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摘要:
Forchheimer方程作为非达西渗流中广泛应用的基本方程之一，方程中$ A $、$ B $系数的确定一直是孔隙介质渗流领域中的热点及难点，不同学者根据渗流试验结果提出了不同的Forchheimer方程$ A $、$ B $系数的经验公式，但对于均质以及混合粒径的非均质条件下评价各经验公式适用性的研究较少。因此在渗流阻力试验的基础上，采用归一化目标函数和线性回归法评价了Forchheimer方程经验公式的适用性，为不同孔隙介质条件下Forchheimer方程经验公式的选取提供参考。结果表明：对于均质孔隙介质，Sidiropoulou公式对水力梯度有着很好的预测效果；对于2种混合粒径孔隙介质，在使用平均粒径的基础上，还应考虑混合粒径的质量比和大小因素，Macdonald公式的预测效果受混合粒径的质量比和大小影响较小，Kadlec and Knight公式对于水力梯度的预测结果较为稳定；对于5种混合粒径孔隙介质，使用d60作为特征粒径进行预测的效果较好，Kadlec and Knight公式对于系数A的预测效果较好，Ergun公式对于系数B的预测效果较好。研究结果能够为工程中均质及非均质松散砂砾石孔隙介质渗流计算的Forchheimer方程的选取提供依据。
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关键词:
- 孔隙介质 /
- Forchheimer方程 /
- 非达西流 /
- 量化评价 /
- 经验方程
Abstract:The Forchheimer equation is one of the basic equations widely used in non-Darcy seepage. The determination of coefficients A and B in the equation has always been a hotspot and difficulty in the field of porous media seepage. Different studies have proposed various empirical formulas for the coefficients A and B from seepage experiments. However, there are few studies evaluating the applicability of each empirical formula under homogeneous condition and heterogeneous with mixed particle size. In this study, to provide basic information for selecting the empirical formula of the Forchheimer equation under different porous media conditions, the normalized objective function (NOF) and linear regression method are used to evaluate the applicability of the empirical formula of the Forchheimer equation, on the basis of the seepage resistance experiment. The results show that: for homogeneous porous media, Sidiropoulou’s formula has a good prediction effect on hydraulic gradient. For the porous media mixed two kinds of particle size, the mass ratio and size factors of the mixed particle size should be considered based on the average particle size. The prediction effect of the Macdonald formula is slightly affected by the mass ratio and the mixed particle size; while the predicted hydraulic gradient from the Kadlecan and Knight formula is relatively stable. As to the porous media mixed five kinds of particle size, the predictive effect of using d60 as the characteristic particle size is fine.The Kadlecan and Knight formula is suitable to predict coefficient A, and the Ergun formula is effective to predict coefficient B. This study can provide a basis for selecting the Forchheimer equation for seepage of homogeneous and heterogeneous loose sand and gravel porous media in engineering.
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Key words:
- porous media /
- Forchheimer equation /
- non-Darcy flow /
- quantitative evaluation /
- empirical equation
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表 2 非均质孔隙介质渗流阻力试验特征值
Table 2. Experimental characteristic value of seepage resistance in the heterogeneous porous media
编号 d50/mm $ J=AV+B{V}^{2} $ 混合粒径/mm 质量比 A B R2 H2-1 1.31 85.74 1106.89 0.9999 1.075，1.850 7∶3 H2-2 1.46 76.98 203.3 0.9999 5∶5 H2-3 1.62 53.88 294.13 0.9999 3∶7 H2-4 1.70 113.27 10424.5 0.999 1.075，3.175 7∶3 H2-5 2.12 97.42 3803.17 0.9998 5∶5 H2-6 2.54 67.84 970.01 0.9995 3∶7 H2-7 2.25 17.18 510.19 0.9989 1.850，3.175 7∶3 H2-8 2.51 13.59 628.74 0.9995 5∶5 H2-9 2.77 12.23 398.62 0.9997 3∶7 H2-10 1.98 28.17 683.36 0.9993 1.475，3.175 7∶3 H2-11 2.32 24.02 728.96 0.9995 5∶5 H2-12 2.66 16.85 808.05 0.9996 3∶7 编号 d50/mm $ J=AV+B{V}^{2} $ d10/mm d60/mm dm/mm s dm-s/mm Cu A B R2 H3-1 1.075 53.68 2323.68 0.9999 0.8 1.15 1.37 0.48 0.9 1.4 H3-2 59.71 1520.56 0.9997 0.8 1.52 1.97 1 0.9 1.9 H3-3 78.21 1043.18 0.9999 0.8 1.85 1.82 0.92 0.96 2.3 H3-4 1.475 31.13 982.2 0.99949 1.12 1.6 2.03 0.78 1.25 1.4 H3-5 45.47 866.62 0.99986 0.75 1.55 2.04 0.91 1.01 1.9 H3-6 55.5 1057.16 0.99986 0.75 1.75 1.64 0.63 1.13 2.3 H3-7 1.85 19.77 776.06 0.9998 1.475 2.05 2.44 0.76 1.68 1.4 H3-8 25.05 866.72 0.9999 1.075 2.05 2.32 0.89 1.55 1.9 H3-9 37.54 853.48 0.9999 0.91 2.05 2.38 0.83 1.43 2.3 注：dm为矩算术法计算的平均粒径；s为标准偏差；dm-s为矩算术法平均粒径与标准偏差的差值的新特征粒径。 
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表 1 均质孔隙介质渗流阻力试验特征值
Table 1. Experimental characteristic value of seepage resistance in the homogeneous porous media
编号 d50/mm $ J=AV+B{V}^{2} $ A B R2 H1-1 1.075 90.411 1973.4 0.9999 H1-2 1.475 57.596 1122.6 0.9997 H1-3 1.850 56.67 397.92 0.9997 H1-4 2.500 39.264 689.46 0.9996 H1-5 3.170 33.163 724.69 0.9998
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表 3 不同质量比和不同混合粒径下各经验公式预测系数
$ A $ 、$ B $ 的归一化目标函数值Table 3. NOF values in the empirical formulas for predicting coefficients
$ \mathit{A} $ and$ \mathit{B} $ under different mass ratios and different mixed particle sizes混合方式 系数 经验公式预测系数A、B的归一化目标函数值 Ergun Irmay Macdonald Kovács Kadlec and Knight Sidiropoulou 不同质量比 7∶3 A 0.775 0.590 0.516 0.827 0.646 1.102 B 4.147 13.324 1.942 2.872 3.556 2.940 5∶5 A 1.138 0.878 0.752 1.207 0.613 1.348 B 1.462 5.451 0.684 0.966 1.224 1.044 3∶7 A 1.062 0.832 0.724 1.124 0.641 1.122 B 0.312 1.482 0.595 0.455 0.368 0.398 不同粒径/mm 1.075，3.175 A 1.325 0.935 0.725 1.422 0.361 1.774 B 4.556 14.923 2.025 3.101 3.883 3.227 1.475，3.175 A 0.308 0.427 0.492 0.279 0.809 0.209 B 0.352 1.048 0.653 0.520 0.427 0.498 1.850，3.175 A 0.479 0.568 0.617 0.457 0.852 0.437 B 0.488 0.579 0.730 0.625 0.551 0.592
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