基于分形理论的页岩气分支水平井产能数学模型

利用分形几何法，研究了页岩储集层内水平井分支数目和长度的分形维度，以及水平井各分支井段压降扰动的传播规律，得到压降漏斗边界随时间变化的关系，考虑了页岩储集层解吸吸附、扩散系数和启动压力梯度的影响，建立了页岩气分支水平井产能数学模型。采用拉普拉斯变换法，求解了水平井外边界定压条件下的不稳定渗流的产能方程。结合四川某海相页岩气藏储集层参数，计算分析了页岩气分支水平井产能特征及影响因素。研究表明:在页岩气生产过程中，同一分形维度下，水平井分支数目越大，气井的产量增大越快；分支井的井段长度越大，对主干井的流量影响越大。吸附页岩气的解吸使得压降漏斗的边界传播速度减慢，地层压力下降缓慢；产气量越大，井底压力随时间下降越快，但是下降的趋势随时间逐渐减缓。

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.