Bell, Andrew

Geiger, N.

Main, Ian

Tait, Daniel

Worton, Bruce

In this paper we investigate nonlinear models for rock failure data.The complexities of estimation and analysis using sample experimental data arestudied when fitting a power-law type model. Both the high frequency of samplingof observations during the experiment, as well as the nature of the model withhigh levels of correlation of the parameter estimators result in some challengingissues in the modelling

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     Edinburgh Research Explorer                                      Nonlinear modelling of experimental rock failure dataCitation for published version:Tait, D, Geiger, N, Worton, B, Bell, A & Main, I 2016, Nonlinear modelling of experimental rock failure data.in Proceedings of the 31st International workshop on Statistical Modelling Volume II. pp. 151-154, 31stInternational Workshop on Statistical Modelling, Rennes, France, 4/07/16.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Peer reviewed versionPublished In:Proceedings of the 31st International workshop on Statistical Modelling Volume IIGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 05. Apr. 2019Nonlinear modelling of experimental rockfailure dataDaniel Tait1, Nadine Geiger1, Bruce J. Worton1, Andrew Bell2,Ian G. Main21 School of Mathematics and Maxwell Institute for Mathematical Sciences, TheUniversity of Edinburgh, Edinburgh, UK2 School of GeoSciences, The University of Edinburgh, Edinburgh, UKE-mail for correspondence: Bruce.Worton@ed.ac.ukAbstract: In this paper we investigate nonlinear models for rock failure data.The complexities of estimation and analysis using sample experimental data arestudied when fitting a power-law type model. Both the high frequency of samplingof observations during the experiment, as well as the nature of the model withhigh levels of correlation of the parameter estimators result in some challengingissues in the modelling.Keywords: Nonlinear regression; Least squares estimation; Likelihood; Power-law model.1 IntroductionWe consider methods for modelling experimental data collected in the labconcerning rock failure. Accelerating rates of foreshocks are often observedprecursory to natural hazards such as earthquakes and volcanic eruptions.Similarly, rock failure in laboratory experiments is preceded by acceleratingstrain rates. In this work we investigate the usage of a damage mechanicsmodel proposed by Main (2000) for the analysis of strain and strain ratedata during the tertiary phase of brittle creep. The model studied consistsof 3 parameters, of which we focus on the failure time and on the power-lawexponent. When examining the likelihood function and the Fisher Informa-tion we find that there is substantial correlation between these parameters.This paper was published as a part of the proceedings of the 31st Inter-national Workshop on Statistical Modelling, INSA Rennes, 4–8 July 2016. Thecopyright remains with the author(s). Permission to reproduce or extract anyparts of this abstract should be requested from the author(s).2 Nonlinear modelling2 ModelMain (2000) develops a damage mechanics model to explain the time-dependent, trimodal behaviour of brittle creep. We consider modelling thestrain using a relationship of the formΩ = ΩI(1 +tmτ1)m + ΩIII(1 − ttf)−v,where for time t, Ω is the strain. The parameters are ΩI , ΩIII , m, τ1, vand tf . There is also interest in modelling strain rate Ω˙, the derivative ofΩ with respect to time t. Here we focus on the accelerating crack growthwhich is associated with the second term on the right-hand-side of theabove equation as it illustrates the complexities of modelling the data, i.e.a strain model of the formΩ = ω(1 − ttf)−v, (1)where tf represents the time of failure which is of particular interest, whilethe exponent parameter v relates to the curvature of the strain relationship.3 EstimationWe assume for the moment that the strain observations are subject to iidexperimental errors with variance σ2, and the expectation at time t hasthe form given in (1) which leads to a nonlinear model (Bates and Watts,1988; Fahrmeir et al., 2013).We apply a nonlinear least squares estimation procedure to fit the modelwith parameters v and ω for given tf to the strain data (Heap et al. (2009)gives details of lab experiments). This seems to be an effective numericalapproach for parameter estimation. Models were fitted for failure times tfover a suitable range of values and the best model selected; this corre-sponded to tf = 138.864. Table 1 presents the estimates of the fitted modelwith this particular failure time value.TABLE 1. Estimation of v and ω from strain for tf = 138.864.Parameter Estimate Std. Errorv 2.147×10−2 4.104×10−5ω 1.741 1.143×10−4σ 0.001628The fitted model and residuals are given in Figure 1. Looking at the leftpanel we can see that the fitted model appears to represent the experimentalTait et al. 3data well, but the residuals show some concerning departures from theassumed model. Firstly there are multiple small waves with a length ofabout ten minutes, and secondly there are two irregular large waves, whichindicate a more severe discrepancy between the fitted values and the data. Apartial autocorrelation analysis of the residuals suggests that residuals areautocorrelated, which may possibly be due to the nature of the experiment.Therefore, we expect the estimates to be reliable, but the SEs may bemisleading.80 90 100 110 120 1301.781.801.821.841.861.88Strain data and fitted regression lineTimeStrain80 90 100 110 120 130−0.004−0.0020.0000.0020.004Residuals for strain model TimeStrainFigure 5.6: Fitted regression line and residuals for the estimation from real strain data.0 5 10 15 20 25 300.00.20.40.60.81.0LagPartial ACFPartial ACF for residuals from strainFigure 5.7: PACF analysis for strain.rameter regions for nonlinear regression models:S(θ) < S(θˆ)(1 +pn− pF (p, n− p, α)) (5.1)S(θ) denotes the residual sum of squares, n is the number of observations and p is thenumber of parameters in the model. For large samples of normal distributed randomvariables, confidence regions obtained by this method are approximately equivalent tolikelihood ratio based confidence regions (see Appendix B).−2(l(θ)− l(θˆ)) = 1σˆ2(S(θ)− S(θˆ)) < χ2p(α) (5.2)To illustrate the confidence regions graphically, we again evaluate the log-likelihood func-tion and the residual sum of squares over a 3-dimensional grid in proximity to the maxi-mum likelihood estimator. We extract the sets of parameters, for which the likelihood-ratiostatistic or the residual sums of squares are below the required critical value. The approx-imate confidence regions for v and tf are created by projecting these sets onto the tf -v51FIGURE 1. Fitted regression line with strain observations (left) and residuals(right) for the estimation from the experimental lab data.The estimated parameters (obtained by minimising RSS over a range of tfvalues, with argmin(tf ) = 143.37) are given in Table 2 for the strain ratedata, and the fitted model is shown in Figure 2, with the residuals. Theestimates for the strain and the strain rate produce very similar fits whenviewed on the strain scale. However, the SEs for the strain rate wouldappear to be more reliable. On investigating the differences between theparameters estimated by the two models, the tf over which the RSS wasminimised is not very well determined as the RSS does not vary greatly overa range of tf -models. Also, the parameter estimators are highly correlatedso changes in tf lead to changes in the other two parameters. Nevertheless,the strain and strain rate models provide a useful representation of theexperimental data.TABLE 2. Estimation of v and ω′ (the constant multiplicative parameter fromΩ˙ model) from strain rate for tf = 143.37.Parameter Estimate Std. Errorv 0.25634 0.05680ω′ 0.13246 0.04688σ 0.0024794 Nonlinear modellingEstimate Std. Error t value Pr(>|t|)v 0.25634 0.05680 4.513 7.05e-06ω 0.13246 0.04688 2.825 0.00481σ 0.002479Table 5.2: Estimation of v and ω from the strain rate for tf = 143.3707The fitted regression line and the residuals can be seen in Figure 5.13. The residuals occurto be more random than the residuals of the fitted model for strain. However, they arenot free of a pattern alike, as they manifest the discrete nature of the rate data.80 90 100 110 120 130−0.010−0.0050.0000.0050.0100.015Rate data and fitted regression lineTimeRate80 90 100 110 120 130−0.010−0.0050.0000.0050.010Residuals for rate modelTimeresFigure 5.13: Fitted regression line and residuals for the estimation from real rate data.Moreover partial autocorrelation is also present among the residuals of the strain rate,although the problem is not as severe as in the case of strain. There is a comparativelystrong negative autocorrelation at a lag of one, and positive autocorrelation at lags between10 and 15.The 99% and 95% confidence regions for v and tf are shown in Figure 5.15. In the plot onthe right-hand side ω is unknown and the confidence regions were created by the projectionof a 3-dimensional set onto a plane of v and tf . On the left-hand side ω is treated as fixedconstant of the size of the maximum likelihood estimator. Compared with the estimationsfrom strain, the estimation from the strain rate leads to much larger confidence regions.The 99 % confidence interval for tf under the assumption of unknown ω spans from 139.9to over 148. For v the interval is even less precise. Even if we assume that ω is fixed,the resulting confidence region is larger than the confidence region given three unknownparameters for strain.55FIGURE 2. Fitted regression line with strain rate observations (left) and residuals(right) for the estimation from the experimental lab data.4 DiscussionIn this paper we have considered the application of nonlinear models foranalysis of experimental rock failure data during the tertiary phase of brit-tle creep. These 3-parameter models have been used to explore differentfeatures of the data and highlight some of t e challenges of analysing suchdata. I future work, where lab experiments are re eated under identicalconditions, it is expected that the use of replication will enable the featureswe ave see to be i vestigated more fully, and provide further insights intothe processes related to n tural hazards such as earthquakes nd volcaniceruptions.Acknowledgments: Daniel Tait is supported by an EPSRC studentship.ReferencesBates, D.M. and Watts, D.G. (1988). Nonlinear Regression Analysis andits Applications. New York: Wiley.Fahrmeir, L., Kneib, T., Lang, S., Marx, B. (2013). Regression: Models, Meth-ods and Applications. Heidelberg: Springer.Heap, M.J., Baud, P., Meredith, P.G., Bell, A.F., and Main, I.G. (2009). Time-dependent brittle creep in Darley Dale sandstone. Journal of Geo-physical Research, 114, B07203.Main, I.G. (2000). A damage mechanics model for power-law creep andearthquake aftershock and foreshock sequences. Geophysical JournalInternational, 142, 151 – 161.

Nonlinear modelling of experimental rock failure data

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Nonlinear modelling of experimental rock failure data

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