Purpose

This paper aims to employ a multivariate nonlinear regression analysis to establish a predictive model for the final fracture area, while accounting for the impact of individual parameters.

Design/methodology/approach

This analysis is based on the numerical simulation data obtained, using the hybrid finite element–discrete element (FE–DE) method. The forecasting model was compared with the numerical results and the accuracy of the model was evaluated by the root mean square (RMS) and the RMS error, the mean absolute error and the mean absolute percentage error.

Findings

The multivariate nonlinear regression model can accurately predict the nonlinear relationships between injection rate, leakoff coefficient, elastic modulus, permeability, Poisson’s ratio, pore pressure and final fracture area. The regression equations obtained from the Newton iteration of the least squares method are strong in terms of the fit to the six sensitive parameters, and the model follow essentially the same trend with the numerical simulation data, with no systematic divergence detected. Least absolutely deviation has a significantly weaker performance than the least squares method. The percentage contribution of sensitive parameters to the final fracture area is available from the simulation results and forecast model. Injection rate, leakoff coefficient, permeability, elastic modulus, pore pressure and Poisson’s ratio contribute 43.4%, −19.4%, 24.8%, −19.2%, −21.3% and 10.1% to the final fracture area, respectively, as they increased gradually. In summary, (1) the fluid injection rate has the greatest influence on the final fracture area. (2)The multivariate nonlinear regression equation was optimally obtained after 59 iterations of the least squares-based Newton method and 27 derivative evaluations, with a decidability coefficient R2 = 0.711 representing the model reliability and the regression equations fit the four parameters of leakoff coefficient, permeability, elastic modulus and pore pressure very satisfactorily. The models follow essentially the identical trend with the numerical simulation data and there is no systematic divergence. The least absolute deviation has a significantly weaker fit than the least squares method. (3)The nonlinear forecasting model of physical parameters of hydraulic fracturing established in this paper can be applied as a standard for optimizing the fracturing strategy and predicting the fracturing efficiency in situ field and numerical simulation. Its effectiveness can be trained and optimized by experimental and simulation data, and taking into account more basic data and establishing regression equations, containing more fracturing parameters will be the further research interests.

Originality/value

The nonlinear forecasting model of physical parameters of hydraulic fracturing established in this paper can be applied as a standard for optimizing the fracturing strategy and predicting the fracturing efficiency in situ field and numerical simulation. Its effectiveness can be trained and optimized by experimental and simulation data, and taking into account more basic data and establishing regression equations, containing more fracturing parameters will be the further research interests.

中文翻译：

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基于混合FEM-DEM的水力压裂参数多元非线性回归分析

目的

本文旨在采用多元非线性回归分析来建立最终裂缝面积的预测模型，同时考虑各个参数的影响。

设计/方法论/途径

该分析基于使用混合有限元-离散元 (FE-DE) 方法获得的数值模拟数据。将预测模型与数值结果进行比较，并通过均方根（RMS）和RMS误差、平均绝对误差和平均绝对百分比误差来评价模型的准确性。

发现

多元非线性回归模型可以准确预测注入量、漏失系数、弹性模量、渗透率、泊松比、孔隙压力和最终裂缝面积之间的非线性关系。最小二乘法牛顿迭代得到的回归方程对6个敏感参数的拟合度较强，模型与数值模拟数据的趋势基本一致，未发现系统性分歧。最小绝对偏差的性能明显弱于最小二乘法。从模拟结果和预测模型中可以得到敏感参数对最终裂缝面积的贡献百分比。注入量、漏失系数、渗透率、弹性模量、孔隙压力和泊松比对最终裂缝面积的贡献率分别为43.4%、-19.4%、24.8%、-19.2%、-21.3%和10.1%，且逐渐增大。综上所述，（1）注液速度对最终裂缝面积影响最大。(2)经过59次基于最小二乘牛顿法的迭代和27次导数评估，最优得到多元非线性回归方程，可判定系数R2 = 0.711代表模型的可靠性，回归方程拟合了泄漏系数的四个参数，渗透率、弹性模量和孔隙压力非常令人满意。模型与数值模拟数据的趋势基本一致，不存在系统性差异。最小绝对偏差的拟合效果明显弱于最小二乘法。(3)本文建立的水力压裂物性参数非线性预测模型可作为现场及数值模拟优化压裂策略、预测压裂效率的标准。其有效性可以通过实验和模拟数据进行训练和优化，考虑更多的基础数据并建立回归方程，包含更多的压裂参数将是进一步的研究兴趣。

原创性/价值

本文建立的水力压裂物性参数非线性预测模型可作为现场及数值模拟优化压裂策略、预测压裂效率的标准。其有效性可以通过实验和模拟数据进行训练和优化，考虑更多的基础数据并建立回归方程，包含更多的压裂参数将是进一步的研究兴趣。