# A Method to Estimate Dynamic Pore Water Pressure Growth of Saturated Sand-Gravel Materials
## 5 Estimation of the Dynamic Pore Water Pressure of the Saturated Sand-Gravel Material  
The sand-gravel material dynamic pore pressure growth conforms to the hyperbolic form seen in Figure 5. The modification of the hyperbolic pore pressure model in this paper can be given by the following Equation (5):  
\[\frac{u_{\text{d}}}{\sigma_{\text{sc}}}=a_{u}\left(\frac{\frac{N}{N_{\text{L}}} }{1+\frac{N}{N_{\text{L}}}}\right)^{b_{u}} \tag{5}\]  
where \(a_{u}\) and \(b_{u}\) are test constants related to the soil properties, \(u_{\text{d}}\) refers to excess pore water pressure, \(\sigma_{\text{sc}}\) is the initial effective confining pressure, \(N\) is the number of dynamic cycles, and \(N_{\text{L}}\) is the number of cycles required to reach liquefaction. In practical engineering, when the axial strain reaches a certain value, the soil can be judged to be liquefied. Therefore, \(a_{u}\) and \(b_{u}\) are identified as independent parameters in this paper.  
The changes in the dynamic pore water pressure and the corresponding fitting curve of Equation (5) for the eight groups of Xinjiang sand-gravel tests are shown in Figure 12. The fitting curves of the [PERSON] [6] pore water pressure model and the Xu [30] pore water pressure model are also shown in Figure 12 for comparison. Under low initial confining pressure conditions, the pore water pressure in the sample increases rapidly during the initial stages of dynamic loading. At lower initial confining pressures, the pore water pressure changes to conform more readily to Equation (5). The initial confining pressure inhibits the accumulation of pore water pressure to a degree. The fitting curves of the [PERSON] (1971) [6] and the [PERSON] (1983) [30] models both overestimate the growth rate of the pore water pressure during the early stages of dynamic loading but underestimate the dynamic pore pressure in the middle and late stages. Therefore, the [PERSON] and [PERSON] (1971) [6] and the [PERSON] (1983) [30] pore water pressure models may provide a dangerous pore water pressure prediction result. In contrast, the pore water pressure model proposed in this paper fits most of the test results well.  
The results of the sand-gravel and the sand dynamic tests conducted by several researchers [26; 27; 42; 49] and the fitting curves obtained by using the pore water pressure model proposed in this paper are shown in Figure 13. The pore pressure growth model in Figure 13 covers the three forms of pore pressure growth defined by [PERSON] [10]. The goodness of fit \(R^{2}\) of the pore pressure model proposed in this paper for different test pore pressure ratio curves is greater than 0.95. This means that the pore pressure model proposed in this paper is not only suitable for simulating the dynamic pore pressure growth  
Figure 12: Pore pressure ratio fitting curve of Xinjiang sand-gravel material. (**a**) Confining pressure is 50 or 100 kPa; (**b**) confining pressure is 200 or 300 kPa.  
in sand-gravel materials but is also suitable for sandy materials and that it can simulate three different pore pressure growth modes.