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NUMERICAL MODELING OF THE ONE MODEL OF FILTRATION IN ELASTIC POROUS ENVIRONMENT

The linear relaxational filtration is described by the conservation law of pulse of resistance force, by the linearized conservation law of fluid mass and determining relations for pulse of resistance forces and fluid mass. After exception of a pulse density of resistance forces (J) and (m ?) this system with respect to pressure (p)and velocity of filtration (W)is

(1)

, (2)

Here and are relaxation kernels of the filtration law and fluid mass. [1].

Let us consider a model of filtration under the elementary nonequilibrium law in the elastic porous environment ? є R3 In this case kernels of relaxation

, (3)

, (4)

We can write the system (1) – (2)

(5)

(6)

Here p (x,t) is pressure of fluid, x є ? є R3, t є [0,T], t is time, ? is time, is time of relaxation, ?0 is a fluid density in unperturbed layer conditions, W is filtration velocity, µ is a fluid viscosity, k is penetrability coefficients, ?w and ?p nonnegative constant relaxation times of filtration velocity and pressure, ? is coefficient of elasticity capacity of the layer, ?= ?c+m0 . ?f, ?c is compressibility coefficient of the porous environment, ?f is compressibility coefficient of the fluid,

is Heaviside function, x is piezoconductivity coefficient of the laye,

[1].

Setting of a problem. Let us consider equation (5). Let for t+0 the initial (Cauchy) conditions are realized, i.e.

(7)

and

(8)

Dirichlet condition is realized on the border д? of domain ?, i.e.

(9)

Now we shall set up a problem for (6). For t=0 the initial (Cauchy) condition is realized, i.e.

(10)

Dirichlet condition is realized on the border д?

(11)

Solution of the problem (5), (7) – (9). Let us assume that,

where and nonnegative constant relaxation times of pressure and filtration velocity . Let us denote by

. (12)

Then from (5) and (6) we obtain

(13)

(14)

We define initial boundary value problem for (12). On the strength of (7) and (8) for t=0 initial condition

(15)

is realized.

We shall calculate . Then we can to define the boundary condition for P(x,t) .

. (16)

Solution of the problem (13), (15) and (16) by Monte Carlo methods. Let us consider initial boundary value problem

(13)

(15)

(16)

where X is prescribed coefficient, is prescribed relaxation time, a(x), b(x), c (x,t), d(x,t) are prescribed functions.

We shall divide [0T] into N equal parts with step

We approximate the problem (13), (15), (16) by implicit scheme only with respect to time variable

(17)

(18)

(19)

Let us transform (17).

Lowering an index j from (17) – (19) we get Dirichlet problem for the Helmholtz equation on a time layer j+1 with respect to P(x)

(20)

(21)

where is known function by virtue of (18), is prescribed boundary condition.

Solution of the problem (20) – (21) is estimated by Monte Carlo methods [2].

Algorithm of Monte Carlo methods (“random walk on spheres” algorithm) for estimation of the solution of the problem (20), (21) in prescribed point x0:

1. Markov chain is simulated from the point x0 with respect to the first hit in ? - vicinity of ? - border д? of the domain ? . Point is defined, x* is the point of border nearest to the last state xL, L is number of a last state of Markov chain;

2. Respectively density in each

sphere S(x) (the surface of ball Ix-x1I <R) the value of function is calculated.

is Green function of operator ? - c1 for ball

? is unit isotropic vector. Weights Qn are calculated: Q0=1 .

3. We get a required estimation by averaging of value on all trajectories.

Note that for random variable the Theorem is correct.

Theorem. Variance of a random variable uniformly bounded with respect to ? , i.e.

Let P(x,t) is a solution of the problem (13), (15) and (16). We substitute this solution in (12). Then from (12), (7) and (9) we obtain initial boundary value problem

, (22)

(23)

(24)

Solution of the problem (22) – (24)

(25)

For definition of filtration velocity vector W(x,t) we shall calculate gradx P(x,t)?Px(x,t) and substitute in (14). Combining the initial boundary value conditions (10) and (11) we obtain the problem for definition of W(x,t)

(26)

(27)

(28)

The solution of the problem (26) – (28)

Initial boundary value problem (13), (15) and (16) is solved by classical numerical methods, Monte Carlo methods and probability-difference method. Then initial boundary value problems (22) – (24) and (26) – (28) are exactly solved. [3], [4], [5].

 

References

1. Molokovich U.M., Osipov P.P. Bases of the theory relaxational filtrations. Kazan, Publishing in Kazan University. 1987. 106 p. (in Russian).

2. Elepov B.S., Kronberg A.A., Michailov G.A., Sabelfeld K.K. Solution of boundary problem by Monte Carlo methods. Novosibirsk: Nauka. 1980. 74 p. (in Russian).

3. Shakenov K.K. Solution of problem for one model of relaxational filtration by probability–difference and Monte Carlo methods. Polish Academy of Sciences. Committee of Mining. Archives of Mining Sciences. Krakow. 2007. V. 52. No. 2. P. 247–255.

4. Shakenov K.K., Issabekova N.A. Solution of problems for model relaxational filtration proceeding under linear Darcy law by Monte Carlo and probability - difference methods. The Bulletin Kazakh National University. Series: mathematics, mechanics, computer science. 2007. No. 1 (52). P. 8195. (in Russian).

5. Shakenov K.K. Solution of one mixed problem for equation of relaxational filtration by Monte Carlo methods. Notes on Numerical Fluid Mechanics and Multidisciplinary Design. V. 93. Advances in High Performance Computing and Computational Sciences. Springer. Springer–Verlag Berlin Heidelberg. 2006. P. 205-210.