Error-in-variables for rock failure envelope

Transcription

1 International Journal of Rock Mechanics & Mining Sciences 4 (3) Technical Note Error-in-variables for rock failure envelope O. Zambrano-Mendoza*, P.P. Valk!o, J.E. Russell Harold Vance Department of Petrol Engineering, Texas A&M University, 336 Tamu, College Station, TX , USA Accepted 6 September. Introduction A failure envelope separates stable and unstable zones of stress state. A closed-form representation of this envelope is of great importance in practical applications. The delimitation of this boundary is the key for modeling various phenomena like subsidence, borehole stability, sanng propensity, just to mention some, relevant for the petroleum industry. Most often, a failure criterion is expressed in terms of the major (s ) and minor (s 3 ) principal stresses, because of the physical limitations of the experimental set-up. In such case, the criterion can be represented by the equation []: fðs * ; s 3 Þ¼: ðþ For every stress state (s ; s 3 ), producing a failure during the triaxial failure experiments, a Mohr circle can be drawn on the s; t plane accorng to s s þ s 3 þt ¼ s s 3 : ðþ The failure envelope is then defined as the curve enveloping (touching from one side) all these circles. The equation of the failure envelope can be represented by f ðs; tþ ¼: ð3þ Because of the experimental errors and stochastic variations in the rock itself, there is no guarantee that such a curve can be created. Then the problem consists of two parts: (i) A suitable algebraic form of Eq. (3) should be selected. Linear, parabolic and hyperbolic equations are frequently used for this purpose. (ii) The unknown parameters of Eq. (3) should be determined from a suitable optimality criterion. *Corresponng author. Tel.: ; fax: address: oz578@spindletop.tamu.edu (O. Zambrano-Mendoza). Several approaches have been suggested in the literature, most often based on the method of least squares, starting with the pioneering work of Balmer []. Any least-squares method requires the stinction between independent and dependent variables with the requirement that the former is known exactly. However, in the case of measured axial and lateral stresses, both quantities are corrupted by error (as well as any new variables derived for them). Therefore, it is not justified to assume that one of them is without any error. This poses an inherent limitation on least-squares-based methods of determining the failure envelope. More recently, Pincus [3] suggested a family of methods to construct the failure envelope piece-wise. Each segment of the envelope is derived from closedform solution for two Mohr circles. While the method provides great flexibility, it has some drawbacks: (i) The resulting envelope is not smoothing (fferentiable) at the intersecting point of two segments. (ii) In construction of the segments, a pre-processing is needed to create average circles, which is fficult to justify from a statistical point of view. In this work, we propose a new and efficient approach based on the error-in-variables (EIV) method [4]. In the following, first we describe the EIV method of curve fitting, then develop a variant suitable for fitting failure envelopes, and finally apply it to a well-documented set of data.. EIV method In the standard approach to parameter estimation, a stinction is made between dependent and independent variables, with the assumption that there are no measurement errors in the independent variable. The presence of the measurement errors in both system variables x and y is taken into account in the EIV //$ - see front matter r Elsevier Science Ltd. All rights reserved. PII: S ()96-5

2 38 O. Zambrano-Mendoza et al. / International Journal of Rock Mechanics & Mining Sciences 4 (3) Nomenclature a b c i CFM d i d ils d pi f ðþ fð * Þ J P j r i tangent vector to the envelope at ð#s i ; #t i Þ vector starting from the center of the Mohr s circle and enng at ð#s i ; #t i Þ center of the ith Mohr s circle closed-form model geometric stance in EIV method vertical stance in least-squares method geometric stance in EIV method parametric function parametric function sum of square stance interception point raus of the ith Mohr s circle x i measured value of the abscissa y i measured value of the ornate #y ils calculated value of the ornate from least square #x i reconciled value of the abscissa from EIV #y i reconciled value of the ornate from EIV Greek letters s i measured axial stress of the ith Mohr circle s 3i measured lateral stress of the ith Mohr circle s i calculated normal stress of the ith Mohr circle #s i reconciled normal stress of the ith Mohr circle t i calculated shear stress of i Mohr circle #t i reconciled shear stress of the ith Mohr circle y % vector of unknown parameters approach, when formulating the objective function of the parameter estimation problem. The model is written in implicit form: f ð #x; #y; yþ¼; ð4þ % where y is the vector of unknown parameters, #x and #y are corrected % state variables. If we denote the measurements by x i and y i ði ¼ ; ; ynþ and their corrected (or reconciled) values by #x i and #y i ; then our goal is to find the optimum parameters at which the sum of necessary corrections squared: J ¼ Xn ð5þ dp i i¼ is minimum, where dp i ¼ð#x i x i Þ þð#y i y i Þ =: ð6þ As shown in Fig., the correction corresponds to the true stance of the point from the envelope, while in trational least squares, only the vertical stance (d ils ) is considered. Deming [4] was the first to formulate the general EIV problem. His primary concern was to obtain approximate solutions, suitable for hand calculations. Until the beginning of the 7s, exact solutions were proposed by other researchers [5 8], concerning straight lines or higher order polynomials. Later on, Britt and Luecke [9] suggested a general algorithm based on the concept of Lagrange multipliers. Peneloux et al. [] and Reilly and Patino-Leal [] provided computational improvements. Schwetlick and Tiller [] separated the parameter estimation and data reconciliation steps and Valk!o and Vajda [3] used a similar decomposition in order to solve the problem by the Gauss Newton Marquardt procedure. Liebman and Edgar [4] and Edgar et al. [5] used nonlinear parameter-estimation (NLP) techniques, not only in the parameter estimation, but also in the y (x i, y i ) d i (x ils, y ils ) (x i, y i ) Fig.. Geometric representation of the stance using least-squares (d ils ) and EIV (d i ) approaches. data reconciliation step. Esposito and Floudas [6] applied a global optimization technique to avoid being trapped in a local minimum. Without some mofication, the EIV method cannot be applied to the failure envelope problem, because in our case the measurement information is represented by a circle, not by a point. Nevertheless, the generalization of the EIV method is straightforward, once the concept of the stance of the Mohr circle from a failure envelope is clarified. 3. Application to failure envelope determination Shown in Fig. are two Mohr s circles and the failure envelope. For circle one, the stance of the Mohr circle from the envelope is denoted by d as shown in Fig.. Circle intercepts the envelope and its stance from the curve, d ; is defined as the stance of the farthest point x d ils

3 O. Zambrano-Mendoza et al. / International Journal of Rock Mechanics & Mining Sciences 4 (3) τ σ 3- P d P d r r σ 3- c c Circle Circle σ - σ - σ and b the vector connecting the center of the Mohr circle to ð#s i ; #t i Þ: b ¼ð#s i c i Þi þ #t i j: ð4þ Because a and b must be orthogonal, a b=: ð#s i c i ð#s i; #t i ; ð#s i ; #t i ; yþ % þ #t i % ¼ At any parameter vector y; the system of Eqs. (5) and (8) can be solved simultaneously % to obtain ð#s; #tþ; and hence the objective function can be evaluated. For the three most frequently used envelope equations, the solutions are given below. Fig.. Schematic representation of the EIV approach applied to failure envelope determination in Mohr space. P of the circle, lying in the unstable half plane. The engineering interpretation is that we have to mofy the raus of the observed Mohr circle by stances d or d to reconcile the measurement with the envelope. Obviously, our goal is to minimize the sum of squares of stances, d i : J ¼ Xn : ð7þ i¼ The reconciled stress state satisfies the equation of the envelope: f ð#s i ; #t i ; yþ¼: ð8þ % Introducing the observed raus r i ¼ s i s 3i ð9þ and center c i ¼ s i þ s 3i ; ðþ we can express the stance as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðc i #s i Þ þ #t i r i : ðþ d i Substituting Eq. () into Eq. (7), we obtain J ¼ Xn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc i #s i Þ þ #t i r i : ðþ i¼ Then minimizing Eq. () subject to constraint (8) constitutes the EIV formulation. Because of the simple form of objective function (), the following algorithm can be used. 4. EIV algorithm Let a denote the tangent vector of the envelope at ð#s i ; #t i Þ: a ð#s i; #t i ; % i ð#s i; #t i ; yþ % ð3þ 5. Application to selected models 5.. Linear envelope equation For this case, Eq. (8) can be expressed as f ð#s i ; #t i ; yþ ¼y þ y #s i #t i ¼ ; ð6þ where y and y are the parameters of the linear envelope. Eq. (5) can be represented as c i #s i y #t i ¼ : ð7þ Solving the system of Eqs. (6) and (7), the solution for the squared stance is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ðy þ c i y Þ ¼ þ y r i : ð8þ Substituting Eq. (8) into objective function (7), we arrive at an unconstrained minimization problem involving two unknown parameters: y and y : For the present works, we programmed the objective functions in Visual Basic and used the GRG optimization code ( Solver ) available in MS-Excel. The starting point of the iterative minimization was obtained by trational (generalized) least squares (that is artificially creating independent and dependent variables). 5.. Parabolic envelope equation For the parabolic approximation of the envelope, Eq. (8) can be written as f ð#s i ; #t i ; yþ ¼y þ y #s i #t i ¼ ; ð9þ where y and y are the parameters. The appropriate form of Eq. (5) is ½ð#s i c i Þþy Š#t i ¼ : ðþ Solving the system of Eqs. (9) and (), we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d i y þ c i y y 4 r i A : ðaþ

4 4 O. Zambrano-Mendoza et al. / International Journal of Rock Mechanics & Mining Sciences 4 (3) Except for the degenerate case of the squared stance, can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y þ c i r i A : ðbþ y 5.3. Hyperbolic envelope equation For a hyperbolic parameters model: f ð#s i ; #t i ; yþ ¼y þ y #s i #t i ¼ ðþ Eq. (5) takes the form ½ð#s i c i Þþy #s i Š#t i ¼ : ð3þ Solving the system of Eqs. () and (3), the squared stance can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y þ c i y þ y y r i A : ð4þ þ y Because this solution is concave upward, it is not compatible with the ðs; tþ data over the full range of s: Instead of using it, we propose a hyperbolic threeparameter equation that is physically meaningful. The parametric equation is f ð#s i ; #t i ; yþ ¼y þ y #s i þ y #s i #t i ¼ : ð5þ Then Eq. (5) becomes ½ð#s i c i Þþðy þ y #s i ÞŠ#t i ¼ : ð6þ For the squared stance, we have three fferent solutions. If #t i ¼ ; there are two solutions of the square stance given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð#s i c i Þ r i ; ð7þ d i where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #s i ¼ y 7 y 4y y y ð8þ and the third solution of the square stance is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð#s i c i Þ þ y þ y #s i þ y #s i r i ; ð9þ d i where #s i ¼ c i y = ð þ y Þ : 6. Results and scussions ð3þ To illustrate the method, the results of a previous ASTM interlaboratory study [7 9] were processed assuming linear, parabolic and hyperbolic envelope models. In the ASTM interlaboratory testing study, the triaxial compressive strength of intact, uniformly oriented cylindrical specimens of Barre granite, Berea sandstone, and Tennessee marble were obtained. Our goal is to obtain all three-failure envelopes for the three rocks, using all the available information provided by the various laboratories. The failure envelopes then will be compared to the ones obtained by Pincus [3]. For Barre granite, 83 Mohr s circles [7,9] were available whereas for both the Berea sandstone and the Tennesse marble, the number of circles was 7. Table contains the optimal parameters for Barre granite. The parameters describe the failure envelope located nearest to the 83 Mohr s circles obtained from data measured in various laboratories under various confining pressures. The possible use of these parameters is within a computational algorithm, where a stress state needs to be tested for failure. Using the listed parameters, one can minimize the likelihood of making a wrong judgment (i.e. declaring a failure state as safe or a safe state as failure). In case of the two-parameter parabolic model, we used an adtional constraint: the envelope s intersection with the #o-axis must exist, which requires that y X: The standard deviation is the sum of the squared stances between the failure envelope and the Mohr s Table Optimal parameters determined from the EIV method Linear Parabolic Hyperbolic f ð#s; #t; % yþ¼y þ y #s #t f ð#s; #t; % yþ¼y þ y #s #t f ð#s; #t; % yþ¼y þ y #s #t y y y y y y Test rock MPa MPa MPa MPa Barre granite a Berea sandstone b Tennesse marble b a 83 Test samples. b 7 Test samples.

5 O. Zambrano-Mendoza et al. / International Journal of Rock Mechanics & Mining Sciences 4 (3) circles (objective function values); the standard deviations are shown in Table. For comparison, we calculated the standard deviation for the closed-form model (CFM) envelopes obtained by Pincus [3]. That calculation is not trivial, because the CFM envelope consists of three segments, and the stance of an invidual Mohr s circle should be calculated either from the orthogonality criterion or at the intersection of two segments, while selecting the valid segment. Notice that similar computational fficulties may arise in calculating the tangent of the envelope, because slope changes from one side to the other of each segment end point. More importantly, for the CFM model proposed by Pincus [3], there are 6 model parameters (for instance describing 3 straight lines for the piece-wise linear model) while the EIV envelope determined here has only either or 3 parameters (one straight line for the linear envelope case.) In spite of the smaller number of parameters, the EIV model provides smaller sum of squared stances for all rocks and all models, except the linear model. Figs. 3 and 4 illustrate the fference between those two approaches. For an arbitrarily selected Mohr circle, the stance from the EIV envelope is uniquely defined (Fig. 3); for the segmented CF envelope we show only two segments with the same Mohr circle in Fig. 4 defining the dot line as segment whereas the solid line as segment. If we consider only the formal equation of segment, we obtain the nearest point P, but it defines a false stance, because it lies outside the vality domain of the segment. Similarly, P defines a false stance. Therefore, the physically meaningful stance of the Mohr circle from the envelope is defined by the intersection point P 3. In adtion to the comparison between EIV and CFM, we suggest a three-parameter hyperbolic model that generates a lower standard deviation. Fig. 5 illustrates the three-parameter model for Berea sandstone inclung the tensile strength data. The standard deviation obtained in this case is about 4.7% compared to 8.3% obtained from the two-parameter parabolic EIV model. We note that the developed program has the option of using weights that depend on whether the envelope crossed the Mohr circle or not. However, the scussion is out of the scope of the present paper. Finally, we note that the EIV approach will reproduce an envelope that resembles the trational failure Table Standard deviation between Mohr s circles and failure envelope from EIV and segmented CF envelope Linear Parabolic Hyperbolic f ð#s; #t; % yþ¼y þ y #s #t f ð#s; #t; % yþ¼y þ y #s #t f ð#s; #t; % yþ¼y þ y #s #t EIV CF EIV CF EIV CF Test rock % % % % % % Barre granite a Berea sandstone b Tennesse marble b a 83 Test samples. b 7 Test samples P EIV 5 5 Fig. 3. Mohr s circle No. 7 for Berea sandstone. Distance from EIV (parabolic) envelope. PEIV: reconciled point of the circle on the envelope.

6 4 O. Zambrano-Mendoza et al. / International Journal of Rock Mechanics & Mining Sciences 4 (3) P 3 P 6 4 P 5 5 Fig. 4. Mohr s circle No. 7 for Berea sandstone. Distance from segmented CF (parabolic) envelope Fig. 5. Hyperbolic three-parameter envelope equation from EIV method for Berea sandstone (tensile strength data from a Brazilian Test is included). The dashed line is an extrapolation of the failure envelope Fig. 6. Hyperbolic three-parameter envelope equation from EIV method for Berea sandstone using average circles. envelope. For this we consider the average circles given by Pincus [3] for Berea sandstone, which are shown in Fig. 6. In this figure we observe the hyperbolic threeparameter envelope perfectly touching the tangential point of each of the four average circles obtained at confining pressures of,, 5 and 4 MPa. The standard deviation found in this case is about.44%. 7. Conclusions The EIV procedure can be applied to virtually any failure-envelope model. It provides exactly reproducible envelope parameters from a given set of data. The resulting parameters correspond to a well-defined optimality criterion that makes more statistical and

7 O. Zambrano-Mendoza et al. / International Journal of Rock Mechanics & Mining Sciences 4 (3) rock-mechanical sense than the trational least-squares approach because the EIV method accounts for measurement errors in both x and y. The smoothness (fferentiability) of the resulting envelope makes it suitable for usage in applications requiring fferentiable failure function. Interpolation within the range of the envelope equation can be done more accurately than with segmented representations for the nonlinear models. Acknowledgements One of us (O.Z.) has pleasure in thanking his colleagues at the rock mechanics research group of Texas A&M University, Mr. Javier Franquet and Mr. Xianje Yi, for their advice and continuous interest. References [] Sheorey PR. Empirical rock failure criteria. Rotterdam: A.A.Balkema, 997. [] Balmer GA. A general analytical solution for Mohr s envelope. ASTM Proc 95;5:6 7. [3] Pincus HJ. Closed-form/least-squares failure envelopes for rock strength. Int J Rock Mech Min Sci ;37: [4] Deming WE. Statistical adjustment of data. New York, NY: Wiley, 943. [5] York P. Least squares fitting of a straight line. Can J Phys 966;44:78. [6] Willians JH. Least squares fitting of a straight line. Can J Phys 968;46:845. [7] O Neil M, Sinclair IG, Smith FJ. Polynomial curve fitting when abscissas and ornates are both subject to error. Comput J 969;:5. [8] Southwell WH. Fitting experimental data. J Comput Phys 969;4:465. [9] Britt HI, Luecke RH. The estimation of parameters in nonlinear, implicit models. Technometrics 973;5:33. [] Peneloux AR, Deyrieux E, Neau E. The maximum likelihood test and the estimation of experimental inaccuracies: application to data reduction for vapor liquid equilibrium. J Phys 976;73: 76. [] Reilly PM, Patino-Leal HA. Bayesian study of the error-invariables model. Technometrics 98;3(3):. [] Schwetlick H, Tiller V. Numerical methods for estimating parameters in nonlinear models with error in the variables. Technometrics 985;7():7. [3] Valk!o P, Vajda S. An extended Marquardt-type procedure for fitting error-in-variables models. Comput Chem Eng 987; (): [4] Liebman MJ, Edgar TF. Data reconciliation for nonlinear processes. Preprint, AIChE Annual Meeting, Washington, DC, 988. [5] Edgar TF, Liebman MJ, Kim IW. Robust error-in-variables estimation using nonlinear programming techniques. AIChE J 99;36(7): [6] Esposito WR, Floudas CA. Parameter estimation in nonlinear algebraic models via global optimization. Comput Chem Eng 998;(Suppl.):S3. [7] Pincus HJ. Interlaboratory testing program for rock properties, round one-longitunal and transverse pulse velocities, unconfined compressive strength, uniaxial elastic modulus, and splitting tensile strength. Geotech Test J 993;6(): [8] Pincus HJ. Addendum to interlaboratory testing program for rock properties, round one. Geotech Test J 994;7():56 8. [9] Pincus HJ. Interlaboratory testing program for rock properties, round two-confined compression: Young s modulus, Poisson s ratio, and ultimate strength. Geotech Test J 996;9(3):3 36.