Download Numerical modelling of the one model of filtration in elastic porous

Numerical modelling of the filtration process in relaxation elastic porous environment by
Monte Carlo methods
Kanat Shakenov
Setting of a problem. Let filtration in relaxationaly–compressed porous environment is
realized by the linear Darcy law. [1]. Then a fluid current in conditions of this model is
characterized by relaxation kernels of the filtration law and fluid mass:

  p
 t 

F (t )   t  (t ), (t )   0     m
  c  exp      (t ),
(1)

m
 m  

 is a fluid viscosity,  is penetrability coefficient, t is time,  (t ) is Heavisid function,
 (t )  1 for t  0 ,  (t )  1/ 2 for t  0 ,  (t )  0 for t  0 ,  0 is a fluid density in the
unperturbed layer conditions,  is elasticity capacity coefficient of the layer, m is the
relaxation time of porosity under the constant overfull of pressure,  p is the relaxation time of
pressure under the constant porosity,  c is compressibility coefficient of the porous
environment. In this case F (0)  0, (0)  0  * , then perturbation extension velocity v0  




 v0   for   0,   0  ,
where *  m0   f   c  p
is
dynamic
 v0 

m


coefficient of elasticity capacity of the layer, m0 is a fluid porosity in the unperturbed layer
conditions,  f is compressibility coefficient of fluid,  

is piezoconductivity coefficient
 
of the layer. It means that perturbations front set instantly passes all the considered domain
instantly making it by filtration domain. In a filtration domain the pressure satisfies
2 
p( x, t )   
p( x, t ) 
(2)
  2  p( x, t )  m 
   p( x, t )  m 
,
x 
t  t 
t 
where m  m 
*
,    c  m0   f . In this case the filtration velocity vector is defined from


(3)
W ( x, t )    grad x p( x, t ).

The starting data for (2) are follows:
p( x, t )  0 for t  0,
(4)
p( x, t )
 0 for t  0.
(5)
t
Mathematical statement of problems. Problem 1. To find in the bounded domain  3
the solution of the equation (2) satisfying the starting data (4), (5) and the boundary condition
(6)
p( x, t )  p1 ( x, t ) for x [0, T ],
where  is a boundary of domain  . (Dirichlet problem).
Problem 2. To find in the bounded domain  3 solution of (2) satisfying the starting data
(4), (5) and the boundary condition
p( x, t )
 p2 ( x, t ) for x    [0, T ],
(7)
n
where n is interior normal to the boundary  . (Neumann problem).
Problem 3. To find in the bounded domain  3 solution of (2) satisfying the starting data
(4), (5) and the boundary condition
p( x, t )
 p3 ( x, t ) for x    [0, T ],
(8)
n
where 1 and 1 are given fixed values. (Mixed problem).
Solution of problems. Let coefficients  , m , m are while positive fixed values. Let us
1  p( x, t )  1 
n  0,1,..., N ,
divide interval t  [0, T ] into N equal parts with step  . So that tn  n  ,
T
  ,   0, and we discretization only with respect to t using implicit scheme. In result
N
taking into account m , we obtain the equation on time layer tn 1
p n1 ( x)  a  p n1 ( x)  f n ( x),
where f n ( x)  b  p n ( x)  c  p n1 ( x)  d  p n1 ( x), a 
b
4  m0   f  m  c   p 

      2   m     c  m   f  .
,
c
m
,
2   m
m0   f    2  m   c    2   p 

d
(9)
,
m0   f   2  m     c   2   p   

,
It is clear that a  0 , as parameters m0 ,  f ,  , m ,  c ,  p ,  are positive. Combining the
starting condition with (9) we obtain
p1 ( x)  p 0 ( x)
0
(10)
p ( x)  0, x  ,
 0, x  ,

which are the difference analogues of the starting data (4) and (5) respectively.
The Problems (9), (10), (6) – (8) (three problems) numerical modelling is implemented by means
of “random walk on spheres” and “random walk on lattices” of Monte Carlo methods algorithms
and probability difference methods. [2], [3], [4], [5], [6], [7].
References
[1]. Y.M. Molokovich, P.P. Osipov. Basics of relaxation filtration theory. Proceeding of Kazan
University, 1987, pages 106. (In Russian).
[2]. K. Shakenov. Solution of one problem of linear relaxational filtration by Monte Carlo
methods. International Conference on COMPUTATIONAL MATHEMATICS (ICCM 2002).
PART ONE. (The International Conference on Computational Mathematics. Proceedings: Part
I). Novosibirsk 2002. Page 276 – 280.
[3]. K. Shakenov . Dispersion of estimation of the solving linearized disturbed differed system
of Navier–Stokes equations. Calculating technologies . 2002. Vol 7, No. 3, Novosibirsk, 2002.
Pages 93 – 97. (In Russian).
[4]. K. Shakenov. Solution of the Neumann problem for Helmholtz equation by Monte Carlo
methods. International Conference on Computational Mathematics. Part 1, Novosibirsk,
2004. Pages 333 – 334. (In Russian).
[5]. K. Shakenov. Solution of one mixed problem for equation of relaxational filtration by
Monte Carlo methods. Notes on Numerical Fluid mechanics and Multidisciplinary Design
(NNFM). Advances in High Performance Computing and Computational Sciences. Volume 93.
Springer-Verlag Berlin Heidelberg 2006. P. 205 – 210.
[6]. K.K. Shakenov. Solution of Problem for One Model of Relaxational Filtration by
Probabilitly – Difference and Monte Carlo methods. Polish Academy of Sciences. Committee of
Mining. Journal “Archives of Mining Sciences”, Vol. 52. Issue 2. Krakow, 2007. P. 247 – 255.
[7]. Kanat Shakenov. Solution of Mixed Problem for Elliptic Equation by Monte Carlo and
Probability – Difference Methods. CP1076, 7th International Summer School and Conference. ©
2008 American Institute of Physics 978-0-7354-0607-0/08/$23.00. P. 213 – 218. (Journal AIP).