水力压裂模型的改进及其算法更新研究(下)

IMPROVED MODEL FOR HYDRAULIC FRACTURING AND ALGORITHM RENEWAL STUDY (2)

文章下篇为上篇的模型方程组提供一个全新的等效求解方法:根据方程组潜在特征,变换自变量(把自变量由位置参数x换为压力参数p),以显式化手段解决了模型方程组极严重的参数耦合问题。针对迭代运算中可能出现的发散,运用Stenffensen 法防散与加速迭代,使运算在数学理论上更严密。迭代运算中采用预报—校准思想,让前步运算的结果作为后步运算的初始迭代值,在滤失计算中,把二维模型中的裂缝延伸缝长—时间关系作为拟三维模型的裂缝延伸速度用,这样在满足工程需要的前提下,大大减少了计算上的复杂性。同时,在初始流量分布假设中,也以二维模型的流量分布代替线形流量分布假设,使迭代初始值尽量接近收敛结果。动态应力强度因子公式和预报-校准思想,使新模型的建立和求解向上挂靠真三维模型,向下联系二维模型,实现三类模型的横向渗透。

Here a new equivalent solution is provided for the equation group of the model set up in the last part. According to the potential characteristics of the equation group, changing the independent variable (from location parameters to pressure parameters), the serious parameters coupling problem is solved with the explicit method. Aiming to the possible divergence in iterative operation, the Stenffensen method is used to prevent divergence and speed up iteration, which makes the operation more rigors on the mathematical theory. The prediction-modification thought is applied in the iteration to take the former result as the later iterative initial value. Taking the fracture extending length-time formula derived from the 2-D model as the fracture extending velocity of the pseudo 3-D model in filtering calculation, it reduces the calculation complexity greatly on the promise of engineering precision. Also, substituting the assuming of linear flow distribution with the flow distribution of the 2-D model, it makes the iterative initial values closing to the final convergent results as much as possible. With the factor formula of dynamic stress intensity and the prediction-modification thought, the building and solution of the new model is related to the real 3-D model upward and the 2-D model downward. The three models are all connected and related transversely.