Rock fractures always influence the hydrological properties of a rock mass. To investigate the seepage characteristics of a rock mass with partly filled fractures, a mathematical model is established. In this model, the clear fluid in fractures is governed by the Navier-Stokes equation, and the fluid both in the porous medium and rock matrix are subjected to the Brinkman-Extended Darcy equation. The analytic solution of an equivalent permeability coefficient for a rock mass with partly filled fractures is solved, and it could be reduced to some special known results. Comparisons with experimental data show good agreement, thus verifying the validity of the present computations.
Incipient motion of non-cohesive sediment, as a branch of water erosion, has been studied for decades, and criteria proposed in predicting incipient motion mainly focus on dimensionless shear stress and shear velocity. The two parameters could provide runoff erosive power, however, are invisible and not easy-measured in experiment. In this paper, critical runoff depth was recognized as another parameter, which is visible, measurable, and closely connected with shear velocity. To get it, a mathematical model coupling surface flow and subsurface flow was established, and they are governed by Navier-Stokes equation and Brinkman-extended Darcy equation, respectively. Velocity derived from the model could be indirectly applied in particle force analysis and finally a new criterion about critical runoff depth was obtained. Accuracy of the criterion was verified by comparison of experimental and theoretical values, e.g. theoretical critical runoff depth ranges from 2.71 $ 1.14 mm with a slope angle range of 20 $ 28 and mean particle diameter of 2.5 mm while the corresponding experiment values ranges from 3 $ 1 mm and the absolute errors are less than 0.3 mm. Compared with Yang's formula, this criterion could also be applied in predicting incipient motion of non-cohesive sediment for its significant advantages of high accuracy and concise formulation.
The critical shear stress is a vital reference indicator for soil erosion. Soil erosion will occur when soil slope suffers from a exceed shear stress, and then causing soil loss and destruction of soil structure. In this work, an equation was proposed based on the force equilibrium of a single particle to estimate the critical shear stress for incipient particle motion of a cohesive soil slope. This formula is characterized by its physical significance, and the critical shear stress for incipient slope soil motion can be easily calculated when the soil properties and the slope angle are known. Moreover, the seepage-runoff coupled model and the excess shear stress equation are introduced in this paper. Two parameters, namely the weight of rushed soil particles and the discharge of water, must be measured in the scouring tests. Through linear regression, the tested τc-values were obtained to validate the τc-values calculated by the formula derived from the critical shear stress. In addition, two other formulas were compared with the derived formulas, which considered more parameters with physical significance. Finally, the influence of all parameters on the critical shear stress was analyzed: the porosity of the soil, the specific gravity of the soil and the slope gradient had less influence on the critical shear stress; the critical shear stress was negatively influenced by the particle diameter and positively influenced by the internal friction angle of the soil.
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