Error-in-Variables for Special Models Describing Brittle-Ductile Transition

• Orlando Zambrano Mendoza Escuela de Ingeniería de Petróleo. Facultad de Ingeniería, Universidad del Zulia, Apartado postal 4011-A-526. Maracaibo, Zulia, Venezuela.
• Peter P. Valko Harold Vance Department of Petroleum Engineering Texas A&M University 3116 TAMU College Station, TX 77843-3116
• James E. Russell Harold Vance Department of Petroleum Engineering Texas A&M University 3116 TAMU College Station, TX 77843-3116

DOI: https://doi.org/10.22209/rt.uv4705

Keywords: cap model, EIV, failure criterion, pore collapse, poroelasticity

Abstract

In this work it shows the application of the method of error-in-variable (EIV), based methodology to fit the envelope in the principal effective stresses plane considering the envelope describing brittle-ductile transition and pore collapse (cap model). The parametric equations of the failure envelope can be obtained introducing the concept of poroelasticity of Biot, representing the state of stress by the effective mean stress   and the equivalent stress . A nonlinear algebraic form represents the envelope equations delimiting the brittle-to-ductile transitional region of the failure or yield envelope. To illustrate the application of this methodology, we use the mechanical data for the brittle strength and compactive yield stress for Bentheim sandstone and the normalized principal stresses for ten different sandstones describing brittle-ductile transition behavior. Results demonstrate that the EIV method provides a simple way to obtain the parametric representation of the envelope in the plane of principal effective stress, describing the deformation mechanism of the tested rock that involves the pore collapse effect.

References

Addis, M. (1987). Material metastability in weakly cemented sedimentary rocks. Memoir of the Geological Society of China, 9, 495.

Anand, A., Kumar, S. (2015). Application of multi-objective optimization techniques to geotechnical engineering problems. M. Tech. Thesis. Rourkela: National Institute of Technology.
Biot, M. (1941). General theory of three-dimensional consolidation. Journal of Apply Physics, 12, 182.

Britt, H., Luecke, R. (1973). The estimation of parameters in nonlinear, implicit models. Technometrics, 15, 233.

Chen, W., Mizumo, E. (1990). Nonlinear analysis in soil mechanics: theory and implementation. New York: Elsevier Science.
Deming, W. (1943). Statistical adjustment of data. New York: Willey.

Edgar, T., Liebman, M., Kim, I. (1990). Robust error-in-variables estimation using nonlinear programming techniques. AIChE Journal, 36(7), 985-993.

Esposito, W., Floudas, C. (1998). Parameter estimation in nonlinear algebraic models via global optimization. Computers and Chemical Engineering, 22, 213-220.

Keles, T. (2018). Comparison of classical least squares and orthogonal regression in measurement error models. International Online Journal of Educational Sciences, 10(3), 200-221.

Keles, T., Altun, M. (2016). Comparison of classical least squares and orthogonal regression in measurement error models. Journal of Measurement and Evaluation in Education and Psychology, 7(2), 296-308.

Klein, E., Baud, P., Reuschle, T., Wong, T. (2001). Mechanical behaviour and faliure mode of bentheim sandstone under triaxial compression. Physical Chemical Earth (A), 26(1-2), 21-25.

Kumar, S., Kumar, P. (2011). Parameter optimization of rock failure criterion using error-in-variables approach. International Journal of Geomechanics, 11(1), 36-43.

Liebman, M., Edgar, T. (1988). Data reconciliation for nonlinear processes. Proceedings of the AIChE Annual Meeting. Washington, DC: American Institute of Chemical Engineers (AIChE), 137.

Nur, A., Byerlee, J. (1971). An exact effective stress law for elastic deformation of rock with fluids. Journal of Geophysical Research, 76, 6414.

O’Neil, M., Sinclair, I., Smith, F. (1969). Polynomial curve fitting when abscissas and ordinates are both subject to error. Computer Journal, 12, 52-56.

Peneloux, A., Deyrieux, E., Neau, E. (1976). The maximum likelihood test and the estimation of experimental inaccuracies: application to data reduction for vapor-liquid equilibrium. Journal of Computers, 73, 706-716.

Recio-López, J. (2021). Regresión lineal ortogonal. Pensamiento Matemático, XI (1), 005015.

Reilly, P., Patino-Leal, H. (1981). Bayesian study of the error-in-variables model. Technometrics, 23(3), 221.

Schwetlick, H., Tiller, V. (1985). Numerical methods for estimating parameters in nonlinear models with error in the variables. Technometrics, 27(1), 17-24.

Smits, R., de Wall, J., van Kooten, J. (1988). prediction of abrupt reservoir compaction and surface subsidence caused by pore collapse in carbonates. SPEFE (March), 340-346.

Southwell, W. (1969). Fitting experimental data. Journal of Computational Physics, 4, 465-474.

Terzagui, K. (1943). Theoretical soil mechanics. New York: Wiley.
Valkó, P., Vajda, S. (1987). An extended Marquardt-type procedure for fitting error-in-variables models. Computational Chemical Engineering, 11(1), 37-43.

Von Mises, R. (1913). Mechanik der festen Körper im plastisch deformablen Zustand, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göettingen. Mathematisch-physikalische Klasse, 1, 582592.

Willianson, J. (1968). Least squares fitting of a straight line. Canadian Journal of Physics, 46, 1845-1847.

Wong, T. E., David, C., Zhu, W. (1997). The transition from brittle faulting to cataclastic flow in porous sandstones: mechanical deformation. Journal of Geophysical Research, 102, 3009-3025.

York, P. (1966). Least squares fitting of a straight line. Canadian Journal of Physics, 44(5), 1079-1086.

Zambrano-Mendoza, O., Valkó, P., Russell, J. (2023). Determination of the shear failure envelope by adjusting with the statistical method of error in variables through the relationship between the principal stresses. Revista Técnica de la Facultad de Ingeniería de la Universidad del Zulia, 46, e234613.

Zambrano-Mendoza, O. (2004). Error-in-variables for failure criteria applied to the near-wellbore region. PhD. Dissertation. College Station: Texas A&M University, 134.

Zambrano-Mendoza, O., Valkó, P., Russell, J. (2003). Error-in-variables for rock failure envelope. International Journal of Rock Mechanics and Mining Sciences, 40(1), 137-143.

Zhang, J., Rai, C., Sondergeld, C. (2000). Mechanical strength of reservoir materials: key information for sand prediction. SPE Reservoir Evaluation & Engineering, 3(2), 127-131.