水力压裂拟三维模型数值求解新方法

Algorithm Renew Study of Hydraulic Fracture Pseudo-three Dimensional Model

水力压裂裂缝垂向不对称延伸的拟三维模型比较复杂,需要多重迭代,循环耦合,仅依靠增加迭代运算的层数,不仅运算量过大,过程迭代甚至可能出现发散。对此,将控制缝高的方程组变换为超越方程,引入Steffensen加速收敛求解缝高,大大提高了收敛速度,避免了常规算法中缝高求解的反复试算。此外,裂缝内压力在靠近尖端时急剧下降,整体加密缝长方向上的计算步长程序复杂度大大提高。采用变步长的龙格库塔方法求解压降控制方程,使得计算步长随着压裂液流动的摩阻大小自动调整,不仅降低了程序复杂度,而且步长疏密度还可以直观展示压裂液流动的沿程摩阻变化趋势。运算结果证明,新的计算思路降低了压裂裂缝拟三维模型数值求解的复杂度,有一定的理论研究价值。

Modeling of asymmetric vertical growth in elongated hydraulic fracture is complex, with the seriousparameters coupling problem in solution. Depending on increasing the number of iteration level not only leads to ex-cessive operations but also results in divergence, the fracture equation group is simplified into transcendental equa-tion. The use of method of Steffensen accelerating convergence is to get the fracture height, improving the rate ofconvergence and avoiding the trial-and-error of convention algorithm. Besides, the pressure in the fracture decrea-ses sharply next to tip. Diminishing every step will result in more complexity. The resolution of differential pressureequation draws support from variable-step Runge-Kutta. The size of step adjusts to fluid resistance, which decreasesthe complexity of program. Grid laxity in the direction of fracture length displays the tendency of the change of frac-ture fluid resistance. It turns out that the new algorithm decreases the complexity of the numerical solution of P3Dand has some theoretical value.