A mathematical model for productivity calculation of shale-gas multilateral horizontal wells based on fractal theory

In this paper, the fractal dimension of length and branch hole amount of horizontal wells in shale reservoirs and the propagation laws of pressure drop disturbance of each branch were investigated by means of fractal geometry. Accordingly, the relationship between the pressure drop funnel boundary and the time was figured out. Then, a mathematical model for calculating the productivity of shale-gas multilateral horizontal wells was developed by analyzing the effects of desorption, adsorption, diffusion coefficient and startup pressure gradient of shale reservoirs. The productivity equation of unsteady seepage while the pressure at the external boundary of horizontal well is constant was solved by means of Laplace transformation method. Finally, the productivity characteristics of shalegas multilateral horizontal wells and their influential factors were calculated and analyzed based on the reservoir parameters of a certain marine shale gas reservoir in Sichuan. It is indicated that in the same fractal dimension, the production rate of gas wells increases in the process of shale gas production as the number of branch holes of horizontal well increases. The longer the branch hole is, the more the flow rate of main hole is affected. Due to the desorption of adsorbed shale gas, the propagation velocity of pressure drop funeral bound-ary decreases and the decline of formation pressure also slows down. The higher the gas production rate is, the faster the bottom hole pressure drops over the time, but the decline tendency slows down gradually as the time goes.