# General descriptions

The relationships between hydraulic gradient and seepage velocity for bimsoil samples with different RBPs are shown in Figure 5.34. As shown in Figure 5.34, samples with various heights were tested from 40 to 200mm. It can be seen that the seepage velocity increases with increasing hydraulic gradient, and the increment rate for bimsoil samples with a RBP of 60% is most evident. In addition, for the samples with various RBPs,

*Figure 5.33* Technical flowchart for study on the slenderness ratio flow tests for bimsoil samples.

the seepage velocity decreased with the increase of sample height. These results implied that the permeability coefficient of bimsoils is variational and not constant, and it depends on the hydraulic gradient. This result is consistent with the study of Wang et al. (Wang et al., 2016a): the permeability law of bimsoils does not comply with Darcy’s law. With the increase of sample height, the curves tended to be stable. This indicates that the seepage field in inhomogeneous bimsoil becomes steady gradually after a certain flow distance; the flow distance is an important factor influencing the flow characteristics. What is more, the critical sample height is different for samples with various RBPs.

# The non-Darcy’s flow of bimsoils

When water flows in bimsoils, with the increase of seepage velocity, the flow characteristic becomes non-Darcian, and the link between the seepage velocity and pressure gradient of Darcy’s law becomes nonlinear (i.e., the bimsoil permeability depends on the seepage velocity). This dependency is influenced by the randomly distributed rock

*Figure 5.34* The relationship between hydraulic gradient and seepage velocity for sampleswithheightfrom40to200 mm.(a-d)TheRBPis30%,40%,50%,and60%, respectively.

blocks in the bimsoil samples. To interpret this phenomenon well, an empirical equation was proposed by Forchheimer (1901) to correct for the nonlinearity of Darcy’s law. During the flow process in bimsoil, each stage of the stable value of the hydraulic gradient and seepage velocity is performed with a binomial fitting, and a modified version of Darcy’s law can be obtained as below (Wang et al. 2015a):

where *p* is the fluid mass density; C_{a} is the acceleration coefficient; / is the volume force of unit mass; *V* is the seepage velocity; *% = dp/dx* is the pressure gradient; and coefficient /? is termed the non-Darcy flow factor, m^{_l}, also known as the inertial coefficient, inertial resistance, or turbulence factor. Both /? and *к* are regarded as material constants of the Forchheimer’s equation in the range of its validity.

When the duration of the path of the fluid flow in bimsoil samples is long enough, the flow reaches stability, so *dVIdt =* 0 . Theoretical analysis shows that when ignoring the compressibility of the liquid, the pressure gradient presents a uniform distribution. As a result, we rewrite the expression of the pressure difference as follows:

where pbase ^{a}nd *Ihop* ^{are} the hydraulic pressure at the outlet and inlet of the bimsoil sample, respectively, and H is the height of the sample, which is the length of the flow path.

Neglecting the mass force, in case the sample is not very large, the expression for Eq. (5.3) can be written as follows:

Using the experimental data above, the polynomial fitting equations for the typical specimens with different specimen heights are listed in Tables 5.10-5.13. Figure 5.35 plots the curve fitting results of sample height from 40 to 200 mm, with an RBP of 30%, 40%, 50%, and 60%, respectively. The correlation coefficient of all equations is good with a correlation coefficient larger than 0.9. From Eq. (5.4), we can obtain the non- Darcy permeability coefficient and the non-Darcy flow /? factor accordingly.

*Table 5.10* The non-Darcy’s flow equations for typical specimens with a rock block percentage of 30%, using the Forchheimer equation

| ||||

Bimsoil 30-l(H = 40mm) |
25.4676 |
57.28886 |
0.03966 |
0.975 |

Bimsoil_30-1 |
27.3673 |
46.06427 |
0.03691 |
0.977 |

Bimsoil 30-l(H = 80mm) |
37.5564 |
36.6585 |
0.02689 |
0.990 |

Bimsoil_30-I(H = 100mm) |
37.7689 |
25.86399 |
0.02674 |
0.977 |

Bimsoil 30-l(H= 120mm) |
37.2458 |
25.04659 |
0.02712 |
0.980 |

Bimsoil 30-l(H = 140mm) |
38.612 |
26.41917 |
0.02616 |
0.981 |

Bimsoil_30-I(H = 160mm) |
38.827 |
27.59595 |
0.02601 |
0.979 |

Bimsoil_30-I(H = 180mm) |
37.0617 |
27.39719 |
0.02725 |
0.980 |

Bimsoil_30-I(H = 200mm) |
37.6457 |
26.13581 |
0.02683 |
0.976 |

*Table 5.11* The non-Darcy’s flow equations for typical specimens with a rock block percentage of 40%, using the Forchheimer equation

| ||||

Bimsoil_40-1 (H = 40 mm) |
33.21934 |
79.41739 |
0.0304 |
0.991 |

Bimsoil_40-1 |
44.2489 |
54.85734 |
0.02283 |
0.957 |

Bimsoil_40-1 |
55.9207 |
46.08953 |
0.01806 |
0.978 |

Bimsoil 40-l(H = 100mm) |
56.86151 |
38.04205 |
0.01776 |
0.986 |

Bimsoil_40-1 (H = 120mm) |
56.6892 |
29.67069 |
0.01782 |
0.976 |

Bimsoil 40-l(H = 140mm) |
58.2059 |
30.95381 |
0.01735 |
0.990 |

Bimsoil 40-l(H = 160mm) |
58.0889 |
30.68529 |
0.01739 |
0.989 |

Bimsoil 40-l(H = 180mm) |
56.72037 |
30.81588 |
0.01781 |
0.957 |

Bimsoil_40-I(H = 200mm) |
57.1004 |
28.84796 |
0.01769 |
0.977 |

*Table 5.12* The non-Darcy’s flow equations for typical specimens with a rock block percentage of 50%, using the Forchheimer equation

| |||||

Bimsoil 50-1 (H = |
40 mm) |
17.45548 |
48.85914 |
0.05786 |
0.996 |

Bimsoil 50-2(H = |
60 mm) |
21.70822 |
36.53996 |
0.04653 |
0.992 |

Bimsoil_50-3(H = |
80 mm) |
27.27899 |
25.15432 |
0.03702 |
0.991 |

Bimsoil 50-4(H = |
100 mm) |
33.96639 |
27.1969 |
0.02974 |
0.991 |

Bimsoil_50-5(H = |
120 mm) |
34.12356 |
26.77677 |
0.0296 |
0.992 |

Bimsoil 50-6 |
140 mm) |
34.85317 |
26.23295 |
0.02898 |
0.990 |

Bimsoil_50-7(H = |
160 mm) |
33.23728 |
27.15362 |
0.03039 |
0.992 |

Bimsoil_50-8(H = |
180 mm) |
34.50031 |
26.9832 |
0.02928 |
0.992 |

Bimsoil_50-9(H = |
200 mm) |
33.54192 |
26.56346 |
0.03011 |
0.993 |

*Table 5.13* The non-Darcy’s flow equations for typical specimens with a rock block percentage of 60%, using the Forchheimer equation

| |||||

Bimsoil_60-1 (H = |
40 mm) |
12.5917 |
30.10298 |
0.08021 |
0.997 |

Bimsoil_60-2(H = |
60 mm) |
16.2290 |
22.59943 |
0.06223 |
0.993 |

Bimsoil_60-3(H = |
80 mm) |
18.3969 |
17.88247 |
0.0549 |
0.988 |

Bimsoil_60-4(H = |
100 mm) |
19.2867 |
17.64004 |
0.05237 |
0.990 |

Bimsoil_60-5(H = |
120 mm) |
21.6305 |
17.5256 |
0.04669 |
0.987 |

Bimsoil_60-6(H = |
140 mm) |
20.9707 |
17.21499 |
0.04816 |
0.989 |

Bimsoil_60-7(H = |
160 mm) |
21.9001 |
16.46893 |
0.04612 |
0.988 |