Numerical Investigation of the Two-Component Suspension Filtration in

a Porous Medium Taking into Account Changes in the Characteristics of

the Porous Medium

BEKZODJON FAYZIEV1, JAMOL MAKHMUDOV1, JABBOR MUSTOFOQULOV2,

TULKIN BEGMATOV1, RAKHMON SAFAROV1

1Department of Mathematical modeling,

Samarkand State University,

Samarkand, 140100,

UZBEKISTAN

2Department of Radioelectronics,

Jizzakh Polytechnic Institute,

Jizzakh, 140300,

UZBEKISTAN

Abstract: The paper explores a mathematical model of the filtration of dual-component suspension within a porous

medium characterized by two distinct zones. This model encompasses mass balance equations of suspended

particles, kinetic equations of deposition formation for both reversible and irreversible deposition types for each

suspension component, and incorporates Darcy’s law. In order to solve the problem, we formulate a numerical

algorithm for computer-based experimentation on the basis of the finite difference method. Through the analysis

of numerical findings, we establish key features of two component suspension filtration within a porous medium.

Furthermore, we examine the effects of model parameters on the transport and deposition of suspended particles

in a two-component suspension within porous media. The polydispersity of the suspension and the multi-stage

nature of deposition kinetics can induce effects that differ from those typically observed in the transport of one-

component suspensions with single-stage particle deposition kinetics.

Key-Words: deep bed filtration, finite difference method, multistage deposition, porous media, mathematical

model, two component suspension

Received: December 16, 2022. Revised: October 24, 2023. Accepted: November 19, 2023. Published: December 22, 2023.

1 Introduction

The phenomenon of suspension filtration, including

transport and deposition of suspended particles are

common occurrences that occur frequently within the

wide scope of industrial and natural processes. In

groundwater, solid particles may be a specific con-

taminant, or the particles may carry solutes with it,

promoting their movement by transport much faster

and longer than normal solute diffusion. Conversely,

in some cases, if their aggregates settle in the envi-

ronment, these particles may solidify and hinder the

migration of flocculating agents, [1], [2]. Filtration

through porous media is primarily used for the re-

moval of micron-sized particles from aqueous flow

in the purification of drinking and wastewater, [3].

In the oil industry, suspended particles in water sent

through reservoirs can accumulate in the well, leading

to a decrease in productivity, [4]. The release of soil

particles during construction can contribute to inter-

nal erosion and subsequent degradation of hydraulic

structures, [5], [6].

The majority of actual suspensions exhibit a multi-

component nature. The solid particles present in these

suspensions can vary in terms of their properties, en-

compassing differences in geometric, physicochem-

ical, and electrokinetic characteristics, [7], [8]. Re-

cent studies have increasingly taken into account the

polydispersity of such suspensions, [9], [10], [11],

[12]. Among multi-component suspensions, the sim-

plest form is the two-component suspension, com-

posed of two dispersed components with distinct con-

centrations and kinetic properties. Consequently, the

focus of research has predominantly been on two-

component suspensions, [11], [12].

To model the transport and deposition of sus-

pended particles developed different approach, such

as, pore-network modeling, [13], [14], particle tra-

jectory analysis models, [15], and empirical models,

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Bekzodjon Fayziev, Jamol Makhmudov,

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[11], [12], [16]. In the network models, a filter is

conceptualized as a porous medium comprising in-

terconnected microscopic units. Estimating the time-

dependent particle distribution within the filter in-

volves assessing the retained particles in each cell. A

significant drawback in the predictability of models

for large-scale particulate systems is the considerable

limitation posed by low particle injection efficiency,

[13], [14]. The initial proposal of the particle trajec-

tory model, alternatively known as the unit bed ele-

ments model, can be traced back to [15]. The preci-

sion of this model is heavily influenced by the choice

of conceptual representations for porous media. Var-

ious existing models serve this purpose, encompass-

ing capillaries, spherical structures, constricted tubes,

and the capillaries-chambers model [15]. The phe-

nomenological model characterizes the macroscopic

filtration process through the formulation of a set of

phenomenological equations and the introduction of a

range of empirically determined coefficients. Numer-

ous researchers have suggested numerical solutions

for the phenomenological model and analytical meth-

ods to determine capturing coefficients under diverse

mechanisms [11], [12], [17].

In this paper, we use phenomenological approach

to formulate a problem of two component suspension

filtration in a dual-zone porous medium. Which is

described as system of partial differential equations.

Therefore, the system of multi-component suspension

filtration equations is a generalization of the single-

component case, and the mass balance equation for

each component of the suspension is (i= 1,2), [12],

[18], [19]

∂c(i)

∂t +v∂c(i)

∂x +∂ρ(i)

∂t +∂ρ(i)

∂t = 0,(1)

where m0is the porosity of the medium, vis the

velocity (m/s),c(i)are concentrations of ith compo-

nent of the suspension (m3/m3),ρ(i)

a, ρ(i)

pare con-

centrations of deposition of ith component formed in

the active and passive zones, respectively (m3/m3),

i= 1,2correspond to the component numbers.

Kinetic equation of reversible deposition forma-

tion

∂ρ(i)

∂t =β(i)

ac(i)−ρ(i)

ρ(i)

c(i)

0(2)

where β(i)

aare kinetic coefficients characterizing the

intensity of deposition formation in the active zone

for the ith component of the suspension, ρ(i)

a0are ca-

pacities of the active zones for ith component of the

suspension.

The equation for the kinetics of irreversible depo-

sition formation:

∂ρ(i)

∂t =β(i)

pϕiρ(1)

p, ρ(2)

pc(i).(3)

where ρ(i)

p0are partial capacities of passive zones. As

in[19] the total capacitance of passive zone of the fil-

ter ρp0is mainly determined by the first fraction, as

well as ρ(1)

p0> ρ(2)

p0. As a result, deposition in the

passive zone stops when the sum ρ(1)

p+ρ(2)

pequals

the total capacitance ρp0. In addition, the deposition

formation for the second fraction ρ(2)

pstops when the

concentration is equal to its partial capacity, although

ρ(1)

p+ρ(2)

p< ρp0the total capacity is not full. Also,

the ”kinetic properties” of the first fraction are higher,

i.e. β(1)

pβ(2)

p>1.

Parameters ϕiρ(1)

p, ρ(2)

pcharacterizing the ”ag-

ing” phenomenon are as follows, [19]

ϕ1ρ(1)

p, ρ(2)

p=









1,0< ρ(1)

p≤ρ(1)

p1,

ρ(1)

p1ρ(1)

p, ρ(1)

p1< ρ(1)

ρ(1)

p+ρ(2)

p< ρp0,

0, ρ(1)

p+ρ(2)

p=ρp0,

(4)

ϕ2ρ(1)

p, ρ(2)

p=









1,0< ρ(2)

p≤ρ(2)

p1,

ρ(2)

p1ρ(2)

p, ρ(2)

p1< ρ(2)

p< ρ(2)

p0,

ρ(1)

p+ρ(2)

p< ρp0,

0, ρ(1)

p+ρ(2)

p=ρp0,

(5)

where ρ(i)

p1are threshold concentrations, ρ(i)

p> ρ(i)

from which the phenomenon of ”aging” begins in the

passive zone.

2 Problem Formulation

We consider semi-infinite porous media filled with

homogeneous liquid (that is, a liquid without sus-

pended particles) and the initial value of porosity is

m0. At the point, x= 0 from the moment t > 0

inhomogeneous liquid containing two different solid

particles with concentration c0=c(1)

0+c(2)

0is injected

into the porous media with the constant velocity v0.

In the balance equation (1), the change in the char-

acteristics of the pore medium as a result of deposition

was not taken into account. But as mentioned in,[12]

deposition significantly change the properties of the

porous medium, so we change this equation as fol-

lows

∂mc(i)

∂t +v∂c(i)

∂x +∂ρ(i)

∂t +∂ρ(i)

∂t = 0,(6)

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where mis the current porosity of the medium. It is

calculated as follows

m=m0−ρ(1)

a+ρ(2)

a+ρ(1)

p+ρ(2)

p(7)

We use the Carman–Kozeny equation to determine

the change in filtration coefficient

K(m) = k0

(1 −m)2,(8)

where k0=const is the fitration coefficient at the

initial moment.

The pressure gradient is determined from Darcy’s

law

v=K(m)|∇p|.(9)

We account for the velocity in both reversible and

irreversible depositiob formation kinetics, i.e. instead

of (2) and (3) we use

∂ρ(i)

∂t =β(i)

avc(i)−β(i)

ρ(i)

c(i)

0(10)

∂ρ(i)

∂t =β(i)

pvϕiρ(1)

p, ρ(2)

pc(i).(11)

So, mathematical model of two component sus-

pension filtration consists of equations (4) - (11).

The initial and boundary conditions are as follows:

c(i)(x, 0) = 0,

ρ(i)

a(x, 0) = ρ(i)

p(x, 0) = 0,

c(i)(0, t) = c(i)

0=const.(12)

3 Problem Solution

We use the finite difference method to solve prob-

lem (4) - (12), [20], [21]. In the area D=

{0≤x < ∞,0≤t≤T}we consider following net

ωhτ ={(xk, tj), xk=kh, k = 0,1, ...,

tj=jτ, j = 0,1, ..., J, τ =T/J}.

Instead of functions c(i)(t, x),ρ(i)

a(t, x),

ρ(i)

p(t, x),m(t, x),∇P(t, x)functions, (xk, tj)

we get the net functions ρ(i)

aj

k,ρ(i)

pj

k,mj

|∇p|j+1

kat the nodes c(i)j

The balance equation (6) is approximated in the

net for each fraction as follows

mj+1

kc(i)j+1

k−mj

kc(i)j

τ+

vc(i)j+1

k−c(i)j+1

k−1

h+ρ(i)

aj+1

k−ρ(i)

aj

τ+

ρ(i)

pj+1

k−ρ(i)

pj

τ= 0.(13)

We get the differential scheme for (10) as follows

ρ(i)

aj+1

k−ρ(i)

aj

τ=

β(i)

avc(i)j

k−β(i)

aρ(i)

aj

ρ(i)

c(i)

0.(14)

The non-washable deposition kinetics equation

(11) becomes the following after approximation

ρ(i)

pj+1

k−ρ(i)

pj

τ=

β(i)

pvϕiρ(1)

pj

k,ρ(2)

pj

kc(i)j

k.(15)

Schemes (13) - (15) look like this after simple sub-

stitutions

c(i)j+1

mj

kc(i)j

k+τv

hc(i)j+1

k−1−

ρ(i)

aj+1

k+ρ(i)

aj

k−ρ(i)

pj+1

k+ρ(i)

pj

k·

mj+1

k+τv

h−1(16)

k= 1, I, j = 0, J −1,

ρ(i)

aj+1

ρ(i)

aj

k+τβ(i)

a



vc(i)j

k−ρ(i)

aj

ρ(i)

c(i)

0



,(17)

k=0, I, j = 0, J −1,

ρ(i)

pj+1

k=ρ(i)

pj

τβ(i)

nvϕiρ(1)

pj

k,ρ(2)

pj

kc(i)j

k,(18)

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k= 0, I, j = 0, J −1.

For equations (7) and (9) we have

mj+1

k=mj

k−

ρ(1)

aj

k+ρ(2)

aj

k+ρ(1)

pj

k+ρ(2)

pj

k(19)

|∇p|j+1

v1−mj+1

k2

k0m0−mj+1

k3,(20)

Initial and boundary conditions (12) are also ap-

proximated on the net ωhτ

ρ(i)

aj

k= 0, k = 0, K, j = 0,

ρ(i)

pj

k= 0, k = 0, K, j = 0,

c(i)j

k= 0, k = 0, K, j = 0,(21)

c(i)j

k=c(i)

0, k = 0, j = 0, J,

Calculations are carried out in the following se-

quence: from relations (10) and (9) the values of

ρ(i)

nj+1

k,ρ(i)

nj

kand ρ(i)

aj

kare ρ(i)

aj+1

kfound

based on the values of c(i)j+1

kand at the correspond-

ing points in the lower layer, then from (8) c(i)j

4 Results and discussion

We take the following numerical values of the param-

eters as initial values: c(1)

0= 0.05,c(2)

0= 0.03,

m0= 0.3,v= 10−4m/s,ρ0= 0.1,ρa0= 0.01,

ρ(1)

a0= 0.007,ρ(2)

a0= 0.003,ρp0= 0.09,ρ(1)

p0= 0.06,

ρ(2)

p0= 0.03,β(1)

a0= 50 s−1,β(2)

a0= 30 s−1,β(1)

p0= 50

s−1,β(2)

p0= 30 s−1.

Let’s go to the analysis of the results. At the cer-

tain points of the porous media the values of c(i),ρ(i)

and ρ(i)

pincrease over time (Fig.1, Fig.2). As passive

zone capacities the values ρ(1)

p0= 0.06,ρ(2)

p0= 0.03

are taken. As can be seen from Figure 1. b, the con-

centration of the first component near the point x= 0

exceeds the capacity of the passive zone for this frac-

tion, i.e. ρ(1)

p> ρ(1)

p0. This shows that the capacity of

the passive zone for the second fraction is filled not

only with concentration ρ(2)

p, but also with ρ(1)

p. At

about t > 900 s, the concentration of the second frac-

tion increases to some extent xand then decreases.

0.00 0.05 0.10 0.15 0.20

0.00

0.01

0.02

0.03

0.04

0.05

(1)

(2)

x, m

= 450

= 900

= 1350

0.00 0.05 0.10 0.15 0.20

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

(1)

(2)

x, m

= 450

= 900

= 1350

0.00 0.05 0.10 0.15 0.20

0.00

0.01

0.02

0.03

0.04

0.05

0.06

(1)

(2)

x, m

= 450

= 900

= 1350

Figure 1: Profiles of c(1), c(2) (a), ρ(1)

a, ρ(2)

a(b),

ρ(1)

p, ρ(2)

p(c) at ρ(1)

p1= 0.015,ρ(2)

p1= 0.006.

Over time the coordinate of the increment of ρ(2)

pbe-

comes larger. If the value of the parameters ρ(1)

p1and

ρ(2)

p1are increased, from t≥450 in the points close

to the point x= 0, the capacity of the passive zone

ρp0is completely saturated with deposition (Fig. 2).

From Fig. 1.a. can be seen that at t≤450scon-

centrations of suspended particles for both types are

reached their maximum value only at inlet point, but

from t≥950sthe maximum value has been reached

at points x≤0.01m, and at t= 1350salmost till to

x≤0.05m. Comparing figures Fig. 1.a. and Fig.

2.a. one can see that increasing the parameters ρ(1)

and ρ(2)

p1leads to further reaching the maximum con-

centrations.

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0.00 0.05 0.10 0.15 0.20

0.00

0.01

0.02

0.03

0.04

0.05

(1)

(2)

x, m

= 450

= 900

= 1350

0.00 0.05 0.10 0.15 0.20

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

(1)

(2)

x, m

= 450

= 900

= 1350

0.00 0.05 0.10 0.15 0.20

0.00

0.01

0.02

0.03

0.04

0.05

0.06

(1)

(2)

x, m

= 450

= 900

= 1350

Figure 2: Profiles of c(1), c(2) (a), ρ(1)

a, ρ(2)

a(b),

ρ(1)

p, ρ(2)

p(c) at ρ(1)

p1= 0.03,ρ(2)

p1= 0.015.

Figure 3 shows the changing dynamics of ρ(i)

pfor

different values of ρ(i)

p1. It can be seen from the graphs

an increase in the values of ρ(i)

p1leads to an increase

in the intensity of deposition formation in the pas-

sive zone at the initial values of time. In the case of

less values of ρ(i)

p1it takes less time to reach maximum

value of ρ(i)

p. At the point x= 0 the maximum val-

ues of ρ(i)

p1reached at approximately 400 and 700 s

(Fig.3.a.), however at the point x= 0.03 at takes ap-

proximately 800 and 1200 s, respectively (Fig.3.b.).

At a fixed value of ρ(i)

p1the change profiles of

|∇P|,mat different times are shown in Figure 4.

On the whole, an increase can be observed for |∇P|,

0 200 400 600 800 1000 1200 1400

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(1)

(2)

t, s

(1)

= 0

015;

(2)

= 0

006

(1)

= 0

030;

(2)

= 0

015

0 200 400 600 800 1000 1200 1400

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(1)

(2)

t, s

(1)

= 0

015;

(2)

= 0

006

(1)

= 0

030;

(2)

= 0

015

Figure 3: Dynamics of ρ(1)

pand ρ(2)

pat the points x=

0m(a) and x= 0.03 m(b), at different values of

ρi

p1.

and a decrease for m. As it can be seen from (7), due

to the formation of deposition of particles in the ac-

tive and passive zones of the medium (which means

the increase of ρ(i)

aand ρ(i)

p), the active porosity of

the medium decreases. As it can be seen from (8),

decreasing the value of current porosity mleads to a

decrease in the filtration coefficient K(m)according

to the Carman-Kozeny law, because the both decreas-

ing the value of numerator of fraction and increasing

the value of denominator of fraction leads to decreas-

ing the value of fraction. Since we are considering a

regime with a constant filtration rate the pressure gra-

dient |∇P|at different points in this medium leads to

an increase. The increasing character of |∇P|of can

be seen in Fig. 4a, and the decreasing character of m

in Fig. 4b.

5 Conclusion

Model of filtration of two-component suspensions

in porous media improved including changes in the

characteristics of the porous media, dynamic factors,

multi-stage deposition formation and diffusion. In

this work the changes of characteristics of porous me-

dia (porosity, permeability) are taken into account di-

rectly in the model. It lead to analyse the role of dif-

ferent types deposition formation in filtration process.

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0.00 0.05 0.10 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

∇

= 450

= 900

= 1350

0.00 0.05 0.10 0.15 0.20

0.18

0.20

0.22

0.24

0.26

0.28

0.30

= 450

= 900

= 1350

Figure 4: Profiles of |∇P|(a) and m(b) at ρ(1)

p1=

0.03,ρ(2)

p1= 0.015.

An effective algorithm has been developed on the ba-

sis of finite difference method. It is shown that tak-

ing into account the multi-stage nature of deposition

formation leads to situations that are not observed in

single-step kinetics. In particular, non-monotonicity

was observed in the dynamics of deposition formation

at certain points.

The authors need to clarify and explain the differ-

ence between the current study with the available lit-

erature, as well as the main contribution of the study

in order to emphasize the main research outcomes of

the paper.

The unique aspect of filtering two-component sus-

pensions in porous media lies in the possibility of the

deposition of first component within the passive zone,

surpassing the environmental capacity designated for

it. This scenario arises as the deposition of the first

component intensifies, not solely due to the presence

of the second component, but also because the capac-

ity reserved for the second component can be partially

occupied by the first component.

In the future works this model will be improved for

ncomponent suspension filtration. Also it will be bet-

ter to conduct laboratory experiments to estimate the

adequateness of the developed mathematical model.

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WSEAS TRANSACTIONS on FLUID MECHANICS

DOI: 10.37394/232013.2023.18.20

Bekzodjon Fayziev, Jamol Makhmudov,

Jabbor Mustofoqulov, Tulkin Begmatov, Rakhmon Safarov

E-ISSN: 2224-347X

220

Volume 18, 2023