From Static to Dynamic Stiffness of Shales: Frequency and Stress Dependence

Abstract

The relation between static and dynamic stiffness in shales is important for many engineering applications. Dynamic stiffness, calculated from wave velocities, is often related to static stiffness through simple empirical correlations. The reason for this is that dynamic properties are often easier to obtain; however, it is the static properties that define the actual subsurface response to stress or pore pressure changes. Rocks are not elastic media, and stiffness depends on the stress state, stress-change amplitude, loading rate, drainage conditions, fluid saturation, and scale. All these factors require consideration when static and dynamic stiffness properties are to be related. Two mechanisms that may have a strong effect on the stiffness of shales were studied in this experimental work: (i) a reduction of undrained static stiffness with an increase in stress amplitude and (ii) a frequency dependence (or dispersion) of dynamic stiffness. Laboratory tests were performed on four fully brine-saturated undrained field shales from different overburden formations. Experiments were conducted using a low-frequency apparatus-a triaxial loading cell with the ability to measure dynamic stiffness at seismic frequencies (1-150 Hz) and ultrasonic velocities (500 kHz). Shale anisotropy was characterized by testing differently oriented core plugs. Static and dynamic stiffness of shales The results demonstrated that all the tested shales exhibited a dispersion of dynamic stiffness from seismic to ultrasonic frequencies. Young's modulus dispersion for the tested shales ranged from nearly 30% to above 100%. Wave velocity dispersion was on the order of 10-20% for P-waves and 20-40% for S-waves. In static tests, the undrained rock stiffness gradually decreased with increasing stress amplitude. For one shale, the static undrained Young's modulus was reduced by 50% when amplitude of the loading-unloading measurement cycle was increased from 1 MPa to 3 MPa. This finding is explained by non-elastic deformations that increase with the stress level. A method of zero-stress extrapolation of static stiffness was used to obtain the purely elastic response. The stiffness for the limit of zero stresschange amplitude agreed well with the dynamic response at seismic frequency, providing a link between static and dynamic stiffness.

Nonlinear Young’s Modulus of New Red Sandstone: Experimental Studies

“…Although the dispersion of Young's modulus has been mentioned in literature [14,16,17], it has not been supported by robust experimental data or mathematical models. Therefore, our studies may facilitate the development of modelling the dynamic characteristics of clastic rocks.…”

“…[13]) reported that Young's moduli of Navajo sandstone, Spergen limestone and Oklahoma granite are independent of the frequency of dynamic loading at the strain below 10 −7 in the frequency range from 4 to 400 Hz. Recent studies on Opalinus Clay (see [14]) and Mancos shale (see [15]) demonstrated an increase in Young's modulus and a decrease in Poisson's ratio with an increase in frequency of the dynamic loading from 1 to 100 Hz at the strain around 10 −6 . The authors [16] showed that with an increase in frequency from 1 to 50 Hz for samples with a strain range between 10 −8 and 10 −6 , both Young's modulus and Poisson's ratio of Donnybrook sandstone increase, i.e.…”

Static and dynamic stiffness measurements with Opalinus Clay

“…9, (dε ax /dσ ax ) 0 0.20 GPa −1 , and C 9×10 −5 MPa −2 . Lozovyi et al (2018) have shown that the equation (10) can be used to model the non-elastic effects in both axial and radial strains, ε ax and ε rad , as function of axial stress change, σ ax . This allows for finding zero-stress extrapolated Young's moduli and Poisson's ratios.…”

“…The model is based on the empirical finding that the rock compressibility increases linearly with stress amplitude. Some rocks, especially shales, exhibit relatively large stiffness dispersion (Duranti, Ewy and Hofmann 2005;Hofmann 2006;Tutuncu 2010;Szewczyk, Bauer and Holt 2016;Lozovyi et al 2018). Quasi-static rock deformations in static tests are usually done with loading rates that correspond to average loading rates of dynamic measurements in the sub-hertz frequency regime.…”

Passive Seismic Imaging of Stress Evolution with Mining-Induced Seismicity at Hard-Rock Deep Mines

“…Stress fields are correlated with stress indicators, such as seismic velocity. Previous studies have shown a significant non-linear increase in seismic velocities with a small rise in the hydrostatic pressure (Mayr and Burkhardt 2006;Lozovyi and Bauer 2019). Additionally, similar studies showed that the increase in velocity is limited once the change in pressure exceeds a certain threshold (Mavko et al 2009;Mavko et al 2009).…”