Int. J. Rock Mech. Min. Sci. & Geomech.Abstr. Vol. 18, pp. 429 to 435, 1981 Printed in Great Britain

0148-9062,81/050429-07502.00/0 Pergamon Press Ltd

Effect of Pore Pressure and Confining Pressure on Fracture PermeabilityJ. B. WALSH*

The effective pressure pc for measurements of fluid permeability ( k ~ is shown to be (Pc - spp) where pc is confining pressure, pv is pore pressure, and s depends on the topography of the fracture surfaces and rock type. Measurements of flow through simulated fractures by Kranz et al. [,1-1 show that s can vary between 0.5 and 1.0. The analysis here suggests that ( k ) ~ should be linearly related to In pc. Data from studies by Kranz et al. I'1], Brat & Stesky [-2], and Jones 1,3]follow the theoretical relationship.

INTRODUCTION The permeability of a fracture to the flow of fluids decreases under increasing compressive stress because of two factors. In the first place, the aperture decreases under increasing compression, and the resistance to flow increases because of the smaller cross-sectional area. Also the number of points of contact between asperities on the fracture surfaces, and their area, increase under higher compressive stresses, and the resistance increases accordingly because of the longer and more tortuous fluid path. Fracture permeability may also vary with fluid pressure. Increasing the fluid pressure causes the fracture to open, and the permeability increases because of the larger aperture and the decreased area of contact between the surfaces. The permeability of fractures and its variation with fluid pressure and tectonic stress is a matter of considerable interest, and several studies, primarily experimental, have been made. A review of the literature on the subject is given by Iwai [4], and Gale 1,5], and experimental work carried out since then is described by Trimmer et al. 1,6], Kranz et al. 1-1-1, and Jaeger & Cook I-7]. A few field experiments on the effect of stress on fluid flow in fractures have been reported (for a survey, see Brace [,81). I have been studying theoretical aspects of fluid flow in fractures with the purpose of determining the fundamental joint parameters which control the process. I took as a starting point a recent analysis 1,9] of 'joint stiffness', i.e. the normal stress required for a unit change in separation between the surfaces. In this analysis, the fracture was assumed to be two rough surfaces, each with random topography. The stiffness of such a fracture model increases as normal compressive stresses are increased because more asperities come into contact. The theory showed that fracture stiffness * Department of Earth and PlanetarySciences, MassachusettsInstitute of Technology,Cambridge,MA 02139, U.S.A.

should increase linearly with increasing normal stress, and this relationship was found to hold to a good approximation in the one field experiment for which we could obtain data. Engelder & Scholz [,21] and Tanoli et al. 1,22] have since compared the theoretical relationship between stress and stiffness with data from experiments in the laboratory, and they find good agreement both for mechanically-prepared surfaces and for natural fractures. Goodman 1,10] on the basis of his examination of experimental data, proposed an empirical relationship which is equivalent to the one that we derived from theory. All available evidence suggests, therefore, that the deformation of fractures is adequately described by the simple model developed by Walsh & Grosenbaugh [-9]. The model is also suited for studying fluid permeability because it includes the essential elements of the process: the open area between the fracture surfaces is continuous and the effects of both the aperture of the flow channels and the obstacles formed by asperities in contact can be studied. Only the effect of applied normal stress on fracture deformation is considered in the simple theory discussed above, and the additional effect of pore pressure must also be considered in practical problems. Effective stress, that is, those combinations of applied stress and pore pressure which produce the same effect irrespective of the individual values of applied stress or pore pressure, was considered by Nur & Byerlee [11] and Robin 1,12]. Their discussion seems to have been neglected in recent descriptions of experimental results, and so I reconsider the theoretical development in the analysis below and apply the results to data published since their article. ANALYSIS

Fluid flow Consider a differential element of the flow channel between the fracture surfaces, as shown in Fig. I. Flow

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J.B. Walshq,r+ ('B %/'b y) dy

tive 'flow conductance' ~,ik ), is (k) = - - k l+:~ (5)

where k is conductance of the fracture with no asperities and :( is the ratio of the contact area to the total area of the fracture. Fluid conductance k in (5) is, by analogy to heat flow,

k = q/(?,p/ax)which, from (1) isFig. 1. Flow is two-dimensional in the x-y plane, and resistance is due to drag on the upper and lower surfaces. Flow q. and qy in the x and y-directions changes with distance because of changes in aperture.

(6)

k = (2aa/3#) Introducing (7) into (5), we find (k) = - + 11--~t

(7)

(2aa/3#).

(8)

resistance at lateral boundaries is neglected because the wetted area of these boundaries is assumed to be much less than the plan area of the flow channels. The equation governing one-dimensional laminar flow qx between surfaces separated by a small (compared with lateral dimensions) distance 2a can be found in elementary texts:

Changing the normal stress on the fracture changes both the aperture a and the contact area, as characterized by :~; differentiating (8) with respect to applied stress, p, and rearranging gives: dp = a ~ . (9)

qx = (2a3/3p)(aP/dx)

(1) Walsh & Grosenbaugh [9] show for a fracture surface having approximately random surface topography that

where (dp/dx) is the pressure gradient and/~ is viscosity. For an incompressible fluid and incompressible rock, we see (refer to Fig. 1) that conservation of mass requires that

da/dp = x,/2 hip

(10)

dqxdxor

aqy+-~-y= 0

(cg/dx)(a 3 dp/dx) + (d/dy)(a 3 dp/Oy) = 0.

(2)

where h is the r.m.s, value of the height distribution. Similarly, following Greenwood & Williamson [14] and Whitehouse & Archard [15] we find that the contact area increases linearly with load for this model; that is (d~t/dp) = b = x/3rr (f/h)/E(1 - v2) (11)

For sufficiently gradual changes in aperture, that is, for O,a/dx, and da/dy sufficiently small, (2) becomes V2p = 0 (3)

Equation (3), Laplace's equation, also describes twodimensional heat flow, and so solutions derived using heat conduction theory can be applied to the problem here. In particular, one can adapt Maxwelrs technique for finding the effective conductivity of a three-dimensional model of a mixture of two conducting phases to the two-dimensional problem here. I find, by following Carslaw & Jaeger [13, p. 426] that the effective conductivity < c> of a sheet having conductivity c containing more or less circular cylindrical inclusions of material with conductivity ci is given by the expression