CC BY
22
основания и фундаменты,
подземные сооружения. механика грунтов
УДК 624.131, 532.546 DOI: 10.22227/1997-0935.2017.11.1278-1283
ASYMPTOTICS OF A PARTICLES TRANSPORT
PROBLEM
L.I. Kuzmina, Yu.V. Osipov*
National Research University Higher School of Economics, 20 Myasnitskaya st., Moscow, 101000, Russian Federation; *Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation
Subject: a groundwater filtration affects the strength and stability of underground and hydro-technical constructions. Research objectives: the study of one-dimensional problem of displacement of suspension by the flow of pure water in a porous medium.
Materials and methods: when filtering a suspension some particles pass through the porous medium, and some of them are stuck in the pores. It is assumed that size distributions of the solid particles and the pores overlap. In this case, the main mechanism of particle retention is a size-exclusion: the particles pass freely through the large pores and get stuck at the inlet of the tiny pores that are smaller than the particle diameter. The concentrations of suspended and retained particles satisfy two quasi-linear differential equations of the first order. To solve the filtration problem, methods of nonlinear asymptotic analysis are used.
Results: in a mathematical model of filtration of suspensions, which takes into account the dependence of the porosity and permeability of the porous medium on concentration of retained particles, the boundary between two phases is moving with variable velocity. The asymptotic solution to the problem is constructed for a small filtration coefficient. The theorem of existence of the asymptotics is proved. Analytical expressions for the principal asymptotic terms are presented for the case of linear coefficients and initial conditions. The asymptotics of the boundary of two phases is given in explicit form. Conclusions: the filtration problem under study can be solved analytically.
KEY WORDS: filtration, porous media, suspension, filtration coefficient, particle blocking, suspended particles, retained particles, mathematical model, phase boundary, asymptotics
FOR CITATION: Kuzmina L.I., Osipov Yu.V. Asymptotics of a Particles Transport Problem. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2017, vol. 12, issue 11 (110), pp. 1278-1283.
АСИМПТОТИКА ЗАДАчИ ПЕРЕНОСА чАСТИЦ
л.И. кузьмина, Ю.В. Осипов*
Национальный исследовательский университет «Высшая Школа Экономики» (НИУ ВШЭ),
101000, г. Москва, ул. Мясницкая, д. 20; *Национальный исследовательский Московский государственный строительный университет t- (НИУ МГСУ), 129337, г. Москва, Ярославское шоссе, д. 26
г
f Предмет исследования: фильтрация грунтовых вод влияет на прочность и устойчивость подземных и гидротехни-
¡^ ческих сооружений.
Цель исследования: изучение одномерной задачи вытеснения суспензии чистой водой в пористой среде. Задача и методы: при фильтрации суспензии некоторые частицы свободно проходят через пористую среду, а другие застревают в порах. Предполагается, что распределения размеров частиц и пор перекрываются. В этом случае ц основной причиной блокировки частиц является механико-геометрический механизм: частицы свободно проходят
через большие поры и застревают на входе в малые поры, размеры которых меньше диаметра частиц. Концентрации взвешенных и осажденных частиц удовлетворяют двум квазилинейным дифференциальным уравнениям первого порядка. Для решения задачи фильтрации используются методы нелинейного асимптотического анализа. q Результаты: в математической модели вытеснения суспензии, учитывающей зависимость пористости и проницае-
мости пористой среды от концентрации осажденных частиц, граница двух фаз движется с переменной скоростью. ^ Асимптотическое решение задачи строится в предположении малости коэффициента фильтрации. Доказана теоре-
Q ма существования асимптотики. Представлены аналитические выражения для основных асимптотических членов
в случае линейных коэффициентов и начальных условий. Асимптотика границы двух фаз получена в явном виде.
2 Выводы: изучаемая задача допускает аналитическое решение.
КЛЮЧЕВЫЕ СЛОВА: фильтрация, пористая среда, суспензия, коэффициент фильтрации, блокировка частиц, I взвешенные частицы, осажденные частицы, математическая модель, граница фаз, асимптотика
О
Ф ДЛЯ ЦИТИРОВАНИЯ: Кузьмина Л.И., Осипов Ю.В. Asymptotics of a Particles Transport Problem // Вестник МГСУ.
М 2017. Т. 12. Вып. 11 (110). С. 1278-1283.
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© L.I. Kuzmina, Yu.V. Osipov
introduction
t = 0: C = C0(x), S = S0(x).
(4)
The problem of filtration of solid particles in porous media is an important part of the underground hydrodynamics [1, 2]. The filtration of groundwater affects the strength and stability of tunnels and hydraulic structures. The slurry of fine particles in fluid forms a suspension moving in the pores of rocks and soil. When the particles are transported through a porous medium, some of them get stuck, forming a deposit. During deep bed filtration filtering occurs not only in the surface layer, but throughout the whole porous medium [3-7]. For different types of porous media and suspensions there are various reasons for particle retention. Retention can be caused by electrostatic forces, diffusion, fluid viscosity, gravity, etc. [8-13]. If the particle and pore sizes are of the same order, the main mechanism for particle retention is size-exclusion [14, 15]. Suspended particles pass freely through the large pores and get stuck in the throat of the pores smaller than the particle diameter. A mathematical model of particles transport and capture in a porous medium comprises the mass balance equation of suspended and retained particles, and the kinetic equation for deposit growth rate, proportional to the concentration of suspended particles [16, 17]. As the porosity and permeability of the porous medium change with increasing deposit the balance equation has variable coefficients [18]. In some important cases, the exact and asymptotic solutions of filtration problems were found [3, 15, 18-24].
The paper deals with the one-dimensional problem of the displacement of suspension by pure water. At the initial moment the porous medium is filled with suspension. The porous medium has the initial deposit. The pure water, displacing the suspension, is supplied at the filter inlet. The suspension passes to the filter outlet and forms an additional deposit. The boundary of the two phases is the pure water front. It moves at a variable velocity, which depends on the retained particles concentration. When the water flow reaches the filter outlet, the formation of a deposit terminates.
mathematical model
For a description of state of a suspension and a porous medium the concentrations of suspended and retained particles C(x, t), S(x, t) are used. In dimension-less variables x, t in the domain W = {0 < x < 1, t > 0} these concentrations satisfy the quasi-linear hyperbolic system of the first order differential equations
d; (g (S )C) + (f (S )C )+dS = 0; (1)
dt dx dt
dS
— = eA( S)C; (2)
dt
with the boundary and initial conditions
x = 0: C = 0; (3)
Here the functions of porosity g(S) and permeability f (S), the filtration coefficient eA(S) and the initial concentrations C0 (x), S0 (x) are continuous and positive, e is a small positive parameter.
The domain W is divided into two subdomains WK and Ws, containing water and the suspension (Figure 1). The boundary r of the two phases is a characteristic of the system, starting at the origin. On the phase boundary the concentration of suspended particles has a jump, and the concentration of retained particles is continuous.
Figure 1. Diagram of the solution to the problem (1) - (4)
If the coefficients of the equation (1) are constant
g (S) = go > 0, f (S) = fo > 0, (5)
the boundary r of the two phases is a straight line
t = ax, a = g„/f0. (6)
In general, when the coefficients g (S ), f (S ) are variable the curvilinear boundary r can not be expressed analytically.
existence and uniqueness of asymptotic solution
The asymptotic solution to the system (1) - (4) is constructed in the form of a series in powers of the parameter e ^ 0
C = c0 ( x, t ) + ec1( x, t ) + O(e2); (7)
S = s0( x, t ) + es1( x, t ) + O(e2). (8)
Substitution of the expansions (7), (8) into (1), (2) and grouping the terms with equal powers of the small parameter e gives the equations for the terms of the asymptotic solutions. The initial conditions follow from (4).
dt (g(S0)C0 ) + dx (f (S0)C0 ) = 0, c It=0 = x); (9)
m
(D
0 T
1
s
*
o y
T
0 s
1
ISJ
B
r
3
y
o *
dSc dt
= 0,
J0 It=0
= S0( x);
(10)
d
— ( g( so)c1 + g'( so) sic0 ) + d
+ & ( f( so)ci + f'( so) sico) +
+A(so)co = ° ci [=o = 0;
ds
-d- = A(s0)c0, si It=0 = 0.
(11) (12)
C + C df (So(x)) = o, C| = C0 (x0).
dx k=° 0 0
(17)
O >
E
tt
CM ^
S o
H >
O
X
s
I h
O
o 10
Consider the Jacobian
d( x, t)
J(Xo,t) =
d( Xo, t)
dx/ dx0 dt/ dx0
dx/ dt dt/ dt
(19)
To calculate the partial derivatives in (19) we integrate (16) with respect to t
dx
1
= t,
Theorem 1. Suppose that g(S), f (S), C0(x), S0(x) are the smooth positive functions for S > 0, 0 < x < 1. Then in W the system (9) - (12) has a unique solution.
Proof. From (10)
so = So( x), (13)
the equation (9) can be written as
g (so) ^ + f(so) dc0 + f(so)s0co = o. (14)
dt dx
In the domain Ws the characteristic equation for the system (14) has the form
i = g (So(x)), t|t=o = 0; (15)
x = f (So(x)), x|t 0 = xo; (16)
f (So( x)) and differentiate the integral over x0
= f (So( x))
dxo f (So(xo))
dx
f ( S0( x) )dx0 f (S0( x0))'
- = 0
.(20)
Integrating (15) with respect to t and differentiating the integral over x0 , we get
^ = 1 g'(So) S0( x) -^d ra.
dx, J
Substituting (20) into (21)
dx„
(21)
£=1 *) S0( x) d«.
(22)
Using the expression for the partial derivatives (15), (16), (20) and (22) the Jacobian (19) can be represented in the form
Here x0 e [0; 1] is a starting point of the characteristic emerging from the x-axis; t is a parameter along the characteristic, x denotes differentiation with respect to t. In particular, the boundary r of the two phases is a characteristic with initial condition x0 = 0 (see Figure 2).
J (xo, t) =
f (So( x)) f (So( xo))
f (So( x))
1 g' (So) S0( x)^ dra g (So (x))
0 J (So(xo))
Figure 2. Characteristics and characteristic coordinates in the domain W
By the Cauchy's theorem, the system (15) - (17) has a unique solution {t(x0,t); x(x0,t); C(x0,t)}.
Let us show that for positive g(S), f (S), S0 it is always possible to express the solution in terms of x, t, i.e. we can solve the equations
t = t (x0, t). (18)
Calculation gives
J (xo = "0 =
: f(x)) ( g(So(x))-1 f (S0(x))g '(S0)S0(x)dra 1.(23) J (So(xo)K 0 )
From (15)
1 f (So( x)) g' (So) S0( x)d ra =
0
t
= 1 g' ( So) S0 (x) xd ra = g ( So (x)) - g ( So (x^ ).
Finally
J(x0,t) = g(So(xo)) f (S0(x))> 0.
f ( So(xo) r 0
(24)
Thus, in the domain Ws the equations (18) are always solvable, and the solution to equations (9) and (10) exists and is unique.
In the domain Ww the characteristics emerge from the time axis (see Figure 2). The initial conditions for the equations (15) - (17) have the form
t| = t0; x| = 0; C| = 0.
k=0 0' k=0 ' k=0
The Jacobian
I( x0,t) =
d( x, t ) d(t0, t)
0 f ( S0( x) )
1 g ( S0( x) )
dx/ dt0 dx/ dt dt/dt0 dt/dz
= - f ( S0( x) )< 0.
S1 = A(s0)J C0(x, p)dp .
(26)
The equation (11) can be represented in the form
dc dc
g( S0) + f (S0) + f '( S0) s0 c +
d d
+g'( s0^ ( s1c0 )+T"( f'( s0) s1c0 ) + dt dx
+A(s0)c0 = 0. (27)
t = A + Ba + Bbx, t = 0.
' lt=0
X = U + Fa + Vbx, x = x„.
' k=0 0
C + VbC = 0, C|t=0 = Q(X0).
Integrating (34) with respect to t and expressing x we get
(33)
(34)
(35)
i.e. the equations x = x(t0,t); t = t(t0, t) are solvable in the domain WK.
Consider the equations (11), (12) for the first terms of the asymptotics. The solution of the equation (12) has the form
Xd t
U + Va + Vbx
x = (U + Va)( 1) + ^.(36)
Vb 0
Substitution of (36) into (33) and integration with respect to t gives
,, ^ (U + Va)
t = ( A + Ba)t---- Bbt +
Vb
,(U+Va+Vbx0) Bb( 1).
(Vb)2
The solution of (35)
C ( X0, t) = C0( xa)e~
(37)
(38)
The existence and uniqueness of the solution to the equation (27) in the domain W is proved similarly. Theorem 1 is proved.
Note. Theorem 1 is valid for the next terms of asymptotics to the problem (1) — (4), since the corresponding equations differ from (11), (12) by right-hand sides only.
According to Theorem 1, the system (36), (37) has a unique solution t(x, t); x0(x, t).
In order to find the boundary r of the two phases we express t( x) from (36) setting x0 = 0
1 1 h Vbx
t =— ln I 1 +-
Vb I U + Va
calculation of the main asymptotic terms
A) A constant initial deposit S0 (x) = S0 = const.
From (10)
S(x, t) = S0 + O(e), (x, t) e W . (28)
The equation (9) takes the form
dC dC
g (S0) dC+f (S0) dC=0.
dt dx
(29)
and substitute it into (37). The main asymptotic term of the boundary r of two phases has the form
tr ( x) =
A + Ba U + Va
Vb (Vb)2
coNcLusioNs
Bb I ln I 1 +
Vbx ^ B
U + Va ) V
I + - x. (39)
The solution of (29)
10, (X, t) £n ,
C(x,t) = J ' w (30)
[ C0( x-vt) + O(e), ( x, t) £Ws.
The formula (30) defines a travelling wave in the domain Ws moving with a constant velocity v = f (S0)/g (S0).
В) Linear coefficients and a linear initial deposit Let
g (S ) = A + BS, f (S ) = U + VS,
S0 = a + bx, C0 = C0( x0). (31)
Then
g(S0) = A + Ba + Bbx, f (S0) = U + Va + Vbx. (32) The equations (15) - (17) have the form
The problem of displacement of the suspension by the flow of water is one of the most important examples of filtration. During deep bed filtration the deposit accumulation is extremely slow, so the filtration coefficient is small. This makes it possible to construct the asymptotic solution of the particle transport problem. In the linear case the asymptotics is defined explicitly.
To prove the existence and uniqueness of the asymptotic solution the equation for the main asymptotic term is associated with characteristic system specifying the dependence of the original variables x, t from the characteristic variables. It is proved that this dependence is one-to-one, i.e. the asymptotic solution can be expressed in terms of the variables of the original problem.
For the two-phase problems, of particular interest is to determine the moving boundary of the two media. When the pure water displaces the suspension in a porous medium the two-phase boundary is a nonlinear function of the initial conditions and equation coeffi-
m
(D
0 T
1
s
o y
T
0 s
1
ISJ
B
r 3
y
o
cients. For linear coefficients of the mass transfer equation (1) and the linear initial distribution of the deposit in the filter the main asymptotic term of the boundary between water and suspension is constructed. Numerical modelling of this problem was performed in the papers [25-27].
Analytical solutions of filtration problem allow us to reduce experimental modelling. Having defined parameters of the solution from experiments, it is possible to extrapolate the solution of the problem and significantly reduce the amount and the cost of laboratory and field experiments [28].
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3. Herzig J.P., Leclerc D.M., Legoff P. Flow of suspensions through porous media - application to deep filtration. Industrial and Engineering Chemistry. 1970, vol. 62, no. 8, pp. 8-35.
4. Elimelech M., Jia X, Gregory J., Williams R. Particle deposition and aggregation: measurement, modelling and X simulation. Oxford, ButterworthHeinemann, 1995.
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^ rous media. Environmental Science and Technology. v 2007, vol. 41, pp. 5666-5671.
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aerosols and hydrosols. 2nd ed. Amsterdam, Elsevier, ^ 2007.
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ments. Environmental Science and Technology. 2014, vol. 48, pp. 6805-6813.
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18. You Z., Bedrikovetsky P., Kuzmina L. Exact solution for long-term size exclusion suspension-colloidal transport in porous media. Abstract and Applied Analysis. 2013, article ID 680693.
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MATEC Web of Conferences, 2016, vol. 86: 5th International Scientific Conference «Integration, Partnership and Innovation in Construction Science and Education», no. 01005.
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«Integration, Partnership and Innovation in Construction troleum Science and Engineering. 2001, vol. 32, no. 3,
Science and Education», no. 03003. pp. 167-177.
Received January 23, 2017.
Adopted in final form on September 22, 2017.
Approved for publication October 26, 2017.
About the authors: Kuzmina Ludmila Ivanovna — Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Applied Mathematics, National Research University Higher School of Economics, 20 Myasnitskaya st., Moscow, 101000, Russian Federation, lkuzmina@hse.ru; ORCID 0000-0002-6551-733X; ResearcherID K-5547-2015; Scopus AuthorID 56104145400;
Osipov Yuri Viktorovich — Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Applied Mathematics, Moscow State University of civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation, yuri-osipov@mail.ru; ORCID 0000-0001-8370-1731; ResearcherID P-7832-2017.
Поступила в редакцию 23 января 2017 г. Принята в доработанном виде 22 сентября 2017 г. Одобрена для публикации 26 октября 2017 г.
Об авторах: кузьмина людмила Ивановна — кандидат физико-математических наук, доцент, доцент кафедры прикладной математики, Национальный исследовательский университет «Высшая Школа Экономики» (НИу ВШЭ), 101000, г. Москва, ул. Мясницкая, д. 20; lkuzmina@hse.ru; ORCID 0000-0002-6551-733X; ResearcherID K-5547-2015; Scopus AuthorID 56104145400;
Осипов Юрий Викторович — кандидат физико-математических наук, доцент кафедры прикладной математики, Национальный исследовательский Московский государственный строительный университет (НИу МГСу), 129337, г. Москва, Ярославское шоссе, д. 26; yuri-osipov@mail.ru; ORCID 0000-0001-8370-1731; ResearcherID P-7832-2017.
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