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Error-in-Variables for Special Models

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Rev. Téc. Ing. Univ. Zulia, 2024, Vol. 47, e244705

Error-in-Variables for Special Models Describing Brittle-

Ductile Transition

Error-in-Variables for Special Models

Orlando Zambrano Mendoza

Peter P. Valko

†

James E. Russell

Escuela de Ingeniería de Petróleo. Facultad de Ingeniería, Universidad del Zulia, Apartado postal 4011-A-526.

Maracaibo, Zulia, Venezuela.

Harold Vance Department of Petroleum Engineering Texas A&M University 3116 TAMU College Station, TX

77843-3116

Autor de correspondencia: ozambrano@fing.luz.edu.ve

https://orcid.org/0009-0006-6895-8102

Recepción: 01 noviembre 2023 | Aceptación: 05 octubre 2024 | Publicación: 09 diciembre 2024.

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.

Keywords: cap model; EIV; failure criterion; pore collapse; poroelasticity.

Error-en-las-Variables para Modelos Especiales

Describiendo la Transición Frágil-Dúctil

Error-en-las-Variables para Modelos Especiales

Resumen

En este trabajo se muestra la aplicación de la metodología basada en el método de error en las variables (EIV)

para ajustar la envolvente en el plano de los esfuerzos efectivos principales, considerando la envolvente y describiendo

la transición frágil a dúctil y el colapso de poros (modelo cap). Las ecuaciones paramétricas de la envolvente de rotura

se obtuvieron introduciendo el concepto de poro elasticidad de Biot, representando el estado de esfuerzos por el

esfuerzo efectivo promedio y el esfuerzo equivalente. Una forma algebraica no lineal representa las ecuaciones de la

envolvente, delimitando la región transicional de frágil a dúctil de la envolvente de rotura o cedencia. Para ilustrar la

aplicación de esta metodología, se usó la información de la resistencia frágil y el esfuerzo de cedencia a la

compactación para la arenisca Bentheim y los esfuerzos principales normalizados para diez diferentes areniscas en las

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Rev. Téc. Ing. Univ. Zulia, 2024, Vol. 47, e244705

que se describe el comportamiento de transición frágil a dúctil. Los resultados demuestran que el método EIV

proporciona una forma sencilla de obtener la representación paramétrica de la envolvente en el plano de tensión

efectiva principal, describiendo el mecanismo de deformación de la roca ensayada que implica el efecto de colapso de

los poros.

Palabras clave: colapso de poro; criterio de falla; EIV; modelo cap; poroelasticidad.

Erro variável para modelos especiais que descrevem a

transição frágil-dúctil

Variável de erro para modelos especiais

Resumo

Este artigo mostra a aplicação da metodologia baseada no método error in variables (EIV) para ajustar o

envelope no plano das principais tensões efetivas, considerando o envelope e descrevendo a transição frágil-dúctil e

o colapso dos poros (modelo cap). As equações paramétricas do envelope de rutura foram obtidas através da

introdução do conceito de elasticidade dos poros de Biot, representando o estado de tensões pela tensão efetiva média

e pela tensão equivalente. Uma forma algébrica não linear representa as equações do envelope, delimitando a região

de transição frágil a dúctil do envelope de quebra ou rendimento. Para ilustrar a aplicação desta metodologia, foram

utilizadas as informações de resistência frágil e tensão de rendimento à compactação para o arenito de Bentheim e as

tensões principais normalizadas para dez arenitos diferentes, nas quais o comportamento de transição frágil para dúctil

é descrito. Os resultados mostram que o método EIV fornece uma maneira simples de obter a representação

paramétrica do envelope no plano de tensão efetivo principal, descrevendo o mecanismo de deformação da rocha

testada que envolve o efeito de colapso dos poros.

Palavras-chave: colapso dos poros; critério de reprovação; EIV; Modelo da PAC; poroelasticidade.

Introducción

A failure criterion is usually a relation between the principal effective stresses components and represents a limit

for instability or failure. In the presence of pore fluids, the mechanical properties of porous rocks depend on both pore

fluid pressure and confining pressure according to the Terzagui effective pressures law (Terzaghi, 1943), which can

be used for many rock types and physical properties in its more general form as , where is a constant rock type

(usually referred to as Biot’s constant (Biot, 1941) that takes on different values for different materials but is always

less than or equal to one (Smits et al., 1988), is the effective pressure, is the confined pressure and is the porous

pressure. Rocks fail under stress in different modes depending on the nature and magnitude of the applied stress and

on the properties of the rock itself. Higher-porosity sedimentary rocks are found to compact and lose porosity under

applied stress. When fluids are produced, the effective stress increases in the reservoir and may exceed the compressive

strength of the rock matrix. The rock matrix collapses into the pore space in a mechanism known as “pore collapse”

(Nur and Byerlee, 1971; Addis, 1987). Initially, lower effective stresses cause little or no pore collapse, and

deformation is recoverable (elastic). But as effective stress increases, the reservoir undergoes unrecoverable pore

collapse, accompanied by elastic deformation, which reduces porosity and permeability of the reservoir. The behavior

of some porous rocks undergoes compaction and elastoplastic constitutive relations can model pore collapse. But the

behavior afterward is not perfectly plastic, and a hardening rule is needed for a completed description. The behavior

of porous sandstones undergoing pore collapse and compaction has been found to fit an approximately elliptical “cap

model” (Chen and Mizumo, 1990).

A cap model consists of a failure surface for a perfectly plastic material response and an elliptic strain-hardening

cap that extends isotropically about the hydrostatic axis (Wong et al., 1997) parametric representation describing the

deformation mechanism of the porous rock can be achieving by a method of curve fitting. The objective of this study

is to create the envelope of a cap-type behavior high porosity sedimentary rock in such a way that the derived values

of the effective mean stress and the equivalent stress are considered having experimental error. If the effect of the

principal intermediate stress is considered having not to influence the rock strength (triaxial test) this failure criterion

Error-in-Variables for Special Models

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Rev. Téc. Ing. Univ. Zulia, 2024, Vol. 47, e244705

can be expressed in terms of the effective mean stress and the equivalent stress, which can be represented by

(Zambrano-Mendoza, 2004).





0, 

eqm



(1)

Since this work is an extension of the use of the statistical method of error-in-variables to fit the failure envelope

in the principal stress plane using experimental data from a triaxial test (Zambrano-Mendoza, 2004; Zambrano-

Mendoza et al., 2023), it was utilized the established formulations to developed an appropriate methodology to

describe the deformation mechanism of those sedimentary reservoir rocks, which involve brittle-transition-ductile

behavior in they deformation mechanism.

In a previous earlier review of the use of the statistical method of error-in-variable found the improvement

achieve in the statistical method of EIV since Deming (1943), first formulation. Previously it was reviewing the least

square method and others some variants of the least square method, such as the orthogonal regression method (Keles

and Altun, 2016; Recio-Lopez, 2021). Even though that improvement of the least square method there is still a

dependency of one of the variables. Here it is summarized the different improvements in the error-in-variable statistical

method used in this work (York, 1966; Willianson, 1968; O’Neil et al., 1969; Southwell, 1969; Britt and Luecke,

1973; Peneloux et al., 1976; Reilly and Patino-Leal, 1981; Schwetlick and Tiller, 1985; Valkó and Vajda, 1987;

Liebman and Edgar, 1988; Edgar et al., 1990; Esposito and Floudas, 1998; Anand and Kumar, 2015; Kumar and

Kumar, 2011) as well as described the EIV method of curve fitting and developed a variant suitable for fitting failure

envelopes in the Mohr plane (Zambrano-Mendoza et al., 2003) and in the principal stress plane (Zambrano-Mendoza,

2004; Zambrano-Mendoza et al., 2023) applying it to a well-established set of data.

In this work, it is proposing to use the statistical method of error-in-variables (EIV) to obtain the parametric

representation of the failure envelope in the principal effective stress plane providing an accurate way of obtaining the

parametric representation of the failure envelope describing the deformation mechanism. Next; it is describing the

application of the EIV method of curve fitting in the principal effective stress plane to a previously established set of

data (Klein et al., 2001, Zhang et al., 2000).

Materials and Methods

Application of EIV method to fit the failure envelope in the principal effective stress plane

The group of parametric equations of the failure envelope developed in the principal stress plane can be modified

in a very simple way by introducing the concept of poroelasticity of Biot (Biot, 1941). The state of stress can be

represented by the equivalent stress and the effective mean stress as stress invariants that are independent of

coordinate rotation. These two invariants can be written as (Biot, 1941):





pm p



 3/2 31 (2)

And from von Mises (1913):

231 3J

eq 



(3)

Where ( p

p) is reservoir pore pressure, ( 1



) the major principal stress and ( 3



) the minor principal stress and

ma c

 1



( ma

c) is the matrix compressibility; ( b

c) is the bulk compressibility. For porous sandstones:

bma cc , so 1





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Rev. Téc. Ing. Univ. Zulia, 2024, Vol. 47, e244705

Equations 2 and 3 represent the mean effective stress and the equivalent stress respectively, and ( 2

J) in MPa2,

is the second invariant of the stress deviator tensor. Using those equations and the EIV approach, it was generating an

algebraic solution for the material that will satisfy the equation of the envelope given in implicit form as (Zambrano-

Mendoza, 2004):





ˆ





h (4)

Where







ˆ is the reconciled equivalent stress and







ˆreconciled effective mean stress and







 is the

unknown parameter.

A similar procedure of the EIV approach used in





13 ,



stress plane could be applied. The goal is to find the

optimum parameters, at which the sum of necessary corrections squared:



2

dJ (5)

(J) is minimum, expressing the squared distance





d in terms of the mean effective stress and the equivalent

stress as follow:









2ˆˆ eqieqimimi



 (6)

With







ˆis the ith measurement of the reconciled equivalent stress and







ˆ the ith measurement of the

reconciled effective mean stress, while







is the ith measurement of the equivalent stress and







the ith

measurement of the effective mean stress.

Because the failure envelope is fitting in the principal effective stress plane the EIV algorithm used for the

principal’s stress plane (Zambrano-Mendoza et al., 2023), which can be modified as follows:

EIV algorithm

Thus, vector a, can be expressed as:









jia

eqimi

eqimi hh

























(7)

While vector b is expressed as:









jib eqieqimimi



 ˆˆ (8)

The orthogonality between vector a and b is expressed as:

 





 





ˆ

















eqimi

eqieqi

eqimi

mimi













(9)

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Rev. Téc. Ing. Univ. Zulia, 2024, Vol. 47, e244705

The system of Equations 4 and 9 can be solved simultaneously using either excel VBA or Mathematics Software,

to obtain the reconciled stress





eqm



ˆ at any parameter vector





which have been already defined, evaluating the

objective function. A nonlinear function is considered to represent the failure envelope at the principal effective stress

plane.

Applying the method to special model suitable for describing brittle-ductile transition in yield envelope

Using the concept of effective stresses state in the plane or invariants, can be obtain the failure or yield envelope

considering pore collapse effect. From the EIV approach it can be generating an algebraic solution for the brittle-

ductile transition behavior of the material that will satisfy Equation 4. The parametric representation proposed to fit

the failure envelope in the principal effective stress plane is given by a four parameters equation fitting the envelope

in such a way that describe the deformation mechanism drawing a curve, which connect the brittle region to the ductile

region defining the cap model. Then, Equation 4 can be written as:





ˆˆˆˆ

ˆ2

210 

















 eqieqimimieqimi



(10)

















ˆˆ

2321 









 mimieqieqieqimi



(11)

The 0



 , 1



 ,2



 and 3



 are the unknown coefficients, while the rest of parameters have been already defined

previously. Solving the system of Equations 10 and 11 it can be obtaining the squared distance by Equation 6. Due to

the cumbersome nature of i3



roots the squared distance, 2

d is not showed for the equations.

Results and Discussions

To illustrate the applicability of the EIV method to fit the failure envelope for special model describing brittle-

ductile transition with pore collapse effect in the principal effective stress plane, here it is used the previously set

experiments (Klein et al., 2001; Zhang et al., 2000). A nonlinear parametric function was considered to fit the

envelope. The goal was to obtain the failure envelopes for those rocks in the principal effective stress plane, using all

the available information. Table 1 contains the optimal parameters for normalized data for 10 sandstones. The

parameters describe the failure envelope located nearest to the respective pairs of the mean-effective and equivalent-

stress measurements obtained from the data measured under various confining pressures for the critical pressure

normalized data for 10 different sandstones, while Table 2 contains the optimal parameters for Bentheim sandstone.

The parameters describing the yield envelope located nearest to the respective pairs of the mean-effective and

equivalent-stress measurements obtained from the measured data, including critical stress at the onset of dilatancy and

compactive yield stress at the onset of shear-enhanced compaction (11 test samples) and considering only compactive

yield stress defining a cap model (7 test samples).

Table 1. Optimal parameters determined from the EIV method considering pore collapse effect for a

normalized data.

Nonlinear 4 parameters model





ˆˆˆˆ

ˆ2

110 eqeqmmeqm

























”0



”1



”2



”3

Test rock

(

)

(

)

(

)

(

)

Normalized data

from 10 sandstones

5.1E-13 20.7 -18.2 10.4

35 test samples,







= reconciled measured mean effective stress,







ˆ= reconciled measured equivalent

stress,



”0,



”1,



”2,



”3= unknown coefficients.

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Rev. Téc. Ing. Univ. Zulia, 2024, Vol. 47, e244705

Table 2. Optimal parameters determined from the EIV method for bentheim sandstone considering pore

collapse effect.

Nonlinear 4 parameters model





ˆˆˆˆ

110 eqeqmmeqm

























”0



”1



”



”3

Test rock

MPa

(

)

MPa

Bentheim

sandstone

7.9E-13 1717.4 -3.7 652

Bentheim

sandstone

1.4E+5 629.6 -1.9 562.4

a: 11 test samples, b: 7 test samples,







= reconciled measured mean effective stress,







ˆ= reconciled

measured equivalent stress,



”



”



”



”

= unknown coefficients.

The possible use of these parameters is within a computational algorithm, where a stress state needs to be tested

for failure. The listed parameter, can minimize the likelihood of making a wrong judgment (i.e. declaring a failure

state as safe or a safe state as failure). The standard deviation is the sum of the squared distances between the failure

envelope and each pair of measurements (objective function values); the standard deviation is given by:

.

n

(12)

The standard deviations result to be 0.06 for the normalized data for 10 different sandstones, while the standard

deviation for Bentheim sandstone is 12.2 MPa for whole envelope and 10.2 MPa for the case when considering only

compactive yield stress to define a cap model.

Figure 1 shows the parametric representation of the envelope of the normalized principal stresses describing

brittle/ductile transition behavior for 10 different sandstones (Zhang et al., 2000). The solid black line represents the

optimal parametric representation of the yield envelope.

Error-in-Variables for Special Models

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Rev. Téc. Ing. Univ. Zulia, 2024, Vol. 47, e244705

Figure 1. Parametric representation of the failure envelope in the principal stress plane considering pore

collapse effect using EIV method for normalized data for 10 sandstones (nonlinear four-parameters model). Where:

effective mean stress

 



and the equivalent stress

 



Figure 2 shows the parametric representation of the envelope including critical stress at the onset of dilatancy

and compactive yield stress at the onset of shear-enhanced compaction (11 test samples) and considering only

compactive yield stress defining a cap model (7 test samples) for Bentheim sandstone (Anand and Kumar, 2015). It

should be emphasizing that according with some previous work a 400 MPa it is the critical pressure.

Figure 2. Parametric representation of the failure envelope in the principal stress plane considering pore

collapse effect using EIV method for Bentheim sandstone (nonlinear four-parameters model) . Based on

conventional triaxial test data. (11 test samples) dark curve and (7 test samples) ligth curve. Where: effective mean

stress

 



and the equivalent stress

 



Conclusions

The EIV method can provide a simple way to obtain the parametric representation of the envelope in the principal

effective stress plane describing the deformation mechanism of the tested rock involving pore collapse effect. The

parameters describing the yield envelope located nearest to the respective pairs of the mean-effective and equivalent-

stress measurements.

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Editor Asociado: Wagdi Naime Yehia

Departamento de Ingeniería vial de la

Facultad de Ingeniería de la Universidad Central de Venezuela (UCV)

Caracas, 1041, Zulia, Venezuela

wagdin@gmail.com

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Volumen 47. Año 2024, Edición continua

Esta revista fue editada en formato digital y

publicada en diciembre 2024, por el Fondo

Editorial Serbiluz, Universidad del Zulia.

Maracaibo-Venezuela

www.luz.edu.ve

www.serbi.luz.edu.ve

www.produccioncient

ificaluz.org