Let´s consider the using a method of fictitious area for the system of not evolutional type, which will be a model for us in filtering problem of two-phase incompressible fluid taking with capillary forces.

Let *D* - a certain plain simply connected area with enough smooth interface. In the cylinder *D _{T}* = {

*D*[0 <

_{x}*t*≤

*T*]}, with side surface

*S*= {γX[0 <

*t*≤

*T*]}, there is being searched the solution of mixed Cauchy problem:

(1)

In (1) *k* = *k*(*x*) > 0, *f _{i}* =

*f*(

_{i}*x*,

*t*), λ

*= const > 0, and necessary for justification of the method of fictitious areas additional requirements for input data of the problem (1) will be specified in the process of explanation. First of all we mention that if*

_{i}
2*R* = *u*_{1} - *u*_{2},

*P* = λ_{1}*u*_{1} - λ_{2}*u*_{2} (2)

The initial problem (1) is decomposed into two independent problems:

div(*k*grand *P*) + *f*_{1} + *f*_{2} = 0, *P*_{|s} = 0 (3)

and

2*R*(*x, *0) = ψ(*x*), *R*_{|s} = 0 (4)

For all that the time t comes in the problem (3) as a parameter Therefore, justification of the method of fictitious areas at the differential level might be carried out for each of the problems (3), (4). Also, it is necessary to note that if in the initial problem (1) instead of uniform boundary condition of the first type there will be examined the uniform boundary condition of the second type, so instead of *P*_{|s} = 0 и *R*_{|s} = 0 in (3) and (4) accordingly we will have

(5)

For the problem (3), (4), if , , true estimates are:

(6)

In accordance with the method of fictitious areas, let´s add the initial area D with a area *D*_{1 }up to the area *D*_{0} = *D u **D*_{1}, with boundary Γ = ∂*D*_{1} *u* γ, *S*^{0} = {ΓX[0 < *t* ≤ *T*]}. In the compound area *D*_{0} let´s study additional problems:

(7)

(8)

On the break curve S we lay down fitting conditions

(9)

ε > 0 - series expansion parameter, *Q* - parameter, possessing the value: *Q* = 1 or *Q* = -1,

(10)

For the problem (3), (4) α < 0, and for the same problems, but with boundary conditions (5) α > 0. Through the normal derivative has been represented. Further , and the signs of minus and plus mean that counterpart is a limiting value by the tending of *x* to γ inside or outside of *D*. The auxiliary problems (6)-(8) have the transparent physical sense. The absolute permeability is small (α > 0) or big (α < 0) depending on the type of boundary condition of the initial problem in fictitious area. As regards the input data in the fictitious area *D*_{1}, *R* is an analog of capillary pressure and that´s why equality *R* to zero means that in the fictitious area there is not only displacing phase. The fitting condition in (6), (7) means that for the transfer through γ (γ - line of factors´ break) phase pressures and phase rates are continuous.

For the solution of the problem (8)-(10), (7), (9), (10) the true estimates are:

(11)

(12)

Where и , и correspond to the solution of the problems (8)-(10), (7), (9), (10) by *Q* = 1 and *Q* = -1. Further the problem (8)-(10) is called the problem I, problem (7), (9), (10) - II.

Now the solution of the problem I we will search in the form of power series on the parameter ε, α = -1.

Let в *D _{T}*, в .

Where we put formally in the problem I, so then we will get the system relatively to *V _{m}* and

*W*:

_{m}(13)

By *m* ≤ 1

Let´s suppose that functions in (13) are met the conditions of , , *k* = 1, 2, ..., so the following theorem is right.

Theorem 1. Let *f* € *L*_{2}(*D _{T}*), , so then ε

_{0}will be found this 0 < ε < ε

_{0}, that series

*В*

_{1}and

*В*

_{2}are absolutely converging in and and so correspondently the equalities are true:

(14)

Where *R*_{ε} - solution of the problem I.

Theorem proving. From the theory of uniform boundary problems and conditions of matching we have

(15)

where constants *C*_{1}, *C*_{2} depend on areas *D*, *D*_{1} and factors of initial problem and don´t depend on ε.

Applying the theory of trails in Sobolev spaces

Then from (6) and (15) it follows

(16)

where *C*_{5} = *C*_{1}·*C*_{2}·*C*_{3}·*C*_{4}.

Let , thenseriesВ_{1}is absolutely converging in . For getting equalities (14) we multiply (13) by ε* ^{m}* and sum on

*m*, we have:

(17)

L - operator in the left part (13)

Si it follows that *R*_{ε} = *B*_{1} in *D _{T}*,

*R*

_{ε}=

*B*

_{2}in by 0 < ε < ε

_{0}.

From the theorem it follows unique existence of the solution of the additional problems (13), and the estimates are:

..(18)

Here it is by *Q* = 1, by *Q* = -1, from absolute convergence of the series *B*_{1} and *B*_{2} it follows (11), let´s bring it.

Theorem 2. If 0 < ε < ε_{0}, so then the estimates are true

(19)

Where *R* - solution (4), , - solution (8) by *Q* = 1 and *Q* = -1, correspondently.

Theorem proving. From the theorem 1it follows the following expansion:

(20)

Here it is , - solution (13) by *Q* = 1.

Applying the theorem 1 for it is true:

(21)

Here it is , - solution (13) by *Q* = -1, it is easy to see that - solutions (4).

Let , so the function satisfies the following problem:

So from this we have

or

Let´s suppose that

,

so then the function is the solution of the following problem:

From this we have

, then

Even it supposes that

, ,

we will get

, .

Continuing this process by *m* ≥ 2 we have:

, if m - even , if *m* - uneven (22)

The from (21), using (19), (20), we will get in *D _{T}*

(23)

Using the expansion (22) and estimates (17) we have come to (11):

Estimate (12) is got in much the same way.

Accuracy of received bilateral approximation in this case is limited by the estimate (11), (12). If only to get bilateral estimates *R*, *P*, with specified accuracy ε* ^{P}*, we will use the idea of Richardson extrapolation.

Let´s make extrapolated solutions , being a linear combination , with some weight:

(24)

Concrete view off coefficients β* _{m}* depends on choice of sequence ε > ε

_{1}> ... > ε

*> 0 and accuracy figure*

_{p}*p*. The more spread choice is:

(25)

By which coefficients β* _{m}* are in the explicit form

(26)

And it satisfies the conditions

By this way of the task ε* _{m}*, β

*we find that*

_{m}

where .

In much the same way it is

Let *p* - uneven, then and, it means,

(27)

With the help of this statement the theorem has been proved.

Theorem 3. Let *f* € *L*_{2}(*D _{T}*),

*R*- solution (4), - solution (8) corresponds the choice

*Q*= 1 и

*Q*= -1. So then for all

(

*x*,

*t*) €

*D*and 0 < ε < ε

_{T}_{0}it has a place the asymptotic point wise bilateral inequality:

(28)

Where *p* - uneven, and are defined on the formula (26).

The same estimates are received for the function* Р*, and power (2) for the functions *u*_{1}, *u*_{2}.

The received results allow simulating the processes of oil extraction with the use of production and forcing wells for water blockage of formation under test.