 in tehnologij, in tehnologija. Je vse, da vsem bilo razodaj, da se organizujevajte jezni konferenci za posljegu vsega začenja in da se vse vsega vsega vsega vsega vsega. Taj nečil. meje koleg Anton Voloshin za samih institutov. Vse vse je vse bolje, da moj tudi tudi opravljaj, je tudi, da je vse zrešenja vzrednjaval modem, vzrešenja vzrešenja vzrešenja vzrešenja vzrešenja vzrešenja vzrešenja vzrešenja vzrešenja vzrešenja vzrešenja vzrešenja vzrešenja. v tem vsevih vsevih vsevih kandorov modelu. Tudi, kaj mi največ, da so prijezati, tudi je tudi 3 vsevih vsevih in nekajm vsovih modela. Vse je največ nekaj največ in razvahšen na Bar and Blood, če je bolj tega preljavna pralj, in prijezati pralj, tako ne čel iz vsevih vsevih, na to, češnji parametri. Češnji parametri je model Hasanjizade, kaj je zelo vzvečila dinemikovosti, in nekaj češnji model je vzvečil kandaurav. V tem modelu, v tem modelu, Relativne permeabilitve in pripravljenje vse predstavljenje na zelo začetnjenje in tudi tako, nekaj nekaj parametrem, kaj pripravljenje vse zelo začetnjenje. Zelo začetnjenje je standa. Zelo začetnjenje je zelo začetnjenje, in zelo poslednjamo do homogenizacije. Zelo vzelo je klasko vzelo v poslednjih medijah. Vzelo se, da vzelo vzelo vzelo, vsega je vsega izgleda kondaurov nenekelibram parametrem. Taj parametrem izgleda vsega vsega konečnja. Taj parametrem izgleda vsega konečnja. Now we can present the equations of the flow, they are maybe rather known, except this new parameter, which appears in the equations. Now we start, we pass to the homogenization problem, which is the main subject of the talk. First of all, we consider a porous reservoir omega, with a complicated, rather complicated microstructure. So it is made of epsilon periodically distributed cubes, the matrix block, and the connected set of the fissures. Due to the second parameter delta, one can see that the measure of the fissures set is small, is asymptotically small, as two parameters go to zero. This justifies the geometry with, really, with sin fissures. So if delta is fixed, we have no this relation, but if delta goes to zero, we have that the fractures are very thin asymptotically. Now in the classical double porosity homogenization, it is assumed that the absolute permeability tensor is of order one in the fissures system, and it is like epsilon square in the matrix blocks. In our case we have two parameters, I repeat. In our case we have the following relation. So in the matrix block we have that this permeability tensor is of order epsilon square, epsilon delta square. Now we rewrite the system of equations of the kondauer of equations, like the equations with rapidly oscillating coefficients. No comments, it is a very complicated system. So now we are going to study the asymptotic behavior of the model that is the solution to this model as epsilon and delta go to zero. Now the first step we can show that the model admits homogenization, that is we can pass to the limit as epsilon goes to zero, and we assume in addition that the known equilibrium phenomena occur in the matrix block only, but not in the fissures part. So using the method of two scalar asymptotic expansions, we obtain the following homogenized, but delta model. Why delta model? Because of the coefficients depend on the parameter delta. The terms in the right side of the equations, the additional source terms are defined as follows. Where small as delta is the solution of the following local problem. This problem is formulated in terms of the phase pressures and the saturation. In the first proposition, which is mathematically proved that we can replace this local problem by the following boundary value problem for the non-equilibrium matrix in b-b-shin equation, which is really more simple sample than the previous one. Now we have to pass to the limit as delta goes to zero, and we make a purely heuristic step, we linearize this in b-b-shin equation. The idea belongs to Todar Bogast and was proposed in 90s, I think. We do not justify mathematically this step, but the numerical tests show that the solutions to these equations become more and more close as delta goes to zero. Because the solutions are of boundary layer type solutions. So the non-linear of equation plays no role if delta is rather small. So the linearized system reads, it has the following form, and we want to pass to the limit in the simplified model. Our a priori information is the asymptotical behavior of the exchange term, we know this. We also know the asymptotical behavior of the porosity function and the global permeability tensor. Then the final step, as the final step passing to the limit as delta goes to zero. So we obtain the following system of equations where the exchange terms are defined only in terms of the given data of the mesoscopic model. Now a few words about the references. As for the Kandorov's model, I refer here to Kandorov's paper in the journal of applied mathematics and also the papers by Konychov and Tarakanov. As for the homogenization results in double porosity media, the first non-equilibrium to phase flow in double porosity media, the first result is obtained by Salimian Bruining. Then this result was revisited in our paper, in our paper with Pamphilov and Damazian. As for the homogenization results concerning the homogenization of Kandorov's double porosity model, we have the following papers. Finally, the homogenized models were obtained in the following papers. The first one and the second one concern the equilibrium to phase flow in double porosity media. Thank you for your attention. Here we have the nonlinear equation like this. We replace this function f by its mean value and consider the linearized one. If we put delta rather small, we see that there is no difference between the solutions. Maybe it is rather clear, but this small term in front of the differential operator kills the solution except the boundary layer. Vsledno je tako vse. We are working on it. We have verified that this works. As for the papers by Arbogast, there is no small parameter in front of the differential operator. Arbogast justify only the qualitative behavior of the solution. We can linearize, but in this case we have a difference between the solutions. In this case, we linearize and there is no loss of accuracy. Real accuracy. Pardon? I think that really the presentation is purely a mathematical one. But it is possible. A lot of material to present, so you see it is only a scheme.