Time-dependent stresses and strains in shales are related by the anisotropic creep and relaxation functions. Laboratory measurements of time-dependent creep are often limited to specific composition and mineral distribution. Moreover, creep data acquired for anisotropic rocks using techniques like triaxial loading or nanoindentation usually correspond to specific boundary conditions, and the measured data may not be sufficient to predict creep or relaxation under other modes of deformation. Invoking the correspondence principle, we estimate effective creep and relaxation functions for anisotropic rocks using the effective medium theory. This method can be applied to study the possible links between time-dependent creep/relaxation and other geophysical measurables, such as ultrasonic measurements, well log data, and seismic data. Using well log data for the Eagle Ford shale formation, we quantitatively study the time-dependent relaxation of horizontal stresses upon sudden change in vertical stress.
Experimental studies, e.g., Hagin and Zoback, (2007), Sone and Zoback (2013a, 2013b, 2014), have found that when loaded suddenly, gas shale reservoir rocks exhibit timedependent mechanical creep. It was reported that various shale rocks exhibit power-law creep, i.e., J = Btn , where J is the measured creep, t is time in seconds, "B" is the intercept (i.e., creep at t = 1 second), and "n" is the time exponent. Figure 1 shows the measured creep values for a vertical Haynesville shale sample and a horizontal Barnett shale sample (Sone and Zoback, 2014). The measured values of "B" and "n" were shown to be positively correlated with the amount of viscoelastic phases such as clay and organic matter. From these measurements, Sone and Zoback (2014) inferred that the rock composition may influence the in situ time-dependent differential stress magnitudes more strongly than the formation's geological loading history. Thus ignoring the time-dependent nature of deformation in shales can lead to significant errors in planning geomechanical applications, including analysis of time-dependent rock deformation, analysis of well bore stability, estimation of stress relaxation due to production, planning hydraulic fractures, predicting time-dependent closure of proppant filled fractures, and targeting optimal zones for production. Independent measurements by Chang and Zoback (2009) had also previously found that shales creep. Sone and Zoback (2013) report both acoustic and time-dependent creep properties of shales to be strongly anisotropic. Other studies have previously reported that acoustic properties of shale rocks are strongly anisotropic (Vernik and Nur, 1992; Sondergeld et al., 2000).