By Robert W. Zimmerman
This booklet is a accomplished remedy of the elastic volumetric reaction of sandstones to diversifications in rigidity. the idea and knowledge awarded practice to the deformations that ensue, for instance, because of withdrawal of fluid from a reservoir, or as a result of the redistribution of stresses as a result of the drilling of a borehole. even though the emphasis is on reservoir-type sandstones, effects and strategies mentioned also are acceptable to different porous rocks. half One issues the impact of pressure on deformation and discusses porous rock compressibility coefficients. Elasticity conception is used to derive relationships among the porous rock compressibility coefficients, the porosity, and the mineral grain compressibility. Theoretical bounds at the compressibility coefficients are derived. the idea that of potent tension coefficients is tested, as is the built-in type of the stress-strain relationships. Undrained compression and brought about pore pressures are taken care of in the similar basic framework. half One is concluded with a short, trouble-free advent to Biot's conception of poroelasticity. all of the leads to half One are illustrated and established with vast references to released compressibility info. half offers with the connection among pore constitution and compressibility, and provides equipment that allow quantitative prediction of the compressibility coefficients. - and third-dimensional types of tubular pores, spheroidal pores, and crack-like "grain boundary" voids are analyzed. A serious evaluate is made from numerous tools which were proposed to narrate the potent elastic moduli (bulk and shear) of a porous fabric to its pore constitution. equipment for extracting pore element ratio distributions from stress-strain information or from acoustic measurements are offered, besides functions to genuine sandstone information. half 3 is a short precis of experimental ideas which are used to degree porous rock compressibilities within the laboratory. the knowledge contained during this quantity is of curiosity to petroleum engineers, particularly these concerned with reservoir modeling, petroleum geologists, geotechnical engineers, hydrologists and geophysicists.
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Additional info for Compressibility of Sandstones
This can be seen G, = 3( 1 - 2v, )/2C, (1 + v, ), from which it follows that by recalling that aG, /av, = -9/2( 1 + v,)*C, < 0. Since application of a hydrostatic confining pressure to the outer surface of a porous body leads to both a dilatation and a shear in the solid material, it is understandable that pore compressibilities will be affected by both the bulk and the shear modulus of the mamx material. Since the lower bounds are increasing functions of v,, bounds which do not depend on v, can be found by setting v, = 0 in eqns.
9990. The pore compressibility Cpc is then found by differentiating eqn. 8) with respect to P, : cPc= A +(B/F)e-PJP . 01 0 0 10 20 30 40 CONFINING STRESS, pC (MPa) 50 Fig. 1. , 1986aJ. Curves are found from least-squares fits to functions of the form given in eqn. 8). Since only changes in the pore strain were measured, the curves and the data points have been shifted so as to vanish when the differential pressure is zero. This was done for purposes of comparison, and has no effect on the computed compressibilities.
Various bounds and relationships can be found for these effective stress coefficients, however, thus greatly constraining their possibly range of values. C, and Cr are both positive, as is Cbc -Cr, so eqns. 0. This indicates that the pore and bulk volumes are each more sensitive to changes in confining pressure than to changes in pore pressure. Since ‘tb is a decreasing function of Cbc, the Hashin-Shtrikman bounds on Cbc/Crlead immediately to the following bounds for nb (Fig. 4 C 3 POROSITY, cp Fig.