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Article Highlights Elastic outer boundary conditions are introduced to the seepage model of shale gas reservoir for the first time. Elastic outer boundary conditions include three special outer-boundary conditions (closed, constant pressure, infinite), and it is more general. The introduction of elastic outer-boundary conditions provides a new perspective and method to make a more complete theoretical chart.

This study uses similar construction method of solution (SCMS) to solve mathematical models of fluid spherical flow in a fractal reservoir which can avoid the complicated mathematical deduction. The models are presented in three kinds of outer boundary conditions (infinite, constant pressure, and closed). The influence of wellbore storage effect, skin factor, and variable flow rate production is also involved in the inner boundary conditions. The analytical solutions are constructed in the Laplace space and presented in a pattern with one continued fraction—the similar structure of solution. The pattern can bring convenience to well test analysis programming. The mathematical beauty of fractal is that the infinite complexity is formed with relatively simple equations. So the relation of reservoir parameters (wellbore storage effect, the skin factor, fractal dimension, and conductivity index), the formation pressure, and the wellbore pressure can be learnt easily. Type curves of the wellbore pressure and pressure derivative are plotted and analyzed in real domain using the Stehfest numerical invention algorithm. The SCMS and type curves can interpret intuitively transient pressure response of fractal spherical flow reservoir. The results obtained in this study have both theoretical and practical significance in evaluating fluid flow in such a fractal reservoir and embody the convenience of the SCMS.

Considering the characteristic of the outer boundary of the two-region composite reservoir has elasticity, firstly, a new two-region composite reservoir seepage model which considers the wellbore storage and effective well radius is established to describe the seepage laws of the two-region composite reservoir more accurately. Then, by combining the Laplace transform and the similar construction method, the solution of dimensionless bottom hole pressure is obtained in the Laplace space. Meanwhile, the solution is also inverted into real space by means of the Stehfest numerical inversion method. Finally, the characteristic curves of dimensionless bottom hole pressure and its derivative are drawn, and the influence of elastic coefficient and reservoir parameters on the characteristic curves is also analyzed. The results reveal that the characteristic curves have affine similarities, and the characteristic curves under the elastic outer boundary condition are always between the closed outer boundary condition and constant pressure outer boundary condition. It can be seen that the conventional three idealized outer boundary conditions (closed, constant pressure and infinite) of two-region composite reservoirs are actually three special cases of elastic outer boundary condition. The introduction of elastic outer boundary condition not only expands the range of data interpretation reasonably, but also shows the actual state of the outer boundary of the two-region composite reservoir better. The similar construction method can avoid complicated and trivial operation processes, and greatly accelerate the model solving speed. The findings of this study can help for better understanding of the real situation of the outer boundary of the two-region composite reservoir, which improves the matching degree between the theoretical model and the actual reservoir, and provides convenience for the development of related well test analysis software. [ABSTRACT FROM AUTHOR]