Parameters Quantification of Forchheimer Equation and Critical Point of Transition from Darcian to Non-Darcian Flow
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摘要: 为探究岩体裂隙中水流的运动规律，基于真实岩体材料建立裂隙渗流模型，对裂隙中的渗流状态及渗流参数进行了研究. 区别于水泥、玻璃、亚克力、钢材等常见非石材类材质，选用天然大理石岩块为基材构建单裂隙渗流模型，开展不同隙宽（0.77、1.18、1.97、2.73 mm）的渗流试验，考察压力损失与流量的关系，探讨达西—非达西流转变的临界点及Forchheimer方程的参数量化问题. 研究结果表明：隙宽为0.77 mm时压力梯度与流量基本呈线性达西关系，随着隙宽和流量的增大，二者呈现出明显的非达西特征，可用Forchheimer方程描述；Forchheimer方程的粘滞项和惯性系数均可表达为隙宽的幂函数，引入雷诺数对惯性项系数进行修正可以减少误差；提出以压力梯度-流量曲线的斜率变化特征来判断达西—非达西流临界点的方法，并在本试验中得到了验证.
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关键词:
- 裂隙 /
- 非达西 /
- Forchheimer方程 /
- 雷诺数 /
- 临界点
Abstract: In order to explore the movement law of water flow in rock mass fissures, a fracture seepage flow model was established with real rock mass materials to study the seepage flow state and parameters. The single-fracture seepage flow model was constructed using natural marble blocks, instead of the common non-stone materials such as cement, glass, acrylic, and steel. Based on the model, seepage experiments with different fissure widths (0.77, 1.18, 1.97, 2.73 mm) were conducted to investigate the relationship between pressure loss and flow rate, the critical point of transition from Darcian to non-Darcian flow, and the quantification of parameters in the Forchheimer equation. Results show that the relationship between pressure gradient and flow rate is governed by linear Darcy’s law when the fracture width is 0.77 mm, but obvious non-Darcy characteristics are observed with the increase of the fracture width and flow rate, which can be described by Forchheimer equation. The coefficient of viscosity and inertia term of the Forchheimer equation can be expressed as a power function of fracture width, and the error can be reduced by introducing Reynolds number to correct the inertia coefficient. Besides, the method of judging the critical point of transition from Darcian to non-Darcian flow via the slope characteristics of pressure gradient-flow curve proved feasible in this experiment.-
Key words:
- fracture /
- non-Darcy /
- Forchheimer equation /
- Reynolds number /
- critical point
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图 3 实验值与计算值（
$- \nabla {P_{1i}} $ ）比较Figure 3. Comparison between experimental value and calculated value (
$ - \nabla {P_{1i}} $) 
图 4
$ - \nabla {P_{1i}} $ 相对误差分布Figure 4. Relative error distribution of model
$ - \nabla {P_{1i}} $ 
图 5 试验值与计算值（
$ - \nabla {P_{2i}}$ ）比较Figure 5. Comparison between experimental value and calculated value (
$- \nabla {P_{2i}} $ )
图 6
$- \nabla {P_{2i}} $ 相对误差分布Figure 6. Relative error distribution of model
$- \nabla {P_{2i}} $ 表 1 压力梯度与流量的拟合关系式
Table 1. Fitted formulas between pressure gradient and flow rate
类别 隙宽/mm 拟合方程 R2 线性方程 0.77 $ { {\rm{ - } }\nabla P_1 = 22.539\;6Q}$ 0.997 3 1.18 $ { {\rm{ - } }\nabla P_2 = 8.793\;7Q}$ 0.879 3 1.97 $ { {\rm{ - } }\nabla P_3 = 2.918\;2Q}$ 0.864 9 2.73 $ { {\rm{ - } }\nabla P_4 = 1.079\;3Q}$ 0.887 1 Forchheimer方程 0.77 $ { {\rm{ - } }\nabla P_1 = 20.722\;7Q + 0.090\;8{Q^2} }$ 0.999 1 1.18 $ { {\rm{ - } }\nabla P_2 = 1.739\;6Q + 0.176\;1{Q^2} }$ 0.995 6 1.97 $ { {\rm{ - } }\nabla P_3 = 0.477\;7Q + 0.032\;9{Q^2} }$ 0.999 6 2.73 $ { {\rm{ - } }\nabla P_4 = 0.310\;9Q + 0.009\;8{Q^2} }$ 0.997 3 
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表 2
$ - \nabla {P_{{\rm{1}}i}}$ 模型参数计算结果Table 2. Calculated parameters of model
$ - \nabla {P_{{\rm{1}}i}}$ c1 n1 c2 n2 6.848 4 −3.091 1 0.177 1 −2.718 7
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表 3
$ - \nabla {P_{{\rm{2}}i}}$ 模型参数计算结果Table 3. Calculated parameters of model
$ - \nabla {P_{{{\rm{2}}i}}}$ c3 n3 c4 n4 9.112 2 −2.819 1 0.000 12 −3.452 6
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表 4 不同隙宽的临界流量及雷诺数
Table 4. Values of critical flux and critical Reynolds number for different fracture widths
隙宽/mm ${E_{P{_ {i} } } }$/% 临界流量/（ml•s−1） 临界雷诺数 0.77 18 23.07 614.55 1.18 20 28.44 741.30 1.97 28 42.11 1 054.30 2.73 33 54.09 1 304.72
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