M. M. Sharma, “A Model for Predicting Injectivity Decline in Water-Injection Wells,” SPE Paper 28489, Vol. 12, No. 3, 1997, pp. 194-201. 220LPDVp pCCVCxx00Cx Clim 0xCx  (26)  T. Iwasaki, “Some Notes on Sand Filtration,” Water Works Association, 1937, pp. 1591-1602, The following semi-infinite medium boundary condi-tions are:  J. W. A. Foppen and J. F. Schijven, “Evaluation of Data from the Literature on the Transport and Survival of Es-cherichia coil in Aquifers under Saturated Conditions,” Journal of Water Research, Vol. 40, No. 3, 2006, pp. 401-426. doi:10.1016/j.watres.2005.11.018 (27)  J. F. Schijven and S. M. Hassanizadeh, “Removal of Vi-ruses by Soil Passage: Overview of Modeling Processes, and Parameters,” Critical Reviews in Environmental Sci-ence and Technology, Vol. 30, No. 1, 2000, pp. 49-127 The solution which is shown in dimensional terms is derived in appendix: 0expC xCx (28)  R. Farajzadeh, “An Experimental Investigation into In-ternal Filtration and External Cake Build Up,” MSc. The-sis, Delft University of Technology, 2004. where 11142PLPLP PVDDV  J. P. Herzig, D. M. Leclerc and P. Le Goff, “Flow of sus-pension through porous media-application to deep filtra-tion,” Journal of Industrial and Engineering Chemistry, Vol. 65, No. 5, 1970, pp. 8-35 2**20DpCCPe****DpPe Cxx**01Cx (29)  M. Elimelech and O’Melia, “Effect of Particle Size on Collosion Efficiency in the Deposition of Brownian Par-ticles with Electrostatic Barriers,” 1990, pp. 1153-1163 With the same boundary conditions: **m 0Cx ***expCx (30)  J. E. Houseworth, “Longitudinal Dispersion in Nonuni-form, Isotropic Porous Media,” Ph.D. Thesis, W.M. Keck Laboratory of Hydraulics and Water Resources, Califor-nia Institute of Technology, Report No. KH-R-45, 1984. *lix (31) and a solution in dimensionless terms:  A.C. Paytakes and C. Tien, “Particle Deposition in Fi-brous Media with Dendritic Pattern: Apreliminary model,” Journal of Aerosol Science, Vol. 7, No. 2, 1976, pp. 85-94. doi:10.1016/0021-8502(76)90067-7  (32) *411PDPPe*12gDdPe (33) H. FALLAH ET AL. 62 Appendix. Solution Derivation Consider the one-dimensional steady-state particle ad-vective-dispersion equation which includes the removal term to account for filtration effects: 220LPDVP PCCVCxx00Cx Clim 0xCx  (A.1) With the following boundary conditions: (A.1.1) For convenience, the x-variab le is allowed to range from negative to positive infinity x , although the equation are only applied for x > 0 this avoids difficulty at x = 0, because small dispersion is allowed. In dimen-sionless form, the transport equ ation becomes: 2**20LP DpCCDPe****DpPeCxx**01Cx**m 0Cx **exp x (A.2) With the same boundary conditions: *lix (A.2.1) In order to derive a solution, try the following as a solu-tion: **Cx* (A.3) Check the equation (A.3) by substituting into Equation (A.2), this results in second- deg ree po lyno mial in ter m of  and two roots of this polynomial are: *1411DPPe*12DPPe *1411DPPe**** **12Bexp x**01AB B 1and 0AB*12DPPe Using these two roots, Equation (A.3) becomes: C xAexpx (A.4) The constants of this Equation can be determined by ap-plying the boundary conditions: Cx**Cx  00A  By substituting these constants into Equation (A.4), the solution to Equation (A.2) becomes: *** *14exp1 12DPDPCx PexPe (A.5)  Nomenclature dg= media grain diameter (L) dp= particle diameter (L) c = suspended particle concentration in carrier fluid σ = particle retained concentration kdet = detachment rate coefficient U = flow velocity US = fluid velocity UP = particle velocity UO = fluid centerline velocity r = radial distance ro = capillary radius ap = particle radius p = dynamic pressure x = longitudinal distance DL = longitudinal dispersion coefficient (L·T–1) D = free fluid molecular dispersion coefficient of solute (L2·T–1) VS = fluid interstitial velocity (L·T–1) Pe = Peclet number = SgV.dDL PeD = dynamic Pec let number = SgV.dDL C = particle concentration (M·L–3) X = longitudinal position (L) λ = filter coefficient (L–1) WS = particle settling velocity VS = fluid interstitial velocity Ρpf = densities of particle and fluid, respectively g = gravitational acceleration H = Hamakar constant (e r gs) NG = gravitional group = ηG DLP = particle longitudinal dispersion coefficient (L2·T–1) DP = particle molecular diffusion coefficient in a free fluid (L2·T–1) VP = particle velocity (L2·T–1) Copyright © 2012 SciRes. 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