Journal of Engineering Geology

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Introduction
Local site conditions considerably influence all characteristics of the ground strong motion including the domain, frequency content, and duration. The level of such an effect could be considered as a function of geometry, properties of the materials embedded in the underlying layers, the site topography, and properties of excitement. Site effect fall into two categories: a) the effect of the surface soft layers triggered by the shear velocity differences between the soil layers and b) the surface and subsurface topography effects that lead to the wave reflection and refraction based on the site geometry, and subsequently enhance the level of amplification.
Since most cities have been constructed in the vicinity of or on sedimentary basins, geotechnical earthquake engineering devotes particular attention to effects of the sedimentary basins. Basin edge curvature deposited with soft soils are capable to trap the body waves and generated surface waves within the deposit layers. Such waves could create stronger and lengthier vibrations than those estimated in a 1D analysis that assumes the shear waves to be vertically propagated.
Although critically important, the 2D effect of the site has not been included in seismic codes and standards of the world. This might be due to the fact that the site effect depends on a number of parameters such as the site geometry, the type of wave excitement, properties of the materials, etc. that in return make it almost out of the question to make predictions about the effect. This study was an effort to compare the responses of four sedimentary basins with hypothetical geometries of rectangular, trapezoidal, elliptical, and triangular shapes in order to examine the effect of the geometrical shape of the basin on its responses and the extent of the response sensitivity to the excitation frequency of the wave. The study assumed the edge to depth proportion to be both constant and equal in all four basins so that the effect of the geometrical shape could be equally examined and compared in all four basins.      
Material and methods
In order to validate the results of the sedimentary basin modeling, firstly, ABAQUS finite element software was used to create a free field motion of a semi-circular alluvium valley in accordance with Kamalian et al. (2006) and Moassesian and Darvinsky (1987).  Then, the results from the model were compared with those from the above mentioned studies. The following descriptions are to present the model in details.
To evaluate the geometrical effect of the sedimentary basin on its response, the authors relied on the software to examine four sedimentary basins with the fundamental frequency (2.04 Hz). The basins enjoyed rectangular, trapezoidal, elliptical, and triangular geometrical shapes with a constant edge to depth proportion (49m to 300m respectively). The implicit method was also applied to perform the dynamic analysis. The materials were all viscoelastic and homogeneous. The soil behavior/treatment model was considered to be of a linear nature.  The Rayleigh damping model was used to specify the damping level. The soil element was a plane strain and SV waves (the Ricker wavelet) were used for seismic loadings in a vertical dispersion. The side boundaries (right and left) of the model were of a combinational type (viscous and free field boundaries); the down side boundary was composed of viscous. To achieve higher levels of wave absorptions, heavy columns were used as the free filed columns.
Next, it was the time to conduct the 1D analysis of the site. Three waves were in use in order to examine the effect of the frequency content of the excitation load on the basin response: 1) a wave with the dominant frequency of 1Hz that was out of the frequency range of all basins (2.04 Hz), a second wave with the dominant frequency of 2Hz that was close to the fundamental frequency of all basins, and a third wave with the dominant frequency of 4Hz. The waves were applied to a 2Dmodel. The results were compared with those obtained from a 1Dmodel in terms of the timing.
Then, the basin responses to all three waves (1, 2, and 4 Hz) were subjected to an individual analysis in order to examine the sensitivity of each basin response to its geometrical shape. Results indicated that while the responses of the rectangular and trapezoidal basins were significantly more sensitive to the excitation frequencies, the elliptical and triangular basins showed more stable behaviors to such frequencies. The final stage of the study was dedicated to examine the site 2D effect during the ground motion.
Results and Conclusions
According to the results of the present study, it could be suggested that the geometrical shape of the sedimentary basin has a significant effect on the responses of the field of seismic waves and that it could result in so different responses from the ones attained after a 1D analysis of the site. In addition, the pattern of the seismic waves’ responses is highly dependent on the geometrical shape and the frequency content of the seismic load. Also, the location where the maximum horizontal acceleration occurs along with the sedimentary basin depends on the excitation wave and varies accordingly. Further, it could be suggested that the site 2D effect results in both considerable amplification and an increase in the length of ground motion.
The results of the 2D analysis showed remarkable differences with their 1D counterparts: a 1.45 larger response for the rectangular basin, a 1.28 larger response for the trapezoidal basin, a 1.22 larger response for the elliptical basin, and a 1.19 larger response for the triangular basin.
With the frequency of 1 Hz where the excitation frequency is out of the basin range (i.e. the excitation frequency is below the lowest frequency of basin), the sedimentary basin did not show any signs of amplification and chaos (unlike two other frequencies); instead, it was a cause for de-amplification.
The frequency of 2 Hz that is subject to resonance resulted in amplifications (absent in 1D analysis) and there are traces of a reduction in the acceleration responses near to the edges of the basins. The proportion of the amplification (in the center of the basins) in 2D to 1D analysis was 1.4 for the rectangular basin, 1.28 for the trapezoidal basin, 1.22 for the elliptical basin, and 1.15 for the triangular basin.
 

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Introduction
Hydraulic fracturing is used in the oil industry in order to increase the index of production and processing in wells whose efficiency has been dropped due to long-term harvest or the rocks around the well are low permeable. Since the hydraulic fracturing operation is costly, it is of special importance to determine the pressure required for hydraulic fracturing and the suitable pump for this operation to the project managers.
The hydraulic fracturing technique refers to the process of initiation and extension of fractures in rocks caused by the hydraulic pressure applied by a fluid. This technique was developed by Clark (19). Haimson and Fairhorst (20) continued the research on the initiation and extension of fracture. Hubbert and Willis conducted comprehensive studies on the mechanics of hydraulic fracturing to determine the direction and condition of principal stresses using the hydraulic fracturing process. Since then, numerous studies and modellings have been conducted to investigate the factors effecting the hydraulic fracturing.
The present research is important because experimental and numerical modeling were used to calculate the hydraulic fracturing pressure for different conditions and to select the suitable pump for the operation.
These simulations are aimed to investigate the fracture pressure in Loshan sandstone to determine a relationship between the pressure needed for fracturing and the confining pressure.
Material and methods
The specimen examined in this study is the Loshan sandstone. Sandstone is a sedimentary rock which is formed in all geological periods and is mainly consisted of fine sand particles, different minerals and has various colors. This rock is mainly formed in the shallow seas, deltas, along the coasts, and in hot deserts. Moreover, materials such as clay and silicon oxide contributed to the cementation of its particles.
The rock sample of Loshan sandstone is a calcareous sandstone with a limestone-silica structure whose cement is calcareous (Figure 1). The main and secondary minerals in this rock include calcite, feldspar alkaline, quartz, and opaque minerals. The diagenesis of this rock includes sericitization, chertization, and calcification. The main shapers of this rock are shaped and semi-shaped quartzes with calcite.
The physical and mechanical properties of the specimens are presented in Table 1.
Table 1. Physical and mechanical properties of the Loshan sandstone

Effective Porosity (%)	Dry unit weight (KN/m3 )	Tensile strength (MPa)	Poisson’s ratio	Uniaxial compressive strength (MPa)	Elastic modulus (GPa)
7.5	21.60	6	0.21	54.62	12..22

 
 
Figure 1. Loshan sandstone
Results and discussion
Fracture pressures in the developed models are listed in table 2. The Fracture pressures obtained from numerical modeling had a 10% difference with the experimental modeling results.
Table 2. Experimental ant numerical modeling results

Fracture pressures obtained from experimental modeling	Fracture pressures obtained from numerical modeling	Confining pressure (MPa)	Axial stress (MPa)	Model number
14.58	13.8	2	2.26	1
15.7	15	2.5	2.5	2
11.16	9.9	0	5	3
11.39	9.9	0	7	4

Figure 2 shows the relationship between the pressure required to initiate hydraulic fracturing and confining pressure for Loshan sandstone. There was a linear relationship between fracture pressure and confining pressure. Thus, with an increase of the confining pressure, the pressure required to initiate hydraulic fracturing increased. The relationship between the fracture pressure and the confining pressure for Loshan sandstone is in the form of Equation (1).
P_f = 1.7386 σ₃+ 11.242                                   (1)
 
 
Figure 2. Relationship between fracture pressure and confining pressure
Conclusion
The following conclusions were drawn from this research.
1. The increase of lateral stress led to an increase in the fracture pressure.  
2. Changes in the axial stress did not significantly change the fracture pressure.
3. The results of numerical modellings were well consistent with those of the experimental modellings.
4. Unlike other studies conducted in this field, the numerical modellings in this study were performed without any initial pre-determinations for the crack-less models. Results show that in most cases, cracks initiate from the center and are extended toward both ends of the sample. The crack extension direction was parallel to the borehole axis inside the sample and perpendicular to the lateral stress. This is fully consistent with the observations in the experimental models.
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