May 10, 1973 - strain for several elastic solids, we emphasize that (1) there is no unique law of effective .... cracks in'the rock, opening or closing that itself.
VOL. 78, NO. 14
JOURNAL
OF GEOPHYSICAL
:RESEARCH
MAY 10, 1973
Note on Effective Pressure PiFm•w,-Yvw, s F. ROBIN
Department of Earth and Planetary Sciences,MassachusettsInstitute of Technology Cambridge, Massackusetts 02139
When a pore pressureP• and a confiningpressurePo are applied to a porous solid, various physical properties are affected. When dealing with these properties, the concept of effective pressure is often used. On the basis of the examples of pore volume and of bulk volumetric strain for several elastic solids, we emphasize that (1) there is no unique law of effective pressure, (2) effective pressuremay not be a very useful concept, unless it has the simple form Pe -- Pe -- P,,, and (3) when a property does not vary linearly with pore or confining pressure, effective pressuredoes not in general have any simple analytic expression,unless again it reducesto Pe -- P• -- P•.
Pore pressurewas originally recognizedas an important geologicalvariable by Hubbert and Rubey [1959]. The apparent correlationof earthquakeswith injection of fluids in deep wells [Healy et al., 1968; Raleigh, 1971] and the possibleinterpretationof various phenomena occurringbefore and after earthquakesby variations of pore pressure[Nur and Booker,
1972; Nur, 1972] have recently spurred renewed interest in the more general effects of pore pressureon propertiesof earth materials. A conceptthat is often cited in this context involves 'effective pressure.'Differencesamong various expressionsfor effective pressurehave led to confusion,and the purposeof this note is to emphasizesomeessentialpoints regarding this concept. Consider a property Q, such as density, length, seismicvelocity,porosity,or permeability of a poroussolid,that is a function of confining pressureP• and pore pressurePp:
Q = Q(P•, P•)
(1)
A series of measurementsmade with zero pore
effectivepressureis an expression
Pe = P.(P•,, Pc)
(3)
such that substitutionof (3) into (2) yidds the right value of Q, eventhoughP• is not zero. The interest in such a law is that it should
enableone to predict the value of Q for any combinationof confining and pore pressure, after having made only one seriesof measurements, without pore pressure,to obtain (2). A key questionis: What is the functional rdation (3)?
The first point to be made here is that the form of (3) dependson the property considered, even when dealingwith the same solid. Consider,for example,an isotropiclinear elastic porous solid made up of an isotropic linear elastic material. For this solid to actually be linear elastic, there must be neither significant closure of cracks nor enlargement of contact area between grains over the range of applied confining pressures; the case of a nonlinear elasticsolid made up of a linear elasticmaterial will be considered later.
It
is further
assumed
that all the pores of this solid are interconnected and that the fluid pressurein them is at Q= o) = Qo(3 equilibrium.For sucha solid [Nur and Byerlee, If pore pressureis no longerzero,we take as 1971], as well as for soils,which closelyapproxieffective pressureP• the pressurethat, if ap- mate it [Skempton, 1961], it has been shown plied alone,as a confiningpressure,wouldhave that an effective pressurelaw (EPL) for bulk the sameeffect on property Q as the combina- volumetricstrain is givenby tion of Pe and Pp. In other words, a law of
pressurewill yield a function
P• = P• -- (1 -- K/K•)P•,
Copyright ¸
1973 by the American GeophysicalUnion.
(4)
where K is the bulk modulus of the porous 2434
Rosin:
ErrEc?ivw
solid and K, the intrinsic bulk modulus,i.e., the bulk modulus of the material
of which the solid
is constituted.If, however,insteadof volumetric strain, one considersthe variation of pore volume Au/u of such solid, the EPL becomes(appendix)
PRwssvRw
late an effective pressure from (5) and insert it in (A1). The secondpoint emphasizedhere is thus that the complete functional dependence (1) is or should be known before establishing an EPL; in many cases this law is consequentlyuseless.An exception to the last statement
Be = Pc-
[] -- oK/(Ki-
K)]Pp
(5)
wherep is the porosity.When K is much smaller than K•, (4) and (5) reduce to the usually acceptedeffective pressurelaw [Brace, 1972]
P• = Pc-- P,,
(6)
This equivalenceof (4) and (5) should not be acceptedfor rocks. We can take, for example, the porosity and elastic properties of the Weber sandstoneat low pressure(0-300 bars): p = 0.06; K, = 3.6 X 105bars; K = 1.3 X 105 bars [Nur and Byeflee, 1971, Figure 2]. Equa-
tions4 and 5 then give,respectively, Pe = Pc -- 0. 64P• and
Pe = Pc -- 0.97P•-----Pc -- P• It must be noted that both volumetric
strain
2435
occurs when the EPL
is or reduces
to (6): the simplificationprovided is then a clear advantage. In many rocks, fracture strength, frictional resistance,permeability, and electrical conductivity have been shown experimentally (for a reviewseeBrace [1972]) to obeyeffectivepressure laws closeto (6). This is most probably due to the fact that these properties are all strong functions of the opening or closing of
cracks in'therock,opening or closing thatitself obeysa law closeto (6). There are many other properties of rocks, however,for which the effectsof confiningand pore pressuresare expected to be more complicated. The third point to be made in this note is that, when a property does not vary linearly with either Pc or P•, its EPL has in general no simple analytic expression.The example used for this purposeis again the one of volumetric strain, this time of a nonlinear elastic poroussolid made up of a linear elastic
and pore volume variation would qualify as 'elasticproperties'of the solid.Similar analyses material. for other elastic properties (e.g., variation of Indeed, rocks under most conditions do not crosssection of interconnectingtubular pores, meet the requirementsplaced on the ideal porvariation of volume of the material, AV -- Au, ous elastic solid consideredpreviously. Let us and so forth) will give yet other expressions therefore considera somewhatbetter approxifor their effective pressurelaws. Equations 4 mation: the solid now contains many cracks,
and 5 are sufficient,however,to showthat, even which may close,or grain contacts,which may for 'elastic properties' of an ideally simple enlarge under confining pressure. The cracks poroussolid, there is no singleeffectivepressure and pores remaining open are still assumed law. connected and the pressure in them still at Before going further, it must be made clear equilibrium. If an increasingconfiningpressure that an EPL can always be establishedfor any is applied to the body while maintaininga zero property Q. Indeed, it is always possibleto pore pressure,the volume of the body decreases determinecompletelythe functionaldependence (Figure 1) in a highly nonlinear fashion. (For (1) of Q on Pc and Pp and then identify it with cracks,see Walsh [1965]; for packing of elastic Q: Qo(Pe). In many cases,however,the EPL grains see Thurston and Deresiewicz [1959].) obtained servesno purpose,since Q is already The curve of Figure 1 is in generalnot a simple known for any combination of Pc and Pp. In analytic function of pressure. If pressure is fact, (4) and (5) were derived in exactly this raised, both in the pores and on the outside manner; for example, the complete relation surface of the solid by the same amount P•, (A5) had to be first obtained in order to derive the stress anywhere in the solid is only in(5). It is clear that if one wanted to know the creasedby the constanthydrostatic component variationsAu/u of pore volume with Pc and P•, Pp; the strain is also increasedeverywhere by one would use (A5) directly rather than calcu- a constant component,linearly dependenton P•
2436
i•OBIN:
EFFECTIVE PRESSURE
somehow related
(Pc-•) Pe
F•
Pc
Pressure
Fig. 1. Variation of volume (e -- --/•V/V) with pressure for • nonlinear elastic solid composed of • linear elastic materiLl. The curve 0o (Pc), corresponding to P• -- 0, describes the variation of volume of • jacketed dry specimen. The straight line (Pc -- P• ----0), on the contrary, represents the volume variation of an unjacketed specimen and corresponds to the intrinsic compressibility of the material. When a confining pressure Pc and • smaller pore pressure Pp are applied to the solid, its relative volume change is given by the ordinate of point Q. In general, the effective pressurePc bears no simple analytic relation to Pc and P•.
to the stress and strain fields
under Pc and Pp (point Q). In fact, stressand strain componentsin the final state represented by Q are more closely related to these componentsin the state representedby B: for each component the difference is only a constant throughout, as is explained above. Point A acquires a physical significanceonly if the intrinsic compressibilityis negligiblein comparison to the compressibilityof the porous solid: then A and B coincide,and the EPL again reducesto (6). When pore pressuresare not too large, one could of courseuse a state like the one represented by B as a new referencestate and approximate BA as a straight line. The linear elastic analysis made in the first part of this paper could then apply, taking due account of the new referencelevel of Pc. This is certainly possible; it is, however, a cumbersomeand unnecessaryexercise, since all information required to obtain the soughtvariable, here 0, is containedin curve Oo(P•,-- 0) and (8). A further approximation of a real rock could take into accountthe fact that the compressibility of the mineral is not constant or that some cracks or pores are isolated. In either case, the unjacketed compressibilitycurve (Pc -- P• -- 0) is no longer a straight line as it is in Figure 1. The graphicsolutionof O(P•,,Pc) is still obtainable,but the problemof defining an effective pressureanalytically becomeseven
through the elastic moduli of the material. No crack closesor opens; contactsbetween grains do not change. This result stems from the more insoluble. superpositionprinciple, valid here becausewe Volumetric strain of a poroussolid has been considera linear elasticmaterial. The resulting developedhere at some length, mostly as an additional volumetricstrain of the solid is equal example. More generally,it must be kept in to this uniform volumetric strain in the matemind, when dealing with a property Q of a rial and is givenby poroussolid, that the quantity of interest is Q
0 = --•XF/•
= e•/•:,
(7)
Consequently,when a pore pressureP• and a confiningpressurePc are applied to the solid, the total volumetric strain is found to be (Fig-
itself (equation1) and not the effectivepressure Pt (equation3). In many cases,effective pressureis unnecessary,of little physicalsignificance,and without simpleanalyticalexpression.
ure 1) APPENDIX.
0=
+
(s)
LAW OF EFFECTIVE PRESSURE FOR PORE VOLU•E
By definition the effective pressureis the abWe consider a linear elastic porous solid seissaPc of point A. Clearly, Pc does not bear made up of a linear elastic matehal. Let u be a simple analytic relation to P• and P,. Also, the pore volume. Pc is a misleadingparameter, since it suggests If a uniform hydrostatic confining pressure that the stress and the strain fields in the solid P is applied, the assumptionof linear elasticity under a confiningpressureP, (point A) are allows us to write
ROBIN:
2437
EFFECTIVE PRESSURE
or, using (A2),
Xu/u = where k is a constant coefficient.
ee = Pc-
Betti's reciprocaltheorem [e.g., Sokolniko#, 1956, p. 391] can be applied to the following states of the solid: (1) a pressureP applied as confining pressure only and (2) the same pressureP applied as both pore and confining pressure.
It is then found that
= where K is the effective bulk modulus of the
[1 -- oK/(K,-
K)]P•
(5)
Acknowledgments. Ideas presented here were stimulated by an early draft of their paper presented by A.M. Nur and J. D. Byerlee and by later discussions with A.M. Nur. W. F. Brace, J. B. Walsh, H. C. Heard, and A. G. Johnson critically reviewed the manuscript; C. Goetze and R. M. 8tesky are gratefully acknowledged for many discussions. This work was done with the support of the National Science Foundation under grant GA 18342.
REFERENCES solid, K, is the intrinsic bulk modulus, and p - u/V is the porosity. Brace, W. F., Pore pressurein geophysics,in A combinationof pore pressureP, and conFlow and Fracture o[ Rocks, Geophys. Monogr. Set., vol. 16, edited by H. C. Heard, I. Y. Borg, fining pressurePccan be applied in two stages. N. L. Carter, and C. B. Raleigh, pp. 265-273, In a first stage a pressureequal to P, is estabAGU, Washington, D.C., 1972. lished both in the pores and in the confining Healy, J. H., W. W. Rubey, D. T. Griggs, and medium. The material in the solid is exposed C. B. Raleigh, The Denver earthquakes, Science, 161, 1301-1310, 1968. to a uniform hydrostatic pressure everywhere, so the pore volumeis reducedby the samerela- Hubbert, M. K., and W. W. Rubey, Role of fluid pressurein mechanicsof overthrust faulting, tive amount as the volume of the material; i.e., Bull. Geol. Soc. Amer., 70, 115-205, 1959. Nur, A., Dilatancy, pore fluids and variationsof Xu,/u = -tC, (A3) ts/t• ratios (abstract), Eos Trans. AGU, 53, 1115, 1972. In a secondstage the confiningpressureis inNur, A., and J. D. Byerlee, An exact effective creased,by an amount equal to (PcPp). stress law for elastic deformation of rock with From (A1) the variation in pore volume is fluids, J. Geophys.Res., 76(26), 6414-6419, 1971.
Xu/u = -
-
(A4)
The final confining pressure is then P• and •he final pore pressureP•. Thanks again to the linearity of the response,the total variation of pore volume is
u/u
=
+
=
+
-
To find • law of effective pressure, (A5) mus• be identified
with
Au/u - --Pe/k
(A6)
Nur, A., and J. R. Booker, Aftershocks causedby pore fluid flow?, Science, 175, 885-887, 1972. Raleigh, C. B., Earthquake control at Rangely, Colorado (abstract), Eos Trans. AGU, 52, 344, 1971.
Skempton, A. W., Effective stress in soils, concrete, and rocks, in Pore Pressure and Suction in Soils, p. 1, Butterworths, London, 1961. Sokolnikoff, I. S., Mathematical Theory oJ Elasticity, 2nd ed., McGraw-Hill, 1956. Thurston, C. W., and H. Deresiewicz,Analysis of a compressiontest of a model of a granular medium, J. Appl. Mech., 26, 251, 1959. Walsh, J. B., The effect of cracks on the compressibility of rock, J. Geophys. Res., 70(2), 381, 1965.
The law of effective pressureso obtained is (Received February 23, 1972; revised December 18, 1972.)
