The Mathematical Models And Their Solutions Of Fractal Reservoir

The classical permeation fluid mechanics are based on the Euclidean space, namely the integral dimension space, which the porous medium is approximatively considered as regular collection objects, and each porous medium model takes on so-called"characteristic length". In fact the structure of pore and fissure is quite complicated and the seepage mechanism is a complex dynamic system, and then the classical permeation fluid mechanics become helpless when confronting with stratum which have different scale, poor connectivity, and disorder distribution. The fractal theory serves as a powerful tool to depict roughness, mutability and uncontinuity. The fractal oil reservoir model based on the fractal theory has been extensively investigated and applied.The current investigations of the fractal oil reservoir model are scattered or they have merely considered a certain flow pattern or aimed at a certain well system, without comprehensive discussion of certain fractal oil reservoir model under different definite conditions nor systematical discussion the long-term asymptotic solution and the short-term approximate solution in prior and middle period in the course of well test analysis, let alone detailed mathematical argument about various formation parameter. In addition, for the great deal of complex oil reservoir during later exploitation in our nation, a great many fractal oil reservoirs with special formation parameter are not well considered.On whether considering the influence caused by wellbore storage and skin effect or not, comprehensively making use of the permeation fluid mechanics knowledge and the fractal geometric theory, the circle-source well fractal oil reservoir models,which with two inner boundaries (constant flow rate and constant pressure) and three outer boundaries (infinite, closed and constant pressure) are established in this paper. Moreover, the point-source well fractal oil reservoir models are also established with only constant flow rate inner boundary and three outer boundaries as mentioned above. By using the method of mathematical physics, relative theory of complex function, and properties of some special functions, the semi-analytical solutions or analytical solutions in real space are obtained. By means of asymptotic and approximate formulas of some special functions partly discuss the long-term asymptotic solution and the short-term approximate solution of well test analysis.Through the deduction of different definite problems of unsteady flow fractal oil reservoir models, we not only fully understand the fractal oil reservoir, but also develop the approach to certain extent in solving homothetic problems, which may be applied to other oil reservoir models accordingly.