Estimating Variations of the Minimum Principal Stress in Unconventional Reservoirs Using a Viscous Relaxation Model

Abstract:

It is important to estimate subtle variations of the minimum principal stress (S_hmin) with depth because it controls the propagation of hydraulic fractures in normal and strike-slip faulting environments. We argue that variations of the minimum principal stress can be due to varying degrees of viscous relaxation in different formations following Sone and Zoback (2014). As the viscous relaxation occurs in a formation, the stress field becomes more isotropic, leading to an increase in the minimum principal stress. An important result of this effect is that the formations with high minimum principal stresses tend to arrest the propagation of hydraulic fractures. To quantify this ductile behavior of unconventional reservoir rocks, a power-law creep compliance relationship was developed from laboratory experiments. By taking the Laplace Transform of this creep compliance, the viscous relaxation constitutive law is obtained as, S_V – S_hmin = (ε₀×E×t^-n)/(1-n), where S_V, S_hmin, ε₀, E, t are overburden stress, minimum principal stress, total tectonic strain, Young’s modulus, total geological time, respectively, and n describes the tendency for viscous relaxation. This constitutive law essentially relates the measureable properties of unconventional reservoir rocks to the in situ stress. From this constitutive law, in order to estimate S_hmin in a specific formation, we need to know the relevant parameters. Therefore, S_V is integrated from density logs, ε₀ is assumed in a generic plot but can be fitted by one measured minimum principal stress for a specific case study, and t is not important when it is large enough. With these methods and using E and n measured from rock samples, we estimate variations of the minimum principal stress in different unconventional reservoirs around North America. We also examine variations of the viscous relaxation with depth in a case study from the Permian Basin in West Texas, and compare them to the minimum principal stress measurements from Diagnostic Fracture Injection Tests (DFITs).