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The Modeling of Time-Dependent Deformation and Fracturing of Brittle Rocks Under Varying Confining and Pore Pressures

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Abstract

A numerical hydro-mechanical model for brittle creep is proposed to describe the time-dependent deformation of heterogeneous brittle rock under constant confining and pore pressures. Material heterogeneity and a local material degradation law are incorporated into the model at the mesoscale which affects the mechanical behavior of rocks to capture the co-operative interaction between microcracks in the transition from distributed to localized damage. The model also describes the spatiotemporal acoustic emissions in the rock during the progressive damage process. The approach presented in this contribution differs from macroscopic approaches based on constitutive laws and microscopic approaches focused on fracture propagation. The model is first validated using experimental data for porous sandstone and is then used to simulate brittle creep tests under varying constant confining and pore pressures and applied differential stresses. We further explore the influence of sample homogeneity on brittle creep. The model accurately replicates the classic creep behavior observed in laboratory brittle creep experiments. In agreement with experimental observations, our model shows that decreasing effective pressure, increasing the applied differential stress, and decreasing sample homogeneity increase the creep strain rate and decrease the time-to-failure, respectively. The model shows that complex macroscopic time-dependent behavior can be explained by the microscale interaction of elements. The fact that the simulations are able to capture a similar hydro-mechanical time-dependent response to that of laboratory experiments implies that the model is an appropriate tool to investigate the complex time-dependent behavior of heterogeneous brittle rocks under coupled hydro-mechanical loading.

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Abbreviations

A :

Material constant

D :

Damage variable

e f :

Energy released from a failed element

e ij :

Strain deviator of the elastic strain components

E, E 0 :

Young’s moduli of damaged material and undamaged material

f i :

Body forces per unit volume

F 1, F 2 :

Tensile and shear damage threshold functions

f t0, f c0 :

Uniaxial tensile strength and uniaxial compressive strength

G :

Shear modulus

h :

The convective heat transfer coefficient

k :

The coefficient of permeability

K′, K s :

Bulk modulus of the porous medium, and effective bulk modulus of the solid constituent

m :

A fraction constant

n :

Stress component of greater than one

n i :

The number of failed elements in the ith step

N :

Total number of elements

P, P eff :

Fluid pore pressure and effective pressure

R :

Universal gas constant

r :

Constitutive coefficient

S ij :

Stress deviator tensor of the elastic stress components

T :

Absolute temperature

U :

Creep activation energy

ν e :

The volume of single element

V f, V :

The volume of failed elements, the total volume of all elements

α :

Biot’s efficient

φ :

Porosity

β 1 :

The bulk modulus of fluid

ε :

Total strain

ε 1, ε 3 :

The maximum and minimum principal strain

ε C :

Creep strain

ε e :

Elastic strain

ε ij :

Small strain tensor

ε t0, ε c0 :

The maximum tensile and compressive strain

ε tr, ε cr :

The residual tensile and compressive strain corresponding to residual strength

ε tu :

The ultimate tensile strain corresponding to complete damage

K :

The intrinsic permeability in a general continuum

ρ l :

The fluid density

u, u 0 :

Scale parameter of an element, the average element parameter

u i :

Mechanical parameter of the element i

μ l :

The dynamic fluid viscosity

δ ij :

Kronecker delta

λ :

Lame’s constant

ζ :

Residual strength coefficient

η :

Ultimate tensile strain coefficient

ρ :

Bulk density of medium

χ :

Homogeneity index

ϕ :

Angle of internal friction

ω i :

Random numbers ranging from 0 to 1

ψ :

The ratio of the volume of failed rock to total volume of rock

ξ :

The factor that reflects damage-induced permeability increase

σ e :

Effective stress

σ ij :

Total stress tensor

σ ii :

Average stress

σ 1, σ 2 :

The maximum and minimum principal stress

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Acknowledgements

The support provided by Natural Science Foundation of China (Grant Nos. 51474051, 41672301 and 51761135102), National Basic Research Program (973) of China (Grant No. 2014CB047100), Fundamental Research Funds for the Central Universities of China (N150102002), and the Partenariats Hubert Curien (PHC) Cai Yuanpei Grant (Grant No. 36605ZB) are highly acknowledged. The comments of two reviewers helped improve the clarity of this manuscript.

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Authors and Affiliations

  1. Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang, 110819, People’s Republic of China

    Tao Xu & Guanglei Zhou

  2. Center for Rock Instability and Seismicity Research, Northeastern University, Shenyang, 110819, People’s Republic of China

    Tao Xu & Guanglei Zhou

  3. Géophysique Expérimentale, Institut de Physique de Globe de Strasbourg (UMR 7516 CNRS, Université de Strasbourg/EOST), 5 rue René Descartes, 67084, Strasbourg Cedex, France

    Michael J. Heap & Patrick Baud

  4. State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Shengqi Yang

  5. Geotechnical Institute, TU Bergakademie Freiberg, Gustav-Zeuner-Str. 1, 09596, Freiberg, Germany

    Heinz Konietzky

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  1. Tao Xu
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  2. Guanglei Zhou
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Xu, T., Zhou, G., Heap, M.J. et al. The Modeling of Time-Dependent Deformation and Fracturing of Brittle Rocks Under Varying Confining and Pore Pressures. Rock Mech Rock Eng 51, 3241–3263 (2018). https://doi.org/10.1007/s00603-018-1491-4

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