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СТРОИТЕЛЬНОЕ МАТЕРИАЛОВЕДЕНИЕ
УДК 517.9+531.012
Л.И. Кузьмина, Ю.В. Осипов*
НИУВШЭ, *ФГБОУВПО «МГСУ»
АСИМПТОТИКА ЗАДАЧИ ФИЛЬТРАЦИИ СУСПЕНЗИИ В ПОРИСТОЙ СРЕДЕ
Рассмотрена механико-геометрическая модель фильтрации суспензии в пористой среде. Предполагается, что твердые частицы беспрепятственно проходят через поры большого диаметра и застревают на входе пор, размер которых меньше размера частиц. Концентрации взвешенных и осажденных частиц удовлетворяют квазилинейной гиперболической системе уравнений в частных производных первого порядка. Асимптотическое решение перед фронтом распространения концентрации строится в предположении малости коэффициента фильтрации.
Ключевые слова: суспензия, пористая среда, задача фильтрации, коэффициент фильтрации, квазилинейная гиперболическая система, характеристическая линия, асимптотическое решение.
Фильтрация взвеси мельчайших твердых частиц — суспензии в пористой среде является актуальной задачей для многих областей науки и техники. При долгосрочной оценке прочности фундаментов необходимо учитывать состав грунта и проникающую способность грунтовых вод. В нефтя-
L.I. Kuzmina, Yu.V. Osipov
Higher School of Economics, MGSU
ASYMPTOTICS OF THE FILTRATION PROBLEM FOR SUSPENSION IN POROUS MEDIA
The mechanical-geometric model of the suspension filtering in the porous media is considered. Suspended solid particles of the same size move with suspension flow through the porous media — a solid body with pores — channels of constant cross section. It is assumed that the particles pass freely through the pores of large diameter and are stuck at the inlet of pores that are smaller than the particle size. It is considered that one particle can clog only one small pore and vice versa. The particles stuck in the pores remain motionless and form a deposit. The concentrations of suspended and retained particles satisfy a quasilinear hyperbolic system of partial differential equations of the first order, obtained as a result of macro-averaging of micro-stochastic diffusion equations. Initially the porous media contains no particles and both concentrations are equal to zero; the suspension supplied to the porous media inlet has a constant concentration of suspended particles. The flow of particles moves in the porous media with a constant speed, before the wave front the concentrations of suspended and retained particles are zero. Assuming that the filtration coefficient is small we construct an asymptotic solution of the filtration problem over the concentration front. The terms of the asymptotic expansions satisfy linear partial differential equations of the first order and are determined successively in an explicit form. It is shown that in the simplest case the asymptotics found matches the known asymptotic expansion of the solution near the concentration front.
Key words: suspension, porous media, filtration problem, filtration coefficient, quasilinear hyperbolic system, characteristic line, asymptotic solution.
Filtration of a suspension — a slurry of tiny solid particles in a porous media — is a common phenomenon in nature and in many industrial applications. With long-term assessment of the strength of the construction
ной промышленности перемещение мелких частиц горных пород может привести к значительному падению добычи нефти [1—3]. В разнообразных технических и биологических системах очистки поверхностных и сточных вод и жидких промышленных отходов происходит захват частиц при движении суспензии через фильтрующую среду [4—9].
В работе рассматривается математическая модель фильтрации суспензии, основанная на механико-геометрическом взаимодействии частиц с пористой средой [10—17]. Предполагается, что частицы свободно проходят через пустоты — поры большого диаметра и застревают на входе пор, размеры которых меньше размера частиц (рис.).
bases it is necessary to consider the composition of the soil and groundwater penetrating ability. In petroleum industry, the migration of fine particles in low consolidated natural rocks may lead to significant decline of oil production [1—3]. In various technical and biological treatment systems for surface and waste water and liquid industrial waste solid particles are captured while moving with the suspension through the filter medium [4—9].
In this paper we consider a mathematical model of suspension filtration in porous media based on the mechanical-geometric interaction of particles with pores [10—17]. It is assumed that the particles pass freely through the interstices — pores of large diameter and get stuck at the inlet of pores, which are smaller than the particle size (Fig.).
Поперечное сечение пористой среды с взвешенными и осажденными частицами
The cross-section of porous media with suspended and retained particles
Концентрации взвешенных С(х, 0 и осажденных S(x, 0 частиц удовлетворяют квазилинейной гиперболической системе дифференциальных уравнений первого порядка в частных производных
д(а( S )С + S) д(Ь( S )С)
The concentration C(x, t) of the suspended particles and the concentration S(x, t) of the retained particles satisfy the quasi-linear hyperbolic system of partial differential equations of the first order
dt
dx dS
— = K ( S )C, dt
= 0;
(1)
(2)
где К(£>) — коэффициент фильтрации. Функции а(5), 6(5), К(£!) гладкие и положительные при S > 0.
here K(S) is a filtration coefficient, functions a(S), b(S), K(S) are smooth and positive, when S > 0.
Система уравнений (1), (2) рас- The system of equations (1) and (2) сматривается в области is treated in the domain
W = {0 < x <1, t > 0}. Краевые условия для систем (1), The boundary conditions for the (2) ставятся на входе фильтра x = 0 и в system (1), (2) are placed at the filter начальный момент времени t = 0: inlet x = 0 and at the initial moment t = 0:
С(x, t)|x=0 = p, p > 0; (3)
C ( t ) =0; S ( X, t )| t=0 = 0.
(4)
(5)
Строгий вывод уравнений задачи (1)—(5) на основе микростохастических уравнений фильтрации с последующим макроосреднением содержится в [18]. Решение задачи в интегральной форме приведено в [19]. Однако найти решение аналитически в явном виде удается лишь в простейших случаях [20—22]. В [23] строится асимптотическое решение задачи (1)—(5) вблизи волнового фронта, а в [24] найдена асимптотика вблизи входа фильтра.
Пусть коэффициент фильтрации мал. Представим его в виде
где е — малый положительный параметр. Пусть функции а(5), 6(5), Л(5) регулярны в окрестности 5 = 0 и
а( 5) = а0
A rigorous derivation of the filtration problem (1)—(5) on the basis of micro-stochastic diffusion equations, followed by macro-averaging is contained in [18]. The solution in the integral form is given in [19]. However, the analytical solution in an explicit form can only be found in the simplest cases [20—22]. In [23] an asymptotic solution of the problem (1)—(5) in the vicinity of the wave front was constructed; in [24] the asymptotics near the filter inlet was found.
Let the filtration coefficient is small. We represent it in the form
K(S) = sA(S),
where s is a small positive parameter. Let the functions a(S), b(S), A(S) are regular in a neighborhood S = 0 and
-a1S -h6jS-
b( S) = 60 L( S ) = A0 + A1S
ha2 S2 vb2 S2 - A, 2 S2
-a3 S -63S3
- A3S3
, a0 > 0,
60 > 0,
, A >0.
(6)
(7)
(8)
Below we construct the asymptotic solution of the problem (1)—(5) if e ^ 0.
a
Denote a = —. If e = 0 the exact
60
solution of the problem is
C(x, t) = p, t >ax; C(x, t) = 0, t <ax; S(x, t) = 0, (x, t) eQ.
В общем случае e > 0 прямая t = ax In general case e > 0 the straight line
является характеристикой квазили- t = ax is a characteristic of quasi-linear
нейной гиперболической системы, на hyperbolic system, on which the solution
которой решение имеет разрыв [23]: is discontinuous [23]
Ниже мы построим асимптотику задачи (1)—(5) при е ^ 0.
а
Обозначим а = —-. При е = 0 находим точное решение задачи:
C ( X, t )
1 = 0, t < ax, | = 0, t < ax,
S(x, t) \
[>0, t >ax; [>0, t >ax.
В области t > ax начальные усло- In the domain t > ax the initial con-вия (4), (5) можно заменить одним ус- ditions (4) and (5) can be replaced by
ловием на характеристике: one condition on the characteristic line:
S(^ t)|^ = 0. (9)
Будем искать решение задачи (1)— We seek a solution to the problem (3), (9) в виде рядов по степеням e: (1)—(3), (9) as a series in powers of e:
S (x, t, s) = S5,1( x, t) + s2 s2( x,t) + s3 s3( x,t) + ..., (10)
C (x, t, e) = p + ecj (x, t) + e2c2 (x, t) + ... (11)
Найдем три первых члена асим- In order to find the first three terms
птотики (10), (11). Подставим разло- of the asymptotics (10), (11) we substi-
жения (6)—(8), (10), (11) в уравнения tute the expansions (6)—(8), (10), (11)
(1), (2) и, приравнивая выражения при in the equations (1), (2) and equating the
одинаковых степенях e, составим си- expressions of the same powers of e, we
стемы для определения неизвестных compose the system to determine the
функций sk(x, t), c(x, t). Имеем: unknown functions s(x, t), c(x, t).
d d
—(ao c + ai psi )+dx (b°ci+bi psi )=p; dsi = y p
(12)
—( c2 + aci + a2 psS{ + ax ps2 ) -
d
— (boC2 + blClS1 + b2 pS1 + b1 pS2 ) = -(oC + pS1 ) (13)
dx ôs
—2 = Vi psi; dt
e3 : -3 = A0c2 + A^s + A ps2 + A2psj2. dt
(14)
Определим краевые условия для In order to define boundary condi-(12)—(14), подставляя (10), (11) в tions for the equations (12)—(14), we (3), (5): substitute (10), (11) in (3), (5):
(15)
sA = 0, s J = 0, S3 = 0;
1 lt=ax 7 2 lt=ax 7 3 lt=ax 7
С lx=0 = 0, C2 Ix=0 = 0.
(16)
Последовательно решая методом Successively solving the equations
характеристик уравнения (12)—(14) с (12)—(14) with the conditions (15) and условиями (15), (16), находим (16) by the method of characteristics we
find
Sj = A0 p(t -ax );
(17)
s
ВЕСТНИК
МГСУ-
1/2015
1 = \ix,\i=^(ap^ - pai~ i);
s2 = A0mx(t -ax ) +
A oAi P2
(t -ax )2 ;
Aa
Ba
c2 =-x +--x(t -ax);
2ao ao
A = 2X 0 p^(bja - aj ) - А0ц; B = X0XJp3 (bja - aj ) + 2X2p3 (b2a - a2 ) - X0p(2yjbj + Xjp); A
s3 = —^x(t -ax)(Ax + B(t -ax)) +
(18)
(19)
(20)
lb,
+A0A1 pmx(t -ax)2 + AoP (A12 +2A0A2)(t -ax)3.
(21)
6
Подставим в (10), (11) найденные We substitute in (10), (11) the found
выражения (17)—(21) для членов expressions (17)—(21) for the asymp-
асимптотики. В области t > ax асим- totic terms. In the domain t > ax the as-
птотическое решение задачи (1)—(5) ymptotic solution of the problem (1)—
имеет вид (5) has the form
(
S(x,t,e) = eA0p(t - x) + e A0|x(t -ax) +
A 0Ai P2
Л
(t - ax)1
+e3 ( Ab~ X(t -ax) ( Ax + B(t -ax) ) +A0A1 p|x(t -ax)2 +
(22)
+
A n3 ^
(Aj2 +2A0A2 ) (t -ax)3 +O (e4 ) ;
(
C (x, t, e) = p + e|ix + e2
Aa
Ba
Л
-x +--x(t -ax)
V1a0 a0
O (e3 ).
(23)
Отметим, что для простой систе- Let us note that for a simple system
мы (a(S) = 1, b(S) = 1) при e = 1 пред- (a(S) = 1, b(S) = 1) when e = 1, the ex-
ставления (22), (23) являются главны- pressions (22) and (23) are the main
ми членами разложения в ряд по сте- terms of the expansion in powers of x of
пеням x асимптотики решения вблизи the asymptotic solution near the charac-
характеристики t = x [25]: teristics t = x [25]:
S(x, t) = pA°e~x°x (t - x) + AjA0p2
1
e-2Xox — e-^ox
(t - x)2 +
+P
AiA о +~2~ w 2
e~3A°x - A,2A°e~2Ax + Ai A° A2A2 e-A°x 1 ° 6
Л (24)
C(x, t) = pex + p% (x - ex ) (t - x) + O(t - xf.
(25)
0
В заключение сформулируем важное физическое условие и вытекающее из него условие на коэффициенты уравнения (1). При прохождении через фильтр часть взвешенных частиц суспензии застревает в порах и образует осадок, поэтому концентрация взвешенных частиц в фильтре должна быть меньше концентрации на входе:
Finally, we consider an important physical condition and the ensuing condition on the coefficients of the equation (1). While passing through the porous media a portion of the suspended particles in the suspension gets stuck in the pores and forms a deposit, therefore the concentration of the suspended particles inside the filter should be less than the concentration at the inlet:
C(x, t) < C(0, t) = ^,0 < x < 1, t > 0. (26)
Из (23), (26) следует m < 0 . Получаем From (23) (26) we get m < 0 and obtain условие на производные функций the condition on the derivatives of the a(S), b(S), при S = 0 functions a(S), b(S) at S = 0:
p (abj - aj )< 1. (27)
Из вида главного члена асимпто- The main term of the asymptot-[23] системы (1)—(5) вблизи ics [23] of the system (1)—(5) near the
concentration front also suggests that the physical condition (26) implies (27).
The system of the equations describing the particle motion in a porous medium, in general, has no exact analytical solutions. The asymptotics found makes it possible to obtain an approximate solution for small values of the filtration coefficient and to get an important relationship for the coefficients of the system.
тики
фронта концентраций частиц также следует, что физическое условие (26) влечет (27).
Система уравнений, описывающая движение частиц в пористой среде, в общем случае не имеет точного аналитического решения. Построенная асимптотика дает возможность найти приближенное решение при малых значениях коэффициента фильтрации и получить важное соотношение для коэффициентов системы.
Библиографический список
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математики и вычислительной механики : сб. науч. тр. М. : МГСУ 2014. Т. 17. С. 295—304.
Поступила в редакцию в декабре 2014 г.
Об авторах: Кузьмина Людмила Ивановна — кандидат физико-математических наук, доцент, доцент департамента прикладной математики Московского института электроники и математики, Национальный исследовательский университет Высшая школа экономики (НИУ ВШЭ), 101000, г. Москва, ул. Мясницкая, д. 20, [email protected];
Осипов Юрий Викторович — кандидат физико-математических наук, доцент, профессор кафедры информатики и прикладной математики, Московский государственный строительный университет (ФГБОУ ВПО «МГСУ»), 129337, г. Москва, Ярославское шоссе, д. 26, уип[email protected].
Для цитирования: Кузьмина Л.И., Осипов Ю.В. Асимптотика задачи фильтрации суспензии в пористой среде // Вестник МГСУ 2015. № 1. С. 54—62.
23. You Z., Osipov Y., Bedrikovetsky P., Kuzmina L. Asymptotic Model for Deep Bed Filtration. Chemical Engineering Journal. 2014, vol. 258, pp. 374—385. DOI: http://dx.doi.org/10.1016/jxej.2014.07.05L
24. Kuzmina L.I., Osipov Yu.V. Particle Transportation at the Filter Inlet. International Journal for Computational Civil and Structural Engineering. 2014, vol. 10, iss. 3, pp. 17—22.
25. Kuzmina L.I., Osipov Yu.V. Matematicheskaya model' dvizheniya chas-tits v fil'tre [Mathematical Model of Particle Motion in the Filter]. Voprosy prikladnoy matematiki i vychislitel'noy mekhaniki : sbornik nauchnykh trudov [Problems of Applied Mathematics and Computational Mechanics]. Moscow, MGSU Publ., 2014, vol. 17, pp. 295—304. (in Russian)
Received in December 2014.
About the authors: Kuzmina Lud-mila Ivanovna — Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Applied Mathematics, Moscow Institute of Electronics and Mathematics, Higher School of Economics, 20 Myasnitskaya str., Moscow, 101000, Russian Federation; [email protected];
Osipov Yuri Viktorovich — Candidate of Physical and Mathematical Sciences, Professor, Department of Computer Science and Applied Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe Shosse, Moscow, 129337, Russian Federation; [email protected].
For citation: Kuzmina L.I., Osipov Yu.V. Asimptotika zadachi fil'tratsii suspenzii v poristoy srede [Asymptotics of the Filtration Problem for Suspension in Porous Media]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2015, no. 1, pp. 54—62. (in Russian)
