Elastic properties of microporous cemented grainstones

GEOPHYSICS. VOL. 76, NO. 6 (NOVEMBER-DECEMBER 2011); P. E211–E226, 16 FIGS., 2 TABLES. 10.1190/GEO2011-0047.1

Elastic properties of microporous cemented grainstones

François Fournier1, Philippe Leonide2, Kévin Biscarrat1, Arnaud Gallois1, Jean Borgomano1, and Anneleen Foubert3

cemented grainstones are elastically equivalent to a homogeneous calcitic host with spherical calcitic inclusions comprising spheroidal pores. The best fit is obtained when porous spheres are modelled using the differential effective medium (DEM) approach and the whole composite using the self-consistent (SC) method (DEM-SC model). At lower porosity values ( :

5μc ðK c þ μc Þ 3

¼ μm þ

ð1−f m Þ 2f ðK þ2μ Þ ðμc −μm Þ−1 þ m m 4 m

;

(A-2)

5μm ðK m þ μm Þ 3

where K HS− and K HSþ , respectively, are lower and upper HS bounds for bulk modulus; μHS− and μHSþ , respectively, are lower and upper HS bounds for shear modulus; K c and μc , respectively, are bulk and shear moduli for pure calcite; K m , μm , respectively, are bulk and shear moduli for microporous micritic grains; and f m is the micritic grain volume fraction.

APPENDIX B DEM THEORY The DEM theory models the effective elastic moduli of twophase composites by adding infinitesimal quantities of inclusions to the host phase (Cleary et al., 1980; Norris, 1985; Zimmerman, 1991). In this theory, the effective bulk and shear moduli of the composite, K
ðyÞ and μ
ðyÞ, respectively, are governed by a coupled system of ordinary differential equations (Mavko et al., 1998)

Microporous grainstone petrophysics

d
½K ðyÞ ¼ PðK 2 − K
ÞðyÞ dy d ð1 − yÞ ½μ
ðyÞ ¼ Qðμ2 − μ
ÞðyÞ; dy

(

ð1 − yÞ

and Q ¼ 1∕5ðT ijij − 1∕3 Tiijj Þ;

(B-1)

(B-2)

where the tensor Tijkl relates the uniform far-field strain to the strain within the ellipsoidal inclusion. Tijkl are functions of the inclusion aspect ratio α and of the bulk and shear moduli of the initial host, K 1 and μ1 , respectively, and of the inclusions, K 2 and μ2 , respectively (Mavko et al., 1998). For spherical inclusions, P and Q are given by (Berryman, 1995)

P¼

K c þ 43 μc K m þ 43 μc

(B-3)

and

Q¼

K c þ ξc ; K m þ ξc

with ξc ¼

μc ð9K c þ 8μc Þ 6 ðK c þ 2μc Þ

i Pn þð1−ΦÞK h Pn K nþ1 ¼ ΦKΦP i þð1−ΦÞPh i

(B-4)

APPENDIX C SELF-CONSISTENT APPROXIMATION The SC approximation (Budiansky, 1965; Wu, 1966) allows us to predict the elastic moduli of a composite materials with inclusions. In this approach, the interaction of the inclusions is approximated by replacing the background medium with an as-yet-unknown effective medium and each constituent is treated symmetrically. The SC formulas for bulk K and shear μ moduli of a 2C rock (one host phase, one inclusion phase) are

f i ðK i − KÞPi þ ð1 − f i ÞðK h − KÞPh ¼ 0

(C-1)

f i ðμi − μÞQi þ ð1 − f i Þðμh − μÞQh ¼ 0;

(C-2)

where K h and μh , respectively, are bulk and shear moduli of the host material; K i and μi , respectively, are bulk and shear moduli of the inclusion; f i : volume fraction of the inclusion; and P, Q are geometrical factors where the superscript i (respectively, h) indicates that the factor is for the material of elastic moduli K i and μi (respectively, K h and μh ) in a background medium of elastic moduli K and m.The equations C-1 and C-2 are solved iteratively as follows:

n

h

n

i Qn þð1−ΦÞμh Qn μnþ1 ¼ ΦμΦQ i þð1−ΦÞQh i

with initial conditions K
ð0Þ ¼ K 1 and μ
ð0Þ ¼ μ1 ; where K 1 , μ1 ¼ bulk and shear moduli of the initial host material; respectively; K2, μ2 ¼ bulk and shear moduli of the inclusion; respectively; y ¼ concentration of the inclusions. The coefficients P and Q depend upon the shape of the inclusion and upon the elastic moduli of the host and inclusion phases. For ellipsoidal inclusions of a given aspect ratio α, P and Q are given by (Wu, 1966)

P ¼ 1∕3 Tiijj

E225

n

h

(C-3)

n

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