REVIEW OF SOME POROELASTIC EFFECTS IN ROCK MECHANICS

A. H-D. Cheng •, Y. Abousleiman • and J.-C. Roegiers •

IDepartment of Civil Engineering, University of Delaware, Newark, Delaware 19716 * School of Petroleum and Geological Engineering

the University of Oklahoma, Norman, Oklahoma 73019

INTRODUCTION

The presence of a free-moving fluid in rock modifies its mechanical behavior. In particular, it transforms an otherwise time-independent problem to a time-dependent one through fluid diffusion. The material behaves as a stiffer one under fast than for slow loading. The induced fluid pressure generally brings stress relieve or volume dilation to the solid material. The presence of fluid pressure can also modify the Terzaghi effective stress, which is keenly tied to the onset of some failure mechanisms. The purpose of this paper is to review through examples the manifestation of the pore pressure effect (poroelasticity) in a number practical occasions. A good grasp of these physical mechanisms can equip engineers with the necessary insight to perform prognosis of poroelastic effect for their problems on hand.

POROELASTIC COEFFICIENTS

Based on the continuum mechanics theory it is well established that an isotropic elastic solid can be characterized by two independent material coefficients. For poroelastic mate- rial, a continuum mechanics analysis reveals that there exist four independent constitutive coefficients. Two of the poroelastic coefficients, the shear modulus G and the drained bulk modulus K, are tied to elastic properties of the solid skeleton. The third coefficient, known as the Blot effective stress coefficient a, is also a property of the solid constituent. Only the last coefficient, the Biot modulus M, incorporates the fluid elastic property.

For general anisotropic poroelasticity, there are 28 coefficients. Twenty one of them are elastic constants of the drained skeleton. Six are Biot effective stress coefficients in a

symmetric tensor form aij. The last coefficient M is again the only one which contains fluid property.

Biot effective stress coefficient: It is established that the volumetric deformation of a fluid saturated rock is proportional not to the total compressive stress AP, but to the 'Blot effective stress' Ap -- aAp. It is associated with the bulk modulus of the skeleton, K, and that for the solid constituent, K,, as follows:

K

a- 1-- K'• (1) Hence a measurement of K•, which can be accomplished by an unjacketed test in the labora- tory, gives an indirect estimate of a. This statement is also true for anisotropic material: a measurement of K•, combined with the drained anisotropic elastic moduli, defines the whole cqj tensor. This fact is quite convenient for the laboratory measurement of anisotropic poroelastic properties.

529

Drained and undrained responses: While a drained test allows us to measure the elas- tic properties of the skeleton, an undrained test will sample the combined strength of solid and fluid. An undrained test can be performed by wrapping the sample in an impermeable membrane. Similar to the drained test, we can define a 'undrMned bulk modulus' K• accord- ing to the applied load and the resultant deformation. It is related to the micromechanical quantities of solid and fluid as

(K•-K)• (2) K• = K + Ki Ki(K• - K) + $K•(K• - Ki) where K I is fluid bulk modulus and • the porosity.

Skempton effect: When a specimen is subjected to an incremental average compressive stress AP under undrained condition, the ratio between the pore pressure rise Ap and the average stress is associated with the Skempton pore pressure coet•cient B and the undrained

ß Poisson ratio y• as:

Ap = BAP

zxp = 2B(1 + v)/xp (Sb) 3

Ap = B(l+v,) AP (3c) 3(1 - •.)

respectively for three-dimensional, plane-strain and one-dimensional problems.

Poroelastic stress coefficient: Analogous to thermal stress, there exist a pore pressure induced stress. Under certain conditions [1], the stress generated due to pore pressure can

'be expressed as

axx "1 c ayy "1 c azz -- -4rip (4a) axx -}' ayy -- -2r/p (4b)

axx = -2r/p (4c)

respectively for three-dimensional, plane-strain and one-dimensional problems.

Diffusivity coefficient

The diffusivity coefficient together with a characteristic length scale generally controls the time scale of poroelastic phenomena. Figure 1 illustrate the relevant time scale as a function of geometric size and diffusivity coefficient for a few geomaterials.

EXAMPLES

Mandel problem: The classical Mandel problem involves a specimen sandwiched between two rigid, impermeable plates. As predicted by the Skempton effect, an initial uniform pressure rise results, which will eventually be dissipated. According to a known solution [2], the pressure response near the center is non-monotonic, as demonstrated for a cross-section as shown in Figure 2. The phenomenon of momentary increase in pressure before its full

530

dissipation has been observed in the laboratory as well as in the field. This phenomenon, which is not predicted by the simple pressure diffusion, can be explained from the vantage point of poroelasticity as the effective material softening at the edges and the subsequent load transfer to the center region.

Effective cornpressive stress around borehole: Consider a borehole instantaneously drilled in a porous rock that is characterized by a non-hydrostatic horizontal stress field. The pore pressure, both that existing in-situ, and that generated by the excavation, tends to protect the borehole against compressive stress in the form of a reduced Terzaghi effective stress [3]. Figure 3 demonstrates the amount of 'shielding' provided as compared to the final drained state, at the dimensionless time t* = 0.1.

Effective tensile stress around borehole: The sudden drop of pore pressure in the wellbore can cause a tensile region near the wall. Figure 4 illustrates the tensile region (positive ' t* art ) at = 0.01 as the upper diagram, and the final steady state which is entirely under compression as the lower diagram. This type of poroelastic effect has been utilized for the industry application of coal cavitation.

Shear failure around borehole: Figure 5 presents the ratio of maximum shear stress Sm to the mean effective compressive stress P'•. The tensile regions for cr'r• is blocked out by marking them in light tone. The dark shading corresponds to the area most susceptible to shear failure. We observe that the peak value is somewhere inside the formation.

Hydraulic fracturing: Poroelastic effect in hydraulic fracturing manifests in the form of a poroelastic stress (back-stress), and a pressure dependent leakoff. Figure 6 depicts the fracture length of a P KN fracture simulated by a numerical model [4, 5]. The solid line represents a pressure-dependent leak. The dotted lines are upper and lower bounds estimated based on the classical pressure-independent Carter leakoff.

Acknowledgments--The effort is supported by the National Science Foundation "Rock Mechanics Research Center" grant to the University of Oklahoma and the Rock Mechanics Consortium, with a subcontract to the University of Delaware. A large portion of the review is based on our earlier collaboration with Emmanuel Detournay. His contribution is duly acknowledged.

References

[1] Detournay, E. and Cheng, A.H-D., "Fundamentals of poroelasticity," to appear as Chapter 5, Compre- hensive Rock Engineering: Principles, Practice • Projects, Vol. œ, ed. J.A. Hudson, Pergamon Press, 1993.

[2]

[3]

Cheng, A.H-D. and Detournay, E., "A direct boundary element method for plane strain poroelasticity", Int. J. Numer. Anal. Meth. Geomech., 12, 551-572, 1988.

Detournay, E. and Cheng, A.H-D., "Poroelastic response of a borehole in a non-hydrostatic stress field", Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 25, 171-182, 1988.

[4] Detournay, E., Cheng, A.H-D. and McLennan, J.D., "A poroelastic PKN hydraulic fracture model based on an explicit moving mesh algorithm," J. Energy Resour. Technology, ASME, 112, 224-230, 1990.

[5] Abousleiman, Y.N., "A poroelastic PKN model with pressure dependent leakoff and formation perme- ability determination," Doctoral Dissertation, Univ. Delaware, 1991.

531

10-' 1 10 10 • 1• % 1Q" 10" 10' 10 • 10' to tsec)

Fig. 1. Characteristic time for various rocks.

?'"';:,.;•. o x/a

-- - f ' ':" !::'• • O. S

•'-:%•',' :k. ':. "--'::-' ..... .-.':' .-.'.:' ," ':J 20. 3

%•:•.:?•;:.:.;:.?.]:::;•?;;•:.;?;•:;:?•?•i;•,':?J:?"•½• • • '.•::•'•:•:L:;½:K4•::2:.::::•'•:¾':-:?'.½:'":,"' ' ' ' ' :?:'"

,,S

Fig. 2. Pore pressure history for Mandel problem.

2

--.

o. 4 .•

O.

O.

O.

2

Fig. 3. Evolution of •e as perturbation from the final state. ct/a 2 = 0.1.

1

0.75

0.5

0.25

-0.25

-0.5

-0.75

-1

1

0.75

0.5

0.25

1.2 1.4 1.6 '-'"'TT'8----•2 r/a

r/a

.1•. 6 1.8 2 -0.25

-0.5

-0.75

-1

Fig. 4. Effective radial stress (Cr•rr/Po), t* -- 0.01, c•.

1.5

0.5

.i-.:.:¾.!i!:E:- '-

0 0.5 i 1.5 2

Fig. 5. Distribution of Sm/P•m around the borehole at t* = 0.1.

120

0 0

80.

40-

2000

%.

\

i

,, ,ooo eooo .ooo

t (sec) Fig. 6. Fracture length history from a PKN fracture simulator.

532