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A Simplified Analytical Elastoplastic Embedment Depth Model of Proppant Based on the Drucker–Prager Yield Criterion

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Abstract

In deep shale gas reservoir exploration, elastoplastic embedment plays an increasingly important role in analyses of the conductivity of fracture networks because it is affected by high temperature, high in-situ stress. To obtain a full understanding of proppant embedding behavior, an analytical model of the embedment depth of a single spherical proppant was developed based on the Drucker–Prager (D–P) yield criterion and shale elastoplastic mechanical behavior. Shale indentation tests with rigid balls with diameters of 0.8 mm and 1.0 mm were conducted to validate the reliability of the theoretical model. Compared with the Hertz, Thornton and non-linear models, the new model developed in this paper was in better agreement with embedment experimental data. Each parameter of the new model has clear physical meaning and can be obtained by a series of triaxial compression tests. An Analysis of Variance (ANOVA) method was applied to discuss the sensitivity of the model to the parameters, and the results showed that the strength parameters α and Y contributed the most to, and had nonlinear effects on, the proppant embedment depth.

Highlights

  • According to the shale mechanical properties, the contact area between a single spherical particle and a semi-infinite space can be divided into elastic, strain hardening and fully plastic region approximately.

  • An analytical elastoplastic embedment depth model for a single rigid spherical particle was developed based on the Drucker-Prager yield criterion and total strain theory.

  • The yielding parameters have a dominant and nonlinear effects on the proppant embedment depth.

  • It is more appreciated that applying a suitable yield criterion which can accurately describe the yielding process of rock-like materials to discuss the interaction between proppant and shale.

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Data Availability

All data analyzed during this work are included in this published paper and are available from the corresponding author on reasonable request.

Abbreviations

r :

A radial distance measured from the center of the contact surface

a :

Contact radius

\(p_m\) :

The maximum contact pressure at the center of the contact area

R :

Radius of proppant particle

δ, δ y, δ pp :

Proppant embedment depth at the centerline, the depth at the initial yield state, and the depth at the initial state of the ultimate strength stage

\(\sigma_r\), \(\sigma_\theta\), \(\sigma_z\) :

Normal stresses in radial, circumferential and vertical directions, respectively

\(\tau_{rz}\), \(\tau_{r\theta }\),\(\tau_{z\theta }\) :

Corresponding shear stresses

I 1, J 2 :

Spherical tensor invariant and the second deviator tensor invariant, respectively

α, Y :

Material yield parameters

\(\sigma_{eq}\) :

Equivalent stress under the condition of the D–P yield criterion

\(\xi\), \(\xi_0\) :

Dimensionless embedment depth and initial yield dimensionless embedment depth, respectively

E, v :

Young’s modulus, and Poisson’s ratio, respectively

\(p_y\) :

Initial yield contact pressure

a y, a p ,a pp :

Initial yield contact radius, elastoplastic strengthening region radius, and fully plastic region radius, respectively

\(\beta\), \(\beta_{py}\), d :

Material parameters

\(\sigma_y\) :

Initial yield strength of shale

\(\sigma_i\), \(\varepsilon_i\) :

Equivalent stress and equivalent strain, respectively

\(\varepsilon^*\) :

Virtual strain

F, F y :

Contact force and initial yield contact force, respectively

C v :

\(p_m {/}\sigma_{eq}\)

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Acknowledgements

This work was financially supported by the National Natural Science Foundation (No. 11872258), and Creative Project of Engineering Research Center of Alternative Energy Materials & Devices, Ministry of Education, Sichuan University (Grant No. AEMD 202206). We would like to thank our colleagues for their kind efforts and valuable comments in improving this paper.

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Authors and Affiliations

  1. Institute of New Energy and Low-Carbon Technology, Sichuan University, Chengdu, 610207, China

    Bo He, Lingzhi Xie, Yongxin Che & Jun Liu

  2. Engineering Research Center of Alternative Energy Materials & Devices, Ministry of Education, Sichuan University, Chengdu, 610065, China

    Bo He, Lingzhi Xie & Jun Liu

  3. College of Architecture and Environment, Sichuan University, Chengdu, 610065, China

    Yao Zhang

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  1. Bo He
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Correspondence to Lingzhi Xie.

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He, B., Xie, L., Che, Y. et al. A Simplified Analytical Elastoplastic Embedment Depth Model of Proppant Based on the Drucker–Prager Yield Criterion. Rock Mech Rock Eng 56, 6207–6217 (2023). https://doi.org/10.1007/s00603-023-03394-0

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