By Robert W. Zimmerman
This publication is a entire therapy of the elastic volumetric reaction of sandstones to diversifications in pressure. the idea and information provided practice to the deformations that happen, for instance, as a result of withdrawal of fluid from a reservoir, or end result of the redistribution of stresses brought on by the drilling of a borehole. even supposing the emphasis is on reservoir-type sandstones, effects and techniques mentioned also are acceptable to different porous rocks. half One matters the impact of tension on deformation and discusses porous rock compressibility coefficients. Elasticity concept 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 concept that of potent rigidity coefficients is tested, as is the built-in kind of the stress-strain relationships. Undrained compression and brought on pore pressures are handled in the comparable common framework. half One is concluded with a quick, hassle-free creation to Biot's conception of poroelasticity. the entire ends up in half One are illustrated and demonstrated with broad references to released compressibility information. half offers with the connection among pore constitution and compressibility, and offers tools that allow quantitative prediction of the compressibility coefficients. - and 3-dimensional versions of tubular pores, spheroidal pores, and crack-like "grain boundary" voids are analyzed. A serious evaluate is made up of quite a few equipment which were proposed to narrate the potent elastic moduli (bulk and shear) of a porous fabric to its pore constitution. tools for extracting pore element ratio distributions from stress-strain info or from acoustic measurements are offered, in addition to functions to real sandstone facts. half 3 is a short precis of experimental thoughts which are used to degree porous rock compressibilities within the laboratory. the data contained during this quantity is of curiosity to petroleum engineers, in particular these concerned with reservoir modeling, petroleum geologists, geotechnical engineers, hydrologists and geophysicists.
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And is then multiplied by P d . 18) PRESSURE Fig. 9. Schematic depiction of the difference between the tangent and secant compressibilities. C,, measures the local slope of the stress-strain curve at pressure P , while C,,=c measures the average slope of the curve from 0 to P . 52 ck where iib is a sort of “secant value” of n b . Since is an average of C k , the HashinShtrikman bounds apply to it, and so the associated bounds on nb are applicable to i i b . 19) where Cr iip = 1--. 20) cpc ( P d It should be noted that for low porosity rocks, eqn.
21) then predicts that the in situ strain in the rock grains would be E, = - ( 1 . 184%. 28) The in siru porosity at 3000 m can be calculated by first determining the secant compressibility from eqn. 24), and then using eqn. 22). At a confining pressure of 10,532 psi and a pore pressure of 4232 psi, the differential pressure would be 6300 psi. The secant compressibility is 10-6/psi. Eqn. 22) then shows the porosity at this stress state given by eqn. 24) as 1 . 27~ . - 1 . 4% could prove important when estimating oil reserves.
4 5 POROSITY, p Fig. 1. Lower bound on C,, from eqn. 3). 4 Fig. 2. Lower bound on C, , from eqn. 4). n 0 f P ? 01 E -6 -I v, 5 w? 7JN 0 0 0 73 c 0 cC LOWER BOUND FOR CpJC, 0 ? 4 Fig. 3. Lower bound on Cpcrfrom eqn. 5). 0- 7 LOWER BOUND FOR Cpp/Cr f h. 0- 0 LA ? 0- ??? 4 POROSITY, ‘p Fig. 4. Lower bound on Cpp,from eqn. 6). 5 ? 0 27 These lower bounds are plotted in Figs. 4, for various values of v,, and for porosities up to 50%. This covers the entire range of porosities that would be of practical interest for application to sandstones, although the bounds are actually valid for all O l e < 1.
Compressibility of Sandstones by Robert W. Zimmerman