3rd International Workshop on Rock Physics
th
th
13 – 17 April 2015
Perth, Western Australia
Relating static stiffness to ultrasonic, sonic and seismic velocities
Erling Fjæra, Rune M. Holtb and Anna M. Stroisza.
a
SINTEF Petroleum Research, Trondheim, Norway;
Norwegian University of Science and Technology, Trondheim Norway.
Contact email: [email protected].
b
Summary
This paper outlines the complexity in relations between static stiffness and stiffness derived from
elastic wave velocities, and shows results from modelling efforts aiming to fill some important gaps in
the mathematical description of these relations. The paper also points to some interesting spin-offs of
this modelling effort.
Introduction
Rock physics is to a large extent based on linear elasticity. Within this framework, there is a direct and
simple relation between static stiffness and elastic wave velocities. Numerous observations [1-4] show
however that this relationship is not so simple in real rocks. A consequence of this is that the
estimation of static stiffness on the basis of sonic or seismic velocities – for prediction of reservoir
compaction, for instance – is challenging and burdened with large uncertainty.
Reliable observations of relations between static stiffness and elastic wave velocities are for the most
part done in laboratories, on small samples at ultrasonic frequencies. This implies that we also need to
understand the relation between ultrasonic and sonic or seismic velocities in order to build relations
between static stiffness and sonic or seismic velocities.
Several factors affect the relations between static stiffness and elastic wave velocities, and have to be
controlled and accounted for [5]. The most important ones are: - strain rate, which differs by four to
six orders of magnitude between seismic and ultrasonic frequencies, and also differ largely for
different static deformations; - rock volume involved, which may differ by up to 12 orders of
magnitude or even more, from a cube constrained by the ultrasonic wavelength to the volume of a
reservoir; - drainage conditions, which for elastic waves are always undrained in a fully saturated
rock volume, but are often drained under static deformations; - anisotropy, which implies that
stiffness as well as velocities may differ largely with orientation of the rock; and – non-elastic
processes, which may have a dominating impact on static stiffness, but nearly no impact on the wave
velocities.
A reliable routine for estimation of static stiffness from sonic or seismic velocities requires that all the
factors mentioned above are considered and properly accounted for. Some of these elements are fairly
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3rd International Workshop on Rock Physics, Perth, 13th–17th, April 2015
well understood and described mathematically in well-established models. Still, these effects are often
ignored due to lack of information (as well as lack of attention). Other elements – in particular nonelastic processes and strain rate effects – have not yet reached the same level of understanding and
mathematical modelling.
Strain rate and strain amplitude
The difficulties in measuring dispersion in the range from seismic to ultrasonic frequencies has for a
long time hampered the development of static/dynamic relations, however several laboratories have
now developed equipment and procedures that allow for such measurements [6-9]. Results obtained so
far point towards partial saturation as a major cause for dispersion in this frequency range [5]. The
strain rate of static deformations in laboratory tests is comparable to the strain rate of a seismic wave
[5] (although in field situations the relevant static strain rates may be significantly lower). The strain
rate is therefore primarily a source of dispersion, and to a lesser extent a cause for the differences
between actual static stiffness and stiffness derived from sonic or seismic velocities.
The most important differences between static deformations and the dynamic deformations associated
with a passing elastic wave are the strain amplitude, and the fact that the dynamic deformations are
oscillatory while the static deformations are monotonic. This implies that non-elastic processes may
have a strong impact on static stiffness, while their impact on dynamic stiffness can usually be
ignored. A model describing the impact of such processes during first, monotonic loading [4] has been
shown to give a good description of the relation between static and dynamic stiffness for sandstone
[10] and shale [11], and is used successfully to predict strength and static stiffness from wireline log
data [12]. A recent extension of this model [13] also describes how the creep process may transform
from transient creep through steady state creep into accelerating creep. This model has several
interesting applications, for instance scaling between strain rates in laboratory and field, prediction of
consequences of strain rate changes, and interpretation of complex laboratory tests.
Two types of non-elastic processes are assumed to dominate this behaviour: crushing of pointed
contacts, and frictional slip. Whereas both processes are expected to be active during first loading,
only frictional slip will be significantly active during unloading and reloading [14]. Moreover,
frictional slip activity builds up gradually over a period of monotonous loading or unloading, but starts
from zero again when the loading direction is reversed. A consequence of this behaviour is that the
relationship between static and dynamic stiffness, which changes slowly with monotonously
increasing stress, changes abruptly when the stress path is reversed and maintains a somewhat
complicated behaviour during an unloading/reloading cycle, as shown in Figure 1.
The behaviour during unloading has been utilized for estimation of dispersion in the range from
seismic to ultrasonic frequencies [5]. That method was derived on the basis of observations. Based on
the model by Nihei et al. [15] we here establish a description of how the frictional slip process affects
the static stiffness in the vicinity of a turning point on the stress path. This model predicts a.o. that the
initial unloading gradient of the non-elastic compliance ( S H ) to stress ( σ z ) depends on the stress at
the turning point. Figure 2 shows that these predictions match quite well with observations for a set of
measurements on Berea sandstone. The results support the assumption that only frictional slip is active
during unloading, at least initially.
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3rd International Workshop on Rock Physics, Perth, 13th–17th, April 2015
This model description allows us to subtract the frictional slip process from the static compliance
during loading, using the model calibrated during unloading. The crushing process which is also active
during loading may then be studied separately. This work is in progress.
Static
60
10
40
Unloading
5
First loading
0
40
70
100
20
Continued
first loading
0
160
130
Axial stress [MPa]
80
15
Re-loading
Stiffness [MPa]
100
Dynamic
20
Time [min]
Figure 1. Static and dynamic stiffness, measured in a core plug of Castlegate sandstone.
1.4
∆SH/∆σz [MPa-2]
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
Axial stress at turning point [MPa]
Figure 2. Initial unloading gradient of non-elastic compliance to stress, versus stress at the point
where the stress path is reversed. The data points originate from measurements on Berea sandstone.
Solid line: model predictions.
Conclusions
A model describing the development of frictional slip during unloading appears to give a relevant
description of the relation between static and dynamic stiffness, thus supporting the assumption that
this is the only active non-elastic process during unloading. The model may be used to subtract
frictional slip from the static compliance during loading, thereby allowing for separate studies of
crushing processes that are active during loading.
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3rd International Workshop on Rock Physics, Perth, 13th–17th, April 2015
References
1. Simmons, G. and Brace, W.F. (1965) Comparison of static and dynamic measurements of
compressibility of rocks. Journal of Geophysical Research, 70, 5649-5656.
2. King, M.S. (1970) Static and dynamic elastic moduli of rocks under pressure. Proc. 11th US
symposium on Rock Mechanics. 329-351.
3. Jizba, D. and Nur, A. (1990) Static and dynamic moduli of tight gas sandstones and their
relation to formation properties. SPWLA 31st Annual Logging Symposium, Paper BB.
4. Fjær, E. (1999) Static and dynamic moduli of weak sandstones. In: Amadei B, Krantz RL,
Scott GA, Smeallie PH (eds.) Rock mechanics for industry. Balkema, pp 675-681.
5. Fjær, E., Stroisz, A.M., Holt, R.M. (2013) Elastic dispersion derived from a combination of
static and dynamic measurements. Rock Mech Rock Eng 46:611–618. DOI 10.1007/s00603013-0385-8.
6. Duranti, L., Ewy, R., Hofmann, R. (2005) Dispersive and attenuative nature of shales:
multiscale and multifrequency observations. SEG Expanded Abstracts. 24:1577-1580.
7. Hofmann, R. (2005) Frequency dependent elastic and anelastic properties of clastic rocks.
PhD Thesis, Colorado School of Mines.
8. Batzle, M.L., Han, D.-H., Hofmann, R. (2006) Fluid mobility and frequency-dependent
seismic velocity – Direct measurements. Geophysics 71:N1-N9. doi: 10.1190/1.2159053.
9. Bauer, A., Szewczyk, D., Hedegaard, J., and Holt, R.M. (2015) Seismic dispersion in Mancos
shale. Abstract submitted for 3IWRP, Perth, Australia.
10. Li, L., Fjær, E. (2012) Modeling of stress-dependent static and dynamic moduli of weak
sandstones. Journal of Geophysical Research – Solid Earth, 117, WA103-WA112.
11. Holt, R.M., Nes, O.-M., Stenebråten, J.F., Fjær, E. (2012) Static vs dynamic behavior of shale.
ARMA 12-542.
12. Woerl, B., Wessling, S., Bartetzko, A., Pei, J. and Renner, J. (2010) Comparison of methods
to derive rock mechanical properties from formation evaluation logs. ARMA 10-167.
13. Fjær, E., Larsen, I., Holt, R.M., Bauer, A. (2014) A creepy model for creep. ARMA 14-7398.
14. Stroisz, A.M. (2013): Nonlinear elastic waves for estimation of rock properties. PhD Theses,
NTNU.
15. Nihei, K.T., Hilbert Jr, L.B., Cook, N.G.W., Nakagawa, S., Myer, L.R. (2000) Friction effects
on the volumetric strain of sandstone. Int J Rock Mech & Min Sci 37:121-132.
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