Stress relaxation during creep of rocks around deep boreholes

doi:10.1016/S0020-7225(00)00060-4

International Journal of Engineering Science

Volume 39, Issue 7, May 2001, Pages 737-754

(Communicated by E. SOÓS)

https://doi.org/10.1016/S0020-7225(00)00060-4 Get rights and content

Abstract

The variation of stress during creep convergence of a deep borehole excavated in rock salt is examined. A non-associated elasto/viscoplastic constitutive equation is used to describe both compressibility and/or dilatancy during transient and steady-state creep, as well as evolutive damage possibly leading to failure. An in-house FEM numerical method is used for this purpose. The variation in time of radial convergence of the borehole walls and of the stress state (at various distances from the borehole surface) is illustrated by several figures. The significance of these variations for long-term stability is discussed.

Introduction

The stress distribution around a deep borehole has been studied by very many authors. The problem has a great importance for petroleum industry, mainly for welbore stability [21] as well as for mining industry [19], [20]. Various initial assumptions have been chosen, as well as several mechanical models to describe the behavior of the rocks. Quite often linear elasticity was used together with the assumption of plane stress state (see [6], [10], [16], among others). However, for very deep boreholes, if the two ends are disregarded, the problem can be approached as a plane strain problem, involving all stress components. Also, in order to describe creep and wall convergence rheological models are necessary. Thus, Massier and Cristescu [14] have used simplified viscoelastic and viscoplastic models to describe creep around a borehole. More general viscoelastic model describing volumetric creep as well, has been used by Massier [12], [13]. These models can describe compressibility only. For rocks which are compressible for some stress states, but dilatant for some other stress states, elasto/viscoplastic models were used ([1], [2]; see also [3], [11]).

The stress state just after the drilling is obtained as an elastic instantaneous solution. With a viscoplastic non-associated model this allows us to find out afterwards where around the borehole the rock becomes dilatant, where compressible and where a possible failure can be expected. In the same manner the beginning of the rock creep can be studied, the location where a fast creep will take place as well as where evolutive damage is to be expected (see e.g. [5, Chapter 8] and [17]). A variety of cases were considered in the above-mentioned papers, what concerns the type of rocks, depth, far field stress distribution, inner pressure, etc. The formulae, used to describe creep of a compressible/dilatant viscoplastic rock, with the assumption that the stresses remain constant during creep, are very simple and thus, easy to apply. They may describe well the beginning of the creep process but for a longer period of time and for great depth one has to take into account that during long periods of time following the drilling, the stresses may vary, due to non-uniform stress distribution, the presence of far field stresses and of a borehole which changes geometry.

In the present paper, we are dealing with the stress variation during creep of rock salt around a deep borehole. Various far field stress distributions are considered, several depths and various pressures exerted on the inner surface. Since the system of equations governing this problem is quite involved, numerical methods (FEM) are used, as developed by Ionescu and Sofonea [8] and Paraschiv-Munteanu [18]. Another numerical approach to the same problem, also for rock salt, is due to Glabisch [7] using someother FEM code. A study of stress relaxation in rock salt surrounding a deep borehole using FEM, has been done also by Jin and Cristescu [9]; another non-associated elastic/viscoplastic constitutive equation for rock salt than in the present paper was used, in which only the transient term was taken into account. The stress variation obtained shows a decrease in time of octahedral shear stress with respect to the elastic solution, in the neighborhood of the orifice, but an increase at farer distances.

Finite differences were also used to study the stress variation during creep of other rocks: for andesite [1], granite [2], coal [3], where, however associated elastic/viscoplastic constitutive equations have been used and only some aspects of stress relaxation have been studied. However, as a general conclusion, for very long time intervals one has to take into account the stress variation during creep, in order to correctly predict stability, wall convergence, damage, and possible failure taking place after long time intervals following the drilling.

Section snippets

The constitutive equation

The constitutive equation is of a non-associated elastic-viscoplastic type [3]. The reference configuration, with respect to which the strains must be estimated, is the state in situ before excavation, there where the future excavation is envisaged. Using standard notation we have for the rate of deformation tensor $ε ̇ = σ ̇^{R} 2G + 1 3K − 1 2G σ ̇^{R} 1 +k_{T} 1− W^{I} (t) H(σ) ∂ F ∂ σ (σ)+k_{S} ∂ S ∂ σ (σ),$ where G and K are elastic moduli which may depend on the invariants of stress and strain and possibly on the damage of rock,

Mathematical formulation of the problem

We show shortly how can be found the stress distribution around a circular vertical cavity excavated in rock salt. We formulate the problem as it will be used in the numerical integration described below. If the cavity has the initial radius a and if on its walls $Γ_{1} = (a,θ) | θ∈[0,2 π),$ a pressure p is acting (due to various reasons and which may be constant or variable), the stress state just after excavations is $σ ̃^{S}_{rr} (r)= N ̃ +(1− N ̃) a^{2} r^{2} (p−σ_{h})+σ_{h},$ $σ ̃^{S}_{θθ} (r)= N ̃ −(1− N ̃) a^{2} r^{2} (p−σ_{h})+σ_{h},$ $σ ̃^{S}_{zz} (r)= 3K−2G G+3K N ̃ (p−σ_{h}$

The numerical approach

For the problem , , , , , we determine a numerical solution based on some results presented by Ionescu and Sofonea [8] using a complete implicit method for integration in time (see [18]).

If (u,σ^R) is the solution of the problem , , , , , then we determine: $u =u− u ̃, σ = σ^{R} − σ ̃^{R},$ such that $u (t,ma)=0 (∀)t>0,$ $Div σ (t,r)=0 (∀)t>0 and r∈[a,ma],$ $σ (t,a) n =0 (∀)t>0.$

Thus, we have to solve the problem.

Find the displacement function $u : R_{+} ×[a,ma]→ R$ , the stress function $σ : R_{+} ×[a,ma]→ S_{3}$ and the irreversible stress work function

Numerical results

In order to construct the space V_h, referring to functions $u$ , $σ$ , W^I which depend of the variable r only (these functions are independent on the variable θ), we will consider that the interval [1,m] was divided into 100 linear elements with m=10 or m=15. The time step was considered 1/100 dimensionless time unit or 1/200 dimensionless time unit. The dimensionless time T is time divided by the constant k_S. In numerical calculus we consider $k_{T} =5×10^{−6} s^{−1}$ and $k_{S} =3×10^{−5} s^{−1}$ .

If we assume that after

Conclusions

The conclusions for displacement and stress distribution around a borehole excavated in rock salt are compared: the elastic solution (E), the simplified solution assuming stress constancy during creep (S), and a numerical solution (N) taking into account the stress variation during creep closure of the borehole. It is shown that at relative small depth, and ratio of far field stress n=1, the solutions S and N are quite close and thus the first estimation of the borehole convergence can be

References (21)

N. Cristescu
Viscoplastic creep of rocks around horizontal tunnels
Int. J. Rock Mec. Min. Sci. Geomech. Abstr.
(1985)
J. Jin et al.
An elastic/viscoplastic model for transient creep of rock salt
Int. J. Plasticity
(1998)
N. Cristescu
Fluage, Dilatance et/ou Compressibilité des Roches autour des Puits Verticaux et des Forages Petroliers
Revue Francaise de Geotechnique
(1985)
N. Cristescu
Rock Rheology
(1989)
N. Cristescu, in: N.D. Cristescu, G. Gioda (Eds.), Viscoplasticity of Geomaterials, Visco-plastic Behavior of...
N.D. Cristescu et al.
Time Effects in Rock Mechanics
(1998)
W. Dreyer
The Science of Rock Mechanics. Part 1: The Strength Properties of Rocks
(1972)
U. Glabisch, Stoffmodell fur Grenzzustande im Salzgestein zur Berechnung von Gebirgshohlraumen, Ph.D. Dissertation,...
I.R. Ionescu et al.
Functional and Numerical Methods in Viscoplasticity
(1993)
A.R. Jumikis
Rock Mechanics
(1979)

There are more references available in the full text version of this article.

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Stress relaxation during creep of rocks around deep boreholes

Abstract

Introduction

Section snippets

The constitutive equation

Mathematical formulation of the problem

The numerical approach

Numerical results

Conclusions

Int. J. Rock Mec. Min. Sci. Geomech. Abstr.

Int. J. Plasticity

Fluage, Dilatance et/ou Compressibilité des Roches autour des Puits Verticaux et des Forages Petroliers

Revue Francaise de Geotechnique

Rock Rheology

Time Effects in Rock Mechanics

The Science of Rock Mechanics. Part 1: The Strength Properties of Rocks

Functional and Numerical Methods in Viscoplasticity

Rock Mechanics