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
Here and are relaxation kernels of the filtration law and fluid mass. .
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
We can write the system (1) – (2)
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,
Setting of a problem. Let us consider equation (5). Let for t+0 the initial (Cauchy) conditions are realized, i.e.
Dirichlet condition is realized on the border д? of domain ?, i.e.
Now we shall set up a problem for (6). For t=0 the initial (Cauchy) condition is realized, i.e.
Dirichlet condition is realized on the border д?
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
Then from (5) and (6) we obtain
We define initial boundary value problem for (12). On the strength of (7) and (8) for t=0 initial condition
We shall calculate . Then we can to define the boundary condition for P(x,t) .
Solution of the problem (13), (15) and (16) by Monte Carlo methods. Let us consider initial boundary value problem
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
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)
where is known function by virtue of (18), is prescribed boundary condition.
Solution of the problem (20) – (21) is estimated by Monte Carlo methods .
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
Solution of the problem (22) – (24)
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)
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. , , .
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