Solute Transport with Non-Equilibrium Adsorption in a Non-
Homogeneous Porous Medium
KHUZHAYOROV B. KH1, MUSTAFOKULOV J. A.2, DZHIYANOV T. O.1, ZOKIROV M. S.1
1Department of Mathematical Modeling, Samarkand State University,
REPUBLIC OF UZBEKISTAN
2Department of Physics, Jizzakh Polytechnic Institute, Jizzakh, REPUBLIC OF UZBEKISTAN
Abstract: - In this paper, a solute transport problem with non-equilibrium adsorption in a non-homogeneous
porous medium consisting of two zones, one with high permeability (mobile zone) and another one with low
permeability (immobile liquid zone) are considered. In the mobile zone, there are two zones in both of which
adsorption of solute with reversible kinetics occurs. The results of this approach are compared with known,
traditional approaches. It is shown that this method of modeling the process gives a satisfactory result. By
appropriate selection of the parameters of the source term, one can obtain results close to those of the well-
known bicontinuum approach.
Key-Words: - Adsorption, approximation, fractional derivative, porous media, retardation factor, mobile zone.
Received: July 15, 2021. Revised: September 8, 2022. Accepted: October 11, 2022. Published: November 7, 2022.
1 Introduction
Aquifers, oil, and gas reservoirs, as a rule, have a
heterogeneous structure at the micro- and
macroscale, [1]. Heterogeneous reservoirs on a
macro scale consist of different zones with
different, sometimes very strong, filtration-
capacitive properties, i.e. porosity, permeability,
etc. Zones with well porosity and permeability are
well conductors for liquids and various substances
suspended or dissolved in fluids. A typical example
of heterogeneous formations is fractured porous
media (FPM), [2], [3], the structure of which is
represented as a system of fractures surrounded by
porous blocks.
Colloidal particles suspended in a liquid can
move relatively fast and travel longer distances in
structured porous media than in media with a
homogeneous structure, [8], [12], [14], [15], [39].
The reason for this is the presence of pathways
conducive to the fast movement of substances. In
the simulation of solute transport in FPM, it is
usually assumed that the main ways for moving
liquid and suspended solids (or dissolved
substances) are fractures. Porous blocks in
simplified models are considered impermeable to
liquids, but particles or solutes can penetrate into
them due to the diffusion phenomenon. Thus, two
zones are formed in the medium, one with a mobile
fluid (fractures) and the other with an immobile one
(porous blocks). Between zones, mass transfer
processes occur. The advanced solute transport in a
porous medium can be the result of many factors.
Therefore, there are certain difficulties in the
mathematical modeling of this phenomenon. Some
models in this direction were presented in [10],
[17], [18], [19]. The two-zone approach noted
above was used in these models. Mass transfer
between zones is modeled by a first-order kinetic
equation, [9], [20]. A slightly different approach
combining kinetic and linear mass transfer between
zones was proposed in [13]. A certain modification
of the two-zone approach is an approach that takes
into account fluid motion in both zones, but with
different scales, [10], [17].
When colloidal particles are transported in a
porous medium, they are usually deposited in
pores, the causes of which are varied. Depending
on the nature and location of the interaction of
particles with the surface of the rock skeleton,
deposition can be reversible or irreversible. Given
these factors, transport models naturally become
more complex. Solute transport in double porosity
media taking into account reversible and
irreversible deposition is described by such
complex models. At the same time, it is important
to take into account the texture of the medium in
the models, [6],7], [21]. Mass transfer between two
flow zones is considered as a function of the
deposited volume of the solute in each zone, in
addition, small pores may be excluded from the
transport process, i.e. their locking due to the
deposition of substances, [6], [11], [16].
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Khuzhayorov B. KH,
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In [4] a transport model of colloidal substances
in a medium with double porosity is presented,
taking into account reversible and irreversible
particle retention, as well as first-order mass
transfer between fractures and porous blocks. The
obtained analytical solution was used to describe
experimental results, [5]. A good agreement was
obtained between theoretical and experimental
results. Dispersion and retention parameters were
higher for larger particles; the intensity of
reversible and irreversible particle retention was
higher for a medium with relatively small pores.
In [13] a transport model in a medium with
double porosity was considered taking into account
the reversible and irreversible deposition of colloid
particles in both zones and the first-order
equilibrium mass exchange between the zones. In
each zona, i.e. in fractures and porous blocks, a
reversible and irreversible deposition of particles
with various characteristics occurs, described by
linear equations. An analytical solution to the
problem is obtained, which is used to describe the
results of previous experiments, [6]. Coefficients of
mathematical models are defined as the solution to
the coefficient inverse problems (CIP), known as
identification problems, [26]. It is assumed that the
coefficients of the equation depend on the spatial
coordinates and are independent of time. The
statements of the problems are based on the use of
uniqueness theorems for the solution of the CIP
proved in [25], [29], [30], [33]. To obtain a unique
solution of the CIP, it is required to set an
overdetermined set of boundary conditions on the
boundary of the zone: the function for which the
equation is written or its normal derivative.
Coefficient inverse problems (identification
problems) have become the subject of intensive
study, especially in recent years. Interest in them is
caused primarily by their important applications.
They find applications in solving problems of
designing oil reservoir development (determining
the filtration parameters of reservoirs), [28], [30],
[32], [34], [35], [36], solving environmental
monitoring problems, etc. The standard CIP
statement contains a residual function, which
depends on the solution of the corresponding
problem of mathematical physics, [34]. Methods
for the numerical solution of CIP in connection
with their applications in underground
hydrodynamics were developed in [25], [26], [27],
[29], [31].
In this paper, an inhomogeneous two-zone
medium is considered a single-zone medium with a
source (sink). The second zone is modeled through
the source (sink). This approach is fundamentally
new because, in fact, the bicontinual medium is
presented as a mono-continual one. The validity of
this approach is justified by the convergence of the
results on the basis of the mono-continuous
approach to the corresponding results of the
bicontinuous approach. In the work, this is done by
minimizing the residual function. In addition, it is
assumed that in both parts of the first zone there is
reversible adsorption of particles with the
corresponding kinetic equations. Identification of
parameters in the source (sink) term in the mass
balance equation is carried out by solving the
corresponding CIP using data from [4].
Fig. 1: Scheme of solute transport in a two-zone
medium
2 The Mathematical Model and Its
Numerical Implementation
An inhomogeneous porous medium is considered,
consisting of well-permeable and relatively low-
permeable zones, the diagram of which is shown in
Fig. 1. The parameters in the first zone are
indicated by index 1. There are two sections in
zone 1, in each of which the particle deposition
with reversible nonequilibrium nonlinear kinetics
occurs. It is believed that such processes also occur
in the second zone, but we will not write equations
and conditions for it. With the second zone, there is
an exchange of substance, which we model by the
fractional-order derivative in time of the solute
concentration in the first zone. Consequently, in
contrast to [4], the concentration field in the second
zone is not considered. Note, that the fractional
approach was previously used in [37], [38], [40].
The equations of solute transport in one-
dimensional case are written as
,
1
11
21
2
11
1
2
1
1
21
x
C
v
x
C
D
t
C
a
t
C
t
S
t
Saa
(1)
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where
t
is time, s, x is distance, m,
1
D
is
longitudinal dispersion coefficient,
s
2
m
,
1
v
is the
fluid velocity, m/s,
1
С
is volume concentration of
the solute in the fluid,
1a
S
and
2a
S
are
concentrations of deposited particles,
kg/m3
,
1
is porosity,
33 m/m
,
is medium density,
3
m/kg
,
2
a
is retardation factor related to the mass
exchange between two zones,
1
s
,
is the order
time derivative with respect to time,
10
.
The deposition of particles in each of the
sections of the first zone is reversible with the
different kinetic equations
,
11111
1aada
aSkCk
t
S
(2)
,
22121
2aada
aSkCk
t
S
(3)
where
1a
k
,
2a
k
are coefficients of solute
deposition from the fluid phase to the solid phase,
1
s
,
,
1ad
k
2ad
k
are coefficients of substance
detachment from the solid phase and transition into
liquid,
1
s
.
Let a fluid with a constant solute concentration
0
с
be pumped into the medium initially saturated
with pure (without particles) liquid from the initial
moment of time. Let us consider such time periods
where the concentration field does not reach the
right boundary of the medium,
.x
Under the
noted assumptions, the initial and boundary
conditions for the problem have the form
,0),0(,0),0(,0,0 211 xSxSхС aa
(4)
,)0,( 01 ctС
(5)
.0),(
1
t
x
C
(6)
The problem (1) - (6) although linear, obtaining
an analytical solution is difficult because three
concentration fields must be found at the same
time. Therefore, to solve the problem, we use the
finite difference method. In the considered region
xTtxt 0,0),,(
a uniform grid
was introduced
JjIi
J
T
ihxjtxt ijij
h,0,,0,
,,);,(
,
where I is a sufficiently large integer chosen so that
segment
],,0[ I
x
,hixi
overlaps the area of the
calculated change in the fields C1, Sa1, and Sa2. h is
the grid step in the х direction.
In the open grid area
,1,1,,1,
,,);,(
IiJj
J
T
ihxjtxt ijij
h
equations (1), (2), (3) were approximated as
follows
,
)()(
)()(2)(
))()((
))(
)1((
)()(
)2(
)()(
)()()()(
1
11
1
1
11
2
1
11
1
1
1
11
11
1
01
1
1
1
1
1
1
1
1
2
1
1
1
1
2
1
21
1
1
h
CC
v
h
CCC
D
CC
kj
kj
CC
a
CC
SSSS
j
i
j
i
j
i
j
i
j
i
j
kj
i
j
i
k
i
k
i
j
i
j
i
j
ia
j
ia
j
ia
j
ia
(7)
,)()(
)()( 1
11111
1
1
1
j
iaad
j
ia
j
ia
j
ia SkCk
SS
(8)
,)()(
)()( 1
22121
2
1
2
j
iaad
j
ia
j
ia
j
ia SkCk
SS
(9)
where
j
i
С)( 1
,
j
ia
S)( 1
,
j
ia
S)( 2
are grid values of
functions
),(
1xtC
,
),(
1xtSa
,
),(
2xtSa
at a
given point
),( ij xt
.
From the explicit grid equations (8), (9) we
determine
1
1)( j
ia
S
,
1
2)( j
ia
S
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,)()( 211
1
1b
j
iab
j
ia pSpS
(10)
,)()( 221
1
2b
j
iab
j
ia qSqS
(11)
where
1
11
1
ad
bk
p
,
,)( 1
1
11
2j
i
ad
a
bC
k
k
p
2
11
1
ad
bk
q
,
j
i
ad
a
bC
k
k
q)( 1
2
21
2
.
The grid equations (7) are reduced to the form
,)()()()( 1
1
111
1
11
1
111 j
i
j
i
j
i
j
iFCECBCA
(12)
where
h
v
h
D
A
11
211
1
,
)2(
21
211
211
11
a
h
v
h
D
B
211
1h
D
E
,
k
i
j
k
k
i
j
ia
j
ia
j
ia
j
ia
j
i
j
i
Сkjkj
Сkjkj
a
SSS
SC
a
F
))()()1((
))()()1((
)2(
))()(())(
)(())(
)2(
()(
1
11
1
0
1
1
11
2
2
1
21
1
11
1
2
11
The following procedure of computing is used.
From (10), (11)
1
1)( j
ia
S
,
1
2)( j
ia
S
are determined,
then we solve the system of linear equations (12)
by Thomas’ algorithm in order to calculate
.)( 1
1j
i
C
Since
1, 11
bb qp
, schemes (10), (11) are
stable, and for (12) the stability conditions of the
Thomas’ algorithm are satisfied.
To assess the performance of the proposed
model, it is important to compare the results with
the corresponding results, [4]. To do this, we
compare the source (stock) terms
)( 12 СС
in
[4] and
t
C
a1
2
in (1). To quantify the proximity
of the results based on the curves was calculated
LdxII
0
2
211
(13)
for a given value of t, where L is the conditional
boundary of the medium to which the concentration
profiles extend,
., 1
22121
t
C
aICCI
The proximity of the terms
1
I
and
2
I
should
guarantee the proximity of the concentration fields
1
С
determined using the proposed approach and the
model, [4]. To estimate their proximity, we use the
standard deviation (13), only for
1
C
determined on
the basis of two models, i.e.
LdxCC
0
2
)2(
1
)1(
12 ,
where
)1(
1
C
– concentration field
),(
1xtC
for a
given t, determined according to [4], and
)2(
1
C
– the
same as defined here.
For other moments
t
and
,, 2
а
different
estimates can be obtained for
1
and
2
. In
principle, to approximate the two models, it is
important to set and solve the corresponding
coefficient inverse problems by determining of
,
2
а
for a given value of
or, conversely,
determining of
for a given
and
.
3. Numerical Results and Their
Analysis
In the calculations following initial values of
parameters are used:
,0
0c
sv /m10 4
1
,
l
vD 11
,
,m005,0l
3
/1800 mkg
,
,1,0
1
,105,2,10314
1
14
1 sksk ada
14
2
14
2102,104 sksk ada
and various
.,
2a
Some results are shown in Fig.2. As can be seen
from the figure, the outflow of particles into the
second zone leads to a slow distribution of the
solute concentration profiles in the mobile fluid. As
a consequence of this phenomenon, delays are also
observed in the concentrations of the adsorbed
mass. From this, it is clear that with a decrease in
the index of the fractional derivative
from 1,
both in the solute concentration in the fluid and in
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the concentration of the adsorbed solute in the well-
permeable zone, there is a delay in the distribution.
For a certain set of parameters
,
2
a
and
14 s10
, graphs
21 , II
are shown in Fig.3.
As can be seen from the figure, the patterns of
change in stock terms are similar, which indicates a
qualitative agreement between the results of the
proposed model and the results of the model, [4].
After that, we minimize the functional
T L dtdxIIaФ
0 0
2
212 ),(
, (14)
that characterizes the standard deviation of
1
I
from
2
I
for the entire time period. The calculations
show that the minimum value of
),( 2aФ
is
achieved at
.8,0,0006,0
2a
The proximity of the terms
1
I
and
2
I
should
guarantee the proximity of the concentration fields
,
1
С
determined using the proposed approach and
model, [4]. For this, the corresponding profiles are
plotted for the data, obtained through minimization
of
),( 2aФ
(Fig. 4). As can be seen from the
graphs, the solutions are close to each other.
For a numerical estimation of their proximity,
we use the standard deviation of the type (14), only
for the one determined on the basis of two models,
i.e.
T L dxdtCCaF
0 0
2
)2(
1
)1(
12 ,),(
where
)1(
1
C
is the concentration field
),(
1xtC
for a
given t, determined according to [4].
)2(
1
C
is the
same as defined here. For the cases analyzed above,
the following minimum value of
),( 2aF
was
obtained
76544520,00234738
for
.8,0,0006,0
2а
Fig. 2: Concentration profiles C1 (а), Sа1 (b), Sa2 (c)
at t=3600 s,
C1
а)
х, m
104·Sa2,m3/kg
c)
х, m
104·Sa1,m3/kg
х, m
b)
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The analysis shows that the simpler model
proposed here, with an appropriate choice of
parameters, can satisfactorily describe the results of
a more complex model, [4].
4 Conclusions
In contrast to [4], a new model is proposed where
the presence of the second zone of an
inhomogeneous medium is taken into account in
the form of a source (sink) term in the transport
equation written out for the first zone. The stock
term is presented as a fractional time derivative of
the concentration of the substance in the first zone
with a certain coefficient. Thus, this approach is
mono-continuous, while the bicontinual approach
was used in [4]. The model was implemented
numerically and the effect of mass transfer to the
second zone on the transport characteristics in the
first zone was estimated. It is observed that with a
decrease of the order in the fractional derivative
from 1, both in the concentration of the particles in
the mobile fluid and of the adsorbed substances in
the mobile zone, the dynamics of distribution delay.
A problem of approximation of the results
according to the proposed model with the
corresponding results, [4] was solved. For this,
values of parameters in the stock term, which
ensures close results, are obtained using a
variational approach that minimizes the residual
function. It is shown that for certain values of
parameters
2
a
and
a good convergence can be
achieved. Thus, the fundamental possibility of the
proposed mono-continuum model to describe the
results of the corresponding bicontinuum model is
shown. In addition, as it is shown in [22], the
heterogeneity of porous media can be a cause of
anomalous phenomena in filtration and transport
processes. It is known, [23], [24], fractional time in
filtration and transport laws can model anomalous
phenomena. Therefore one can expect that the
proposed here model can be used to study
anomalous transport phenomena in non-
homogeneous porous media.
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Fig. 3: Comparison of source members
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3,0
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7,0
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8,0
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