Unconventional hydrocarbon resources are now playing a significant role in our energy consumption. In most cases, the development of unconventional resources requires reservoir stimulations, such as hydraulic fracturing. As the viscoelastic properties of rocks from unconventional reservoirs have been verified by previous studies, the proppant embedment after hydraulic fracturing becomes a problem which concerns the width and conductivity of the hydraulic fracture, and even the oil and gas production in the long run. Most of previous studies of proppant embedment are mostly carried out by numerical simulations. In this study, an analytical solution of the depth of the proppant embedment in the viscoelastic formation is deduced. Firstly, a creep experiment of tight sandstone is conducted and the fractional Maxwell model is used to model the viscoelastic response of the rock. And then the Hertz contact theory is utilized to obtain the depth of proppant embedment in the elastic formation. Later, with the correspondence principle the elastic solution of the depth of proppant embedment is extended into a viscoelastic one modeled by the traditional Maxwell model. And a more generally viscoelastic solution is acquired by replacing the traditional Maxwell model with the fractional Maxwell model. Finally, a viscoelastic solution of the depth of proppant embedment is obtained which involves seven parameters, elastic modulus and Poisson's ratio of proppant, and elastic modulus and Poisson's ratio and viscosity of the tight reservoir, and the closure pressure and the fractional order. And the parameter sensitivity analysis of all these seven parameters are conducted and the closure pressure is found to be the most important factor of the proppant embedment. Base on the parameter sensitivity, precautions are also proposed to prevent proppant embedment.
Unconventional hydrocarbon resources are now playing a significant role in our energy consumption. In most cases, the development of unconventional resources requires reservoir stimulations, such as hydraulic fracturing, to create massive hydraulic fractures to expose large surface areas within the formation and facilitate the transportation of hydrocarbons to the wellbore (Alramahi and Sundberg, 2012). Economic production requires sustaining sufficient conductivity in hydraulic fractures through the lifetime of the wells and operators typically try to maintain this conductivity by introducing proppants into the injected fluid -a material such as grains of sand, ceramic, or other particulate, thus preventing the fractures from closing when injection is stopped and pressure removed. Thus, prevention of propping failure is crucial in keeping such high conductivity paths for hydrocarbons. However, several reasons may result in this failure, such as fines migration (Pope et al., 2009), proppant diagenesis (LaFollette and Carmen, 2010), proppant crushing (Terracina et al., 2010), and proppant embedment (Terracina et al., 2010; Huitt and McGlothlin, 1958). Among them, the proppant embedment has been most intensively studied by experiments (Lacy et al.,1997; Lacy et al.,1998; Ghanizadeh et al., 2016; Tan et al., 2017), numerical simulations (Alramahi and Sundberg, 2012; Deng et al., 2014) and analytical modeling (Huitt and Mcglothlin, 1958; Volk et al., 1981; Gao et al., 2012; Li et al., 2015; Guo and Liu, 2015; Chen et al.,2018). Nearly all studies focus on the instantaneous embedment depth of proppant except Guo and Liu's research. However, these deep reservoirs are always located in high stresses and temperatures conditions which make these reservoir rocks may behave in a viscoelastic manner (Xie et al., 2015; Sone and Zoback, 2013; part2). Malan's study (Malan, 1999) has confirmed that significant time-dependent deformations of underground rocks can also be observed even if these rocks are hard. What's more, during the process of hydraulic fracturing, the fracturing fluid in the hydraulic fractures may weaken the surrounding rocks (Akrad et al., 2011; Matthews et al., 2007) and makes those rocks more viscous. And the production of these wells will last for decades and the time-dependent deformations of the reservoir rocks may not be ignored.
