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Equivalent porous medium for modeling of the elastic and the sonic properties of sandstones
Equivalent porous medium for modeling of the elastic and the sonic properties of sandstones S.T. Nguyen, M.-H. Vu, M.N. Vu PII: DOI: Reference:
S0926-9851(15)00179-2 doi: 10.1016/j.jappgeo.2015.06.004 APPGEO 2785
To appear in:
Journal of Applied Geophysics
Received date: Revised date: Accepted date:
8 January 2015 1 June 2015 7 June 2015
Please cite this article as: Nguyen, S.T., Vu, M.-H., Vu, M.N., Equivalent porous medium for modeling of the elastic and the sonic properties of sandstones, Journal of Applied Geophysics (2015), doi: 10.1016/j.jappgeo.2015.06.004
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ACCEPTED MANUSCRIPT EQUIVALENT POROUS MEDIUM FOR MODELING OF THE ELASTIC
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AND THE SONIC PROPERTIES OF SANDSTONES
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S.T. Nguyen(a,b,*), M.-H. Vu(b,c) and M.N. Vu(b)
(a) Euro-Engineering, Pau, France
(b) R&D Center, Duy Tan University, Da Nang, Viet Nam
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(c) CurisTec, 3 rue Claude Chappe, Parc d'affaire de Crécy, 69370 Saint-Didier-au-Mont-
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d'Or, France
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(*) Email: [email protected]
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ACCEPTED MANUSCRIPT ABSTRACT This study is devoted to model the elastic and the sonic properties of sandstones. The main
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difficulty in modeling granular materials like sandstones is the effect of grain-to-grain
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contacts. A new concept of an equivalent porous medium (EPM), which is a porous medium
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of a continuous solid matrix and pore-inclusions with an equivalent porosity that is higher than the porosity of the initial medium, is proposed to avoid this difficulty. A combination of the classical Hashin-Shtrikman (HS) approach and EPM provides an efficient simulation of
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the elastic properties of aggregate materials like sandstones, in comparison with experimental and numerical data in literature. The porosity of EPM of clean sandstones, that
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is calibrated using laboratory data, is about two times greater than that of the initial medium. The effects of clay and organic contents in shaly sandstones are also taken into account by
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introducing a notion of an un-supporting soft-phase. Similarly to the case of clean
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sandstones, the volumetric fraction of the soft-phase of EPM is about two times greater than that of the initial rock. The stress sensitivity and a comparison of this model to the heuristic
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modified Hashin-Strikman model are also discussed at the end of the paper. A power law is proposed for the dependence of the volumetric fraction of the EPM’s soft-phase on the
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effective confining pressure. The proposed concept of EPM is proved to have many practical applications for the interpretation of sonic and seismic data of reservoir rocks. Keywords: sandstone, elastic properties, sonic velocities, Hashin-Shtrikman, EPM
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ACCEPTED MANUSCRIPT INTRODUCTION Rocks are naturally heterogeneous materials which are complex mixtures of solid minerals
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and pores (Willis, 1981; Hashin, 1983; Berryman, 1995). The elastic and the sonic properties
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of rocks are complex functions of the porosity (Wyllie et al., 1956; Raymer et al., 1980), the
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mineralogy, the fluids in the pore space (Gassmann, 1951; Biot, 2005), the effective stress and the microstructure (Giraud et al., 2007). Several empirical formulas were proposed for such relationships (Han et al., 1986; Eberhart-Phillips et al., 1989).
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For many decades, the micromechanical approaches have showed a strong capacity in
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modeling the properties of heterogeneous materials from nano-scale to macro-scale (Eshelby, 1957; Hashin, 1962; Hashin, 1983; Hashin and Shtrikman, 1963; Mori and Tanaka, 1973; Dormieux et al., 2006; Vu, 2012; Nguyen, 2014; Nguyen et al., 2011; Nguyen et al.,
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2014 and Nguyen and Dormieux, 2015). In the spirit of the micromechanical theory, rocks
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are considered as mixtures of multi-phases. The impacts of the mineralogy and the porous
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phase saturated or partially saturated by a single or multi-fluids, the microstructure can be modeled (Giraud et al., 2007; Ortega et al., 2007; Mukerji et al., 1995; Schöpfer et al. 2009). The elastic properties of the heterogeneous media can be accurately estimated if its
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microstructure (shapes, orientations and spatial distribution of grains and pores, grain-tograin contact properties, etc) is well defined. The microstructure of rocks can be captured by digitized 3D images and the elastic moduli can be obtained based on the finite element method (Knackstedt et al., 2003; Madonna, 2012). Empirical formulas can be derived from numerous numerical simulations. However, similarly to other empirical approaches that are based on the measurements, this numericalempirical approach could not be used for rocks of which the microstructure is outside the range of observed samples. Note also that the numerical approach ignores the stress sensitivity due to interface effects because imperfect grain-to-grain contacts cannot be captured by the digitized images.
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ACCEPTED MANUSCRIPT In general, the volumetric fraction of each component (porosity, mineralogy composition, saturation) of rock is the unique available information. The classical micromechanics HS approach (Hashin and Shtrikman, 1963) with the concept of spherical composite, gives
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bounds for elastic properties of rocks. However, due to the contrast between the elastic properties of the void and the solid-skeleton, HS bounds are not close enough for accurate
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estimation of elastic moduli. The Eshelby based Mori-Tanaka, self-consistent and differential effective medium approaches overestimate elastic properties of uncemented sandstones
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(Knackstedt et al., 2003). This overestimation is due to the grain-to-grain contact effect that gives an additional compliance to the medium (Sayer, 2002). Several researchers proposed
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to take into account this effect by considering microracks in their models (Fortin et al., 2007; Walsh, 1965; Sayers and Kachanov, 1995; Hall et al., 2008; Sarout and Gueguen, 2008).
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Beside the actual advanced micromechanical development also allows modelling the
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interface effects (Hashin, 2002; Duan et al., 2006; He et al., 2012). However, usual practical issue when applying the micromechanical approaches with interface effects is the missing
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information on the elastic stiffness and the volumetric fraction of the interface zone. The Hertz-Mindlin theory (Mindlin, 1953; Norris and Johnson, 1997) showed that the elastic
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stiffness of the grain-to-grain contact is stress dependent. This dependency is influenced by the grain size and the average number of contact per grain which are normally not available in practice. Even though the stiffness of the contacts can be well calibrated from measurements of a sample, it is difficult to apply these parameters to other samples where the surface and the number of contact are not identical. In this study, an EPM is proposed to model the elastic and the sonic properties of sandstones. The main objective of this concept is to avoid the common difficulty due to the effect of grain-to-grain contacts by keeping the stress sensitivity. EPM is a porous medium with perfectly connected solid matrix and pore inclusions (Figure 1) that the elastic properties can be accurately estimated by the Mori-Tanaka approach (which is the HS upper bound). We first assume that sandstone is isotropic material and then we define an equivalent 4
ACCEPTED MANUSCRIPT isotropic medium. In order to avoid the complexity of the micromechanical model due to the pore shape, the simplest way is to consider an equivalent medium with spherical pore. The unique parameter is the ratio between the porosity of the equivalent medium and that of the
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initial medium which can be calibrated based on available experimental and numerical data
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founded in prior researches (Han and al., 1986; Knackstedt et al., 2003). In this paper, the case studies are presented in increasing order of complexity. Firstly, the case of clean sandstone (no clay and organic contents) is considered to analyze the effect of
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the porosity on the elastic and the sonic properties. Next, we consider the effect of the clay content on the elastic properties of shaly sandstones by introducing the concept of the soft-
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phase which is a mixture of the porous phase and the un-supporting clay particles (Marion et al., 1992). The stress sensitivity effect on EPM is analyzed for both clean and clayey
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sandstone. Finally, a comparison of our model to the heuristic modified HS model (Walls et
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al., 1998) is discussed.
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CLEAN SANDSTONES
Considering EPM which is a mixture of a solid matrix and spherical pore inclusions (Figure 1, right side), Mori-Tanaka scheme (equivalent to the HS upper bound) (Mori and Tanaka,
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1973; Hashin and Shtrikman, 1963) allows estimating the effective elastic bulk and shear moduli (isotropic case) as: 1
K eff
eq 1 eq 4 Gs 4 4 3 K s Gs K p Gs 3 3
(1)
1
Geff
Where
eq
1 eq eq G 9 K 8Gs as with as s s G a G a 6 K s 2Gs s p s s
(2)
is the equivalent porosity of EPM; Ks and Gs are the elastic properties of the solid
phase; Kp and Gp are the elastic properties of the fluid phase. 5
ACCEPTED MANUSCRIPT For the case of clean sandstone saturated by water, the elastic moduli of the solid phase of EPM are that of quartz given in the works of Mavko et al. (2009): Ks = 37.9 (GPa), Gs=44.3 (GPa) and the elastic moduli of the pores are that of water: Kp=2.3 (GPa) and Gp=0. Note
as:
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4 K eff V p2 Vs2 3
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the bulk density
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that Keff and Geff (dynamic moduli) are related to P-wave, S-wave velocities (Vp and Vs) and
Geff Vs2
eq
(4)
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The equivalent porosity
(3)
of EPM can be calibrated using the measurement of the sonic
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velocities and the density. The calibration of this parameter by fitting the sonic velocity
where
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calculated by (1) to (4) with the measurements of Han et al. (1986) gives:
eq 2
(5)
is the porosity of the initial medium. Figure 2 shows the comparison between the P-
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wave and the S-wave velocities computed by using EPM with
eq
=2 and the real measured
data. The difference between the proposed model and the experimental data (shown on the Fig. 2) may due to the fact that the stress sensitivity (in the measurements of Han et al. (1986), confining pressure varies from 5 to 40 MPa) is not taken into account in the model. This stress dependency will be discussed later in this paper. Knackstedt et al. (2003) simulated numerically (based on digitized 3D images) the bulk and shear moduli of quartz-air and feldspar-air mixtures at different porosity range from 0 to 0.5. These data are used to validate the relationship (5) in dry condition. Based on EPM concept, the effective bulk and shear moduli are calculated by (1) and (2). In case of quartz-air mixture, the elastic parameters of the solid and of the fluid phases are: Ks=37.9 (GPa),
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ACCEPTED MANUSCRIPT Gs=44.3 (GPa), Kp=0.0 (GPa) and Gp=0.0 (GPa). Concerning feldspar-air mixture, the elastic moduli of feldspar are (given in Knackstedt et al., 2003): Ks=37.5 (GPa), Gs=15.0 (GPa). Figure 3 shows a strong consensus between EPM’s model with
eq=2
and the numerical
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simulation based on the finite element method (Knackstedt et al., 2003) for both clean quartz-air mixture and clean feldspar-air mixture. Note also that both the bulk and the shear
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moduli vanish at porosity of about 50% (critical porosity, see Nur et al., 1998).
Note that the porosity of the equivalent media could not excess 100%, thus the porosity of
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the initial media could not excess 50%. Fortunately this is the case for sandstone. Artificial sandstone with same diameter has maximum porosity of about 36% and real sandstone with
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small grains filled between big grains has smaller porosity. The shape of grain and pores can increase the porosity but for sandstones it is normally lower than 40% under null confining
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stress (Mavko et al., 2009).
Moreover, the proposed model is calibrated on the laboratory measurement of Han et al.
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(1986) on well-defined samples under confining stress from 5 to 40 MPa with maximum porosity of 36%. Then we can suppose that our model is validated for sandstone with porosity up to 40%. This range is likely to cover all natural compacted and uncompacted
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sandstones.
The comparison of the model to experimental data and numerical simulation based on 3Ddigitized method suggests that the elastic properties of clean sandstone can be estimated using EPM with equivalent porosity of two times bigger than the initial porosity when the stress sensitivity is ignored. In next sections, the impacts of heterogeneous solid minerals and stress sensitivity on the equivalent porosity of EPM will be discussed.
MINERALOGY AND FLUID CONTENTS EFFECTS Rock with multi-minerals and fluid contents is decomposed into two phases: grains supporting and un-supporting soft-phases. The first phase is composed of hard minerals 7
ACCEPTED MANUSCRIPT (such as quartz, calcite, feldspar, etc.) that form the solid skeleton of rock and support the effective stress. The second phase comprises of soft minerals (such as clay, organic matter) and pore fluids (Marion et al., 1992). Note that this consideration should be applied for shaly
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sandstones with low amount of clay and organic contents (Revil et al., 2002; Marion et al.,
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1992).
In the relationships (1), (2) and (5), the porosity is replaced by the volumetric fraction of the soft-phase. The elastic properties of the solids and the fluids are respectively replaced by the
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overall elastic properties of the supporting grains and that of the soft-phase.
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The volumetric fraction of the soft-phase is the sum of the porosity and of the volumetric fraction of clay and organic matter:
(6)
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Vsoft Vclay Vorganic
where Vclay and Vorganic are respectively the volume of clay mineral and organic matter per unit
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volume of rock.
Similarly to the case of clean sandstones, the volumetric fraction of the soft phase of EPM is
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two times greater than that of the initial medium:
eq Vsoft 2Vsoft
(7)
The overall elastic properties of the supporting grains can be estimated by taking the classical Hill’s average (8) and (9).
Ks
K s K s 2
(8)
Gs
Gs Gs 2
(9)
where the upper and lower bounds are expressed as (Berryman, 1995; Mavko et al., 2009):
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(10)
where the exponent
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G 9 K j 8G j with a j j 6 K j 2G j
means upper or lower bounds and the functions
means
is the relative volumetric fraction of the supported grain in
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minimum or maximum values,
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Gs /
1 fi min/ max aj j i Gi a j
the supporting phase.
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Un-supporting clay and organic constituents are supposed to be suspended in the fluids inside the soft phase. The HS lower bound is appropriate to model the effective properties of
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such a mixture. The overall shear modulus of the soft phase can be approximated to zero.
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Considering an example of a mixture composing of floating clay particles in water, the shear modulus of the mixture can be approximated to zero for volume fraction of clay reach to 80%
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(Figure 4). In this condition, the HS lower bound (Hashin and Shtrikman, 1963) and the
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Reuss average give the same results for the overall bulk modulus of the soft-phase:
Kp
Vsoft
V / K i i
(12) i
Gp 0
(13)
Where Vi and Ki are respectively the volumetric fraction and the bulk modulus of the component i (fluids, gas or unsupported minerals) in the soft-phase. Note that Vsoft=∑Vi. Figure 5 shows the comparison between the P-wave and the S-wave velocity computed using EPM with Vsoft 2Vsoft and the measured data. The deviation between the model and eq
the experimental data are supposed to be due to the stress sensitivity (stress varied from 5 to 40 (MPa)). This stress dependency will be discussed in the next section.
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ACCEPTED MANUSCRIPT STRESS SENSIBILITY We analyze the stress effect on EPM or more precisely the stress sensitivity of the equivalent eq
eq
for the case of clean
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volume Vsoft of the soft-phase of EPM (that is the equivalent porosity eq
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sandstones). The idea is to calibrate the parameter Vsoft to fit better the experimental data
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than that presented in previous section to obtain the ratio between the volume of the softeq
phase of EPM and that of the initial medium ( Vsoft / Vsoft ) for each case of mean effective
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stress. The dependency of the volumetric fraction of the equivalent soft-phase on the mean effective stress is presented by the points in Figure 6 (left side) that are calibrated based on
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the measurements of Han et al. (1986). The power law (14) is found to fit the calibrated points and is proposed to characterize the stress dependency of EPM. Note that in equation . The ratio between the volumetric fraction of
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(14), the unit of the mean effective stress is
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the soft-phase of EPM and that of the initial medium decreases when the confining effective eq
and for high stress condition, Vsoft
tends to 1.5Vsoft .
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stress increases. For free stress condition
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eq Vsoft
Vsoft
1.5 1 p
1 3
(14)
Quartz and calcite grains may be broken under very high stress condition which may affect sonic velocities of sandstone. However, information from measurement does not allow us to calibrate such phenomena. DISCUSSION The heuristic modified Hashin–Shtrikman model is proposed by Walls et al. (1998) to estimate the effective moduli of dry sand with cement deposited out of grain contacts. The material is decomposed by a porous medium with a porosity and shear moduli are then given as: 10
and a solid phase. The bulk
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(16)
is the critical porosity and the Hertz-Mindlin moduli KHM and GHM are defined as:
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where
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Geff
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1
1 0 0 as GHM as Gs as
(15)
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K eff
1 4 0 0 Gs 3 K 4G K 4G s s HM 3 s 3
1
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3 2 3 f s 1 3 f 52 s
(17)
K HM
(18)
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GHM
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K HM
C 2 1 0 2 Gs2 3 p 2 2 18 1 s
contacts;
s
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where C is the average number of contacts per grain; f is the fraction of the perfect-adhesion is the Poisson’s coefficient of solid phase; pꞌis the mean effective stress.
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Replacing Kp=0 and Gp=0 into equation (1) and (2), dry effective moduli are determined. The obtained expression of Keff is compared to equation (15) and the following relationship between the porosity of EPM and the porosity of porous medium is obtained:
K
eq 4Gs K K HM s 30 K s K 4 G HM
3
(19)
s
The obtained expression of Geff is compared to equation (16) and the following relationship between the porosity of EMP and the porosity of porous medium is obtained:
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eq 1 Gs GHM 9 K s 8Gs 60 GHM as K s 2Gs
(20)
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When mean effective stress is very low in comparison to moduli of solid phase, Hertz-Mindlin
K G
1
0
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moduli tend to zero and we obtain:
(21)
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Expression (21) shows that with no loading, the ratio of EPM’s porosity and that of the initial
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material equals inverse of the critical porosity. The theoretical value of critical porosity is the porosity of a random pack of identical spherical grains that is 0.36 (Mavko et al., 2009). However, in reality this parameter depends on the microstructure of material. In fact, the
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critical porosity of sandstone is about 0.40 (Mavko et al., 2009). Using this value of
0,
the
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ratio of EPM’s porosity and that of the initial material is found equal to 2.5 which is in
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agreement with equation (14) (for free stress condition) obtained above from the calibration on experimental data of Han et al. (1986). Equations (19) and (20) also allow confirming a K
and
G
with the mean effective stress.
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decrease of
CONCLUSION
In this paper, the concept of EPM is proposed to model elastic properties and sonic velocities of uncemented sandstones. A scaling factor of 2 is multiplied to the initial rock porosity to obtain EMP porosity in the case of clean sandstone. When extending to shaly sandstone, the model parameters must comprise of the porosity, the mineralogy and the stress effects. In this case, EPM is the porous medium of a supported solid matrix and soft phase inclusions. The solid matrix is the combination of the grains that support the effective stress. The soft inclusions are formed by fluids, gas, unsupported clay and organic constituents. A factor of 2 is again used for defining the volume of EPM soft-phase. However, to take the stress dependency into consideration, this factor can varies between 1.5 and 2.5. The results were 12
ACCEPTED MANUSCRIPT verified against the experimental measures of a sandstone rock (Han et al.1986). This approach shows a strong agreement with the heuristic modified Hashin-Strikman model
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proposed by Walls et al. (1998).
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ACKNOWLEDGEMENTS
The authors wish to express their special thanks to the reviewers for their contribution to
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improve the quality of this paper.
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Solid mineral
Solid matrix
EPM
s
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Pore s
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Interface
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Figure 1. The concept of Equivalent Porous Medium (EPM)
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Equivalent porosity
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5.0
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4.0
3.0
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Computed velocity [km/s]
6.0
2.0
Vs
3.0 4.0 5.0 Measured velocity [km/s]
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2.0
Vp
6.0
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Figure 2. EPM concept for water saturated clean sandstone with eq 2 . Comparison with
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the data measured by Han et al. (1986)
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50
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50
EPM model
35 30 25 Quartz
20 15 10
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Numerical simulation
40
45 40
35
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Effective shear modulus [GPa]
45
Feldspar
25
15 10
0.1
0.2
0.3 0.4 Porosity
0.5
0.6
Feldspar
0
0
0.1
0.2
0.3 0.4 Porosity
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0
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0
Quartz
20
5
5
Numerical simulation
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Effective bulk modulus [GPa]
EPM model
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Figure 3. Comparison between EPM concept for dry clean sandstone with
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numerical data simulated by Knackstedt et al. (2003)
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eq=2
0.5
0.6
and the
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8 6
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4
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2 0 0
0.5
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Shear mdulus (GPa)
10
1
Volumetric fraction of clay in soft phase
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Figure 4. Hashin-Shtrikman under bound for shear modulus of a mixture of floating clay particles in water. The elastic moduli used for the simulation are: clay bulk modulus is 21
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(GPa); clay shear modulus is 7 (GPa); water bulk modulus is 2.3 (GPa) and water shear
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modulus is zero.
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5.0
4.0
3.0
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Computed velocity [km/s]
6.0
2.0
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2.0
3.0 4.0 5.0 Measured velocity [km/s]
Vp
Vs 6.0
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Figure 5. Comparison between EPM concept for water saturated shaly sandstones with
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eq Vsoft 2Vsoft and the data measured by Han et al. (1986)
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3.0
eq Vsoft
eq Vsoft
Vsoft
Vsoft
1.5 1 p
1 3
1.0
Calibrated 0.0 50
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3.0
2.0
3.0 4.0 5.0 Measured velocity [km/s]
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Figure 6. Stress sensitivity of EPM
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Vp Vs
2.0
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10 20 30 40 Mean effective stress [MPa]
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0
4.0
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Power law
5.0
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2.0
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Computed velocity [km/s]
6.0
6.0
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Equivalent porous media is defined for modeling of sonic properties of sandstones. The simulation is compared with laboratory measurements The effect of clay content is accounted The stress sensitivity is also discussed.
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S0926-9851(15)00179-2 doi: 10.1016/j.jappgeo.2015.06.004 APPGEO 2785
To appear in:
Journal of Applied Geophysics
Received date: Revised date: Accepted date:
8 January 2015 1 June 2015 7 June 2015
Please cite this article as: Nguyen, S.T., Vu, M.-H., Vu, M.N., Equivalent porous medium for modeling of the elastic and the sonic properties of sandstones, Journal of Applied Geophysics (2015), doi: 10.1016/j.jappgeo.2015.06.004
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT EQUIVALENT POROUS MEDIUM FOR MODELING OF THE ELASTIC
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AND THE SONIC PROPERTIES OF SANDSTONES
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S.T. Nguyen(a,b,*), M.-H. Vu(b,c) and M.N. Vu(b)
(a) Euro-Engineering, Pau, France
(b) R&D Center, Duy Tan University, Da Nang, Viet Nam
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(c) CurisTec, 3 rue Claude Chappe, Parc d'affaire de Crécy, 69370 Saint-Didier-au-Mont-
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d'Or, France
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(*) Email: [email protected]
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ACCEPTED MANUSCRIPT ABSTRACT This study is devoted to model the elastic and the sonic properties of sandstones. The main
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difficulty in modeling granular materials like sandstones is the effect of grain-to-grain
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contacts. A new concept of an equivalent porous medium (EPM), which is a porous medium
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of a continuous solid matrix and pore-inclusions with an equivalent porosity that is higher than the porosity of the initial medium, is proposed to avoid this difficulty. A combination of the classical Hashin-Shtrikman (HS) approach and EPM provides an efficient simulation of
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the elastic properties of aggregate materials like sandstones, in comparison with experimental and numerical data in literature. The porosity of EPM of clean sandstones, that
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is calibrated using laboratory data, is about two times greater than that of the initial medium. The effects of clay and organic contents in shaly sandstones are also taken into account by
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introducing a notion of an un-supporting soft-phase. Similarly to the case of clean
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sandstones, the volumetric fraction of the soft-phase of EPM is about two times greater than that of the initial rock. The stress sensitivity and a comparison of this model to the heuristic
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modified Hashin-Strikman model are also discussed at the end of the paper. A power law is proposed for the dependence of the volumetric fraction of the EPM’s soft-phase on the
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effective confining pressure. The proposed concept of EPM is proved to have many practical applications for the interpretation of sonic and seismic data of reservoir rocks. Keywords: sandstone, elastic properties, sonic velocities, Hashin-Shtrikman, EPM
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ACCEPTED MANUSCRIPT INTRODUCTION Rocks are naturally heterogeneous materials which are complex mixtures of solid minerals
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and pores (Willis, 1981; Hashin, 1983; Berryman, 1995). The elastic and the sonic properties
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of rocks are complex functions of the porosity (Wyllie et al., 1956; Raymer et al., 1980), the
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mineralogy, the fluids in the pore space (Gassmann, 1951; Biot, 2005), the effective stress and the microstructure (Giraud et al., 2007). Several empirical formulas were proposed for such relationships (Han et al., 1986; Eberhart-Phillips et al., 1989).
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For many decades, the micromechanical approaches have showed a strong capacity in
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modeling the properties of heterogeneous materials from nano-scale to macro-scale (Eshelby, 1957; Hashin, 1962; Hashin, 1983; Hashin and Shtrikman, 1963; Mori and Tanaka, 1973; Dormieux et al., 2006; Vu, 2012; Nguyen, 2014; Nguyen et al., 2011; Nguyen et al.,
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2014 and Nguyen and Dormieux, 2015). In the spirit of the micromechanical theory, rocks
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are considered as mixtures of multi-phases. The impacts of the mineralogy and the porous
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phase saturated or partially saturated by a single or multi-fluids, the microstructure can be modeled (Giraud et al., 2007; Ortega et al., 2007; Mukerji et al., 1995; Schöpfer et al. 2009). The elastic properties of the heterogeneous media can be accurately estimated if its
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microstructure (shapes, orientations and spatial distribution of grains and pores, grain-tograin contact properties, etc) is well defined. The microstructure of rocks can be captured by digitized 3D images and the elastic moduli can be obtained based on the finite element method (Knackstedt et al., 2003; Madonna, 2012). Empirical formulas can be derived from numerous numerical simulations. However, similarly to other empirical approaches that are based on the measurements, this numericalempirical approach could not be used for rocks of which the microstructure is outside the range of observed samples. Note also that the numerical approach ignores the stress sensitivity due to interface effects because imperfect grain-to-grain contacts cannot be captured by the digitized images.
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ACCEPTED MANUSCRIPT In general, the volumetric fraction of each component (porosity, mineralogy composition, saturation) of rock is the unique available information. The classical micromechanics HS approach (Hashin and Shtrikman, 1963) with the concept of spherical composite, gives
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bounds for elastic properties of rocks. However, due to the contrast between the elastic properties of the void and the solid-skeleton, HS bounds are not close enough for accurate
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estimation of elastic moduli. The Eshelby based Mori-Tanaka, self-consistent and differential effective medium approaches overestimate elastic properties of uncemented sandstones
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(Knackstedt et al., 2003). This overestimation is due to the grain-to-grain contact effect that gives an additional compliance to the medium (Sayer, 2002). Several researchers proposed
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to take into account this effect by considering microracks in their models (Fortin et al., 2007; Walsh, 1965; Sayers and Kachanov, 1995; Hall et al., 2008; Sarout and Gueguen, 2008).
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Beside the actual advanced micromechanical development also allows modelling the
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interface effects (Hashin, 2002; Duan et al., 2006; He et al., 2012). However, usual practical issue when applying the micromechanical approaches with interface effects is the missing
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information on the elastic stiffness and the volumetric fraction of the interface zone. The Hertz-Mindlin theory (Mindlin, 1953; Norris and Johnson, 1997) showed that the elastic
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stiffness of the grain-to-grain contact is stress dependent. This dependency is influenced by the grain size and the average number of contact per grain which are normally not available in practice. Even though the stiffness of the contacts can be well calibrated from measurements of a sample, it is difficult to apply these parameters to other samples where the surface and the number of contact are not identical. In this study, an EPM is proposed to model the elastic and the sonic properties of sandstones. The main objective of this concept is to avoid the common difficulty due to the effect of grain-to-grain contacts by keeping the stress sensitivity. EPM is a porous medium with perfectly connected solid matrix and pore inclusions (Figure 1) that the elastic properties can be accurately estimated by the Mori-Tanaka approach (which is the HS upper bound). We first assume that sandstone is isotropic material and then we define an equivalent 4
ACCEPTED MANUSCRIPT isotropic medium. In order to avoid the complexity of the micromechanical model due to the pore shape, the simplest way is to consider an equivalent medium with spherical pore. The unique parameter is the ratio between the porosity of the equivalent medium and that of the
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initial medium which can be calibrated based on available experimental and numerical data
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founded in prior researches (Han and al., 1986; Knackstedt et al., 2003). In this paper, the case studies are presented in increasing order of complexity. Firstly, the case of clean sandstone (no clay and organic contents) is considered to analyze the effect of
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the porosity on the elastic and the sonic properties. Next, we consider the effect of the clay content on the elastic properties of shaly sandstones by introducing the concept of the soft-
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phase which is a mixture of the porous phase and the un-supporting clay particles (Marion et al., 1992). The stress sensitivity effect on EPM is analyzed for both clean and clayey
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sandstone. Finally, a comparison of our model to the heuristic modified HS model (Walls et
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al., 1998) is discussed.
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CLEAN SANDSTONES
Considering EPM which is a mixture of a solid matrix and spherical pore inclusions (Figure 1, right side), Mori-Tanaka scheme (equivalent to the HS upper bound) (Mori and Tanaka,
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1973; Hashin and Shtrikman, 1963) allows estimating the effective elastic bulk and shear moduli (isotropic case) as: 1
K eff
eq 1 eq 4 Gs 4 4 3 K s Gs K p Gs 3 3
(1)
1
Geff
Where
eq
1 eq eq G 9 K 8Gs as with as s s G a G a 6 K s 2Gs s p s s
(2)
is the equivalent porosity of EPM; Ks and Gs are the elastic properties of the solid
phase; Kp and Gp are the elastic properties of the fluid phase. 5
ACCEPTED MANUSCRIPT For the case of clean sandstone saturated by water, the elastic moduli of the solid phase of EPM are that of quartz given in the works of Mavko et al. (2009): Ks = 37.9 (GPa), Gs=44.3 (GPa) and the elastic moduli of the pores are that of water: Kp=2.3 (GPa) and Gp=0. Note
as:
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4 K eff V p2 Vs2 3
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the bulk density
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that Keff and Geff (dynamic moduli) are related to P-wave, S-wave velocities (Vp and Vs) and
Geff Vs2
eq
(4)
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The equivalent porosity
(3)
of EPM can be calibrated using the measurement of the sonic
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velocities and the density. The calibration of this parameter by fitting the sonic velocity
where
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calculated by (1) to (4) with the measurements of Han et al. (1986) gives:
eq 2
(5)
is the porosity of the initial medium. Figure 2 shows the comparison between the P-
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wave and the S-wave velocities computed by using EPM with
eq
=2 and the real measured
data. The difference between the proposed model and the experimental data (shown on the Fig. 2) may due to the fact that the stress sensitivity (in the measurements of Han et al. (1986), confining pressure varies from 5 to 40 MPa) is not taken into account in the model. This stress dependency will be discussed later in this paper. Knackstedt et al. (2003) simulated numerically (based on digitized 3D images) the bulk and shear moduli of quartz-air and feldspar-air mixtures at different porosity range from 0 to 0.5. These data are used to validate the relationship (5) in dry condition. Based on EPM concept, the effective bulk and shear moduli are calculated by (1) and (2). In case of quartz-air mixture, the elastic parameters of the solid and of the fluid phases are: Ks=37.9 (GPa),
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ACCEPTED MANUSCRIPT Gs=44.3 (GPa), Kp=0.0 (GPa) and Gp=0.0 (GPa). Concerning feldspar-air mixture, the elastic moduli of feldspar are (given in Knackstedt et al., 2003): Ks=37.5 (GPa), Gs=15.0 (GPa). Figure 3 shows a strong consensus between EPM’s model with
eq=2
and the numerical
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simulation based on the finite element method (Knackstedt et al., 2003) for both clean quartz-air mixture and clean feldspar-air mixture. Note also that both the bulk and the shear
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moduli vanish at porosity of about 50% (critical porosity, see Nur et al., 1998).
Note that the porosity of the equivalent media could not excess 100%, thus the porosity of
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the initial media could not excess 50%. Fortunately this is the case for sandstone. Artificial sandstone with same diameter has maximum porosity of about 36% and real sandstone with
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small grains filled between big grains has smaller porosity. The shape of grain and pores can increase the porosity but for sandstones it is normally lower than 40% under null confining
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stress (Mavko et al., 2009).
Moreover, the proposed model is calibrated on the laboratory measurement of Han et al.
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(1986) on well-defined samples under confining stress from 5 to 40 MPa with maximum porosity of 36%. Then we can suppose that our model is validated for sandstone with porosity up to 40%. This range is likely to cover all natural compacted and uncompacted
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sandstones.
The comparison of the model to experimental data and numerical simulation based on 3Ddigitized method suggests that the elastic properties of clean sandstone can be estimated using EPM with equivalent porosity of two times bigger than the initial porosity when the stress sensitivity is ignored. In next sections, the impacts of heterogeneous solid minerals and stress sensitivity on the equivalent porosity of EPM will be discussed.
MINERALOGY AND FLUID CONTENTS EFFECTS Rock with multi-minerals and fluid contents is decomposed into two phases: grains supporting and un-supporting soft-phases. The first phase is composed of hard minerals 7
ACCEPTED MANUSCRIPT (such as quartz, calcite, feldspar, etc.) that form the solid skeleton of rock and support the effective stress. The second phase comprises of soft minerals (such as clay, organic matter) and pore fluids (Marion et al., 1992). Note that this consideration should be applied for shaly
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sandstones with low amount of clay and organic contents (Revil et al., 2002; Marion et al.,
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1992).
In the relationships (1), (2) and (5), the porosity is replaced by the volumetric fraction of the soft-phase. The elastic properties of the solids and the fluids are respectively replaced by the
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overall elastic properties of the supporting grains and that of the soft-phase.
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The volumetric fraction of the soft-phase is the sum of the porosity and of the volumetric fraction of clay and organic matter:
(6)
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Vsoft Vclay Vorganic
where Vclay and Vorganic are respectively the volume of clay mineral and organic matter per unit
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volume of rock.
Similarly to the case of clean sandstones, the volumetric fraction of the soft phase of EPM is
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two times greater than that of the initial medium:
eq Vsoft 2Vsoft
(7)
The overall elastic properties of the supporting grains can be estimated by taking the classical Hill’s average (8) and (9).
Ks
K s K s 2
(8)
Gs
Gs Gs 2
(9)
where the upper and lower bounds are expressed as (Berryman, 1995; Mavko et al., 2009):
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(10)
where the exponent
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G 9 K j 8G j with a j j 6 K j 2G j
means upper or lower bounds and the functions
means
is the relative volumetric fraction of the supported grain in
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minimum or maximum values,
(11)
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Gs /
1 fi min/ max aj j i Gi a j
the supporting phase.
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Un-supporting clay and organic constituents are supposed to be suspended in the fluids inside the soft phase. The HS lower bound is appropriate to model the effective properties of
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such a mixture. The overall shear modulus of the soft phase can be approximated to zero.
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Considering an example of a mixture composing of floating clay particles in water, the shear modulus of the mixture can be approximated to zero for volume fraction of clay reach to 80%
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(Figure 4). In this condition, the HS lower bound (Hashin and Shtrikman, 1963) and the
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Reuss average give the same results for the overall bulk modulus of the soft-phase:
Kp
Vsoft
V / K i i
(12) i
Gp 0
(13)
Where Vi and Ki are respectively the volumetric fraction and the bulk modulus of the component i (fluids, gas or unsupported minerals) in the soft-phase. Note that Vsoft=∑Vi. Figure 5 shows the comparison between the P-wave and the S-wave velocity computed using EPM with Vsoft 2Vsoft and the measured data. The deviation between the model and eq
the experimental data are supposed to be due to the stress sensitivity (stress varied from 5 to 40 (MPa)). This stress dependency will be discussed in the next section.
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ACCEPTED MANUSCRIPT STRESS SENSIBILITY We analyze the stress effect on EPM or more precisely the stress sensitivity of the equivalent eq
eq
for the case of clean
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volume Vsoft of the soft-phase of EPM (that is the equivalent porosity eq
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sandstones). The idea is to calibrate the parameter Vsoft to fit better the experimental data
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than that presented in previous section to obtain the ratio between the volume of the softeq
phase of EPM and that of the initial medium ( Vsoft / Vsoft ) for each case of mean effective
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stress. The dependency of the volumetric fraction of the equivalent soft-phase on the mean effective stress is presented by the points in Figure 6 (left side) that are calibrated based on
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the measurements of Han et al. (1986). The power law (14) is found to fit the calibrated points and is proposed to characterize the stress dependency of EPM. Note that in equation . The ratio between the volumetric fraction of
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(14), the unit of the mean effective stress is
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the soft-phase of EPM and that of the initial medium decreases when the confining effective eq
and for high stress condition, Vsoft
tends to 1.5Vsoft .
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stress increases. For free stress condition
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eq Vsoft
Vsoft
1.5 1 p
1 3
(14)
Quartz and calcite grains may be broken under very high stress condition which may affect sonic velocities of sandstone. However, information from measurement does not allow us to calibrate such phenomena. DISCUSSION The heuristic modified Hashin–Shtrikman model is proposed by Walls et al. (1998) to estimate the effective moduli of dry sand with cement deposited out of grain contacts. The material is decomposed by a porous medium with a porosity and shear moduli are then given as: 10
and a solid phase. The bulk
ACCEPTED MANUSCRIPT 1
(16)
is the critical porosity and the Hertz-Mindlin moduli KHM and GHM are defined as:
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where
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Geff
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1
1 0 0 as GHM as Gs as
(15)
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K eff
1 4 0 0 Gs 3 K 4G K 4G s s HM 3 s 3
1
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3 2 3 f s 1 3 f 52 s
(17)
K HM
(18)
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GHM
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K HM
C 2 1 0 2 Gs2 3 p 2 2 18 1 s
contacts;
s
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where C is the average number of contacts per grain; f is the fraction of the perfect-adhesion is the Poisson’s coefficient of solid phase; pꞌis the mean effective stress.
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Replacing Kp=0 and Gp=0 into equation (1) and (2), dry effective moduli are determined. The obtained expression of Keff is compared to equation (15) and the following relationship between the porosity of EPM and the porosity of porous medium is obtained:
K
eq 4Gs K K HM s 30 K s K 4 G HM
3
(19)
s
The obtained expression of Geff is compared to equation (16) and the following relationship between the porosity of EMP and the porosity of porous medium is obtained:
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eq 1 Gs GHM 9 K s 8Gs 60 GHM as K s 2Gs
(20)
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When mean effective stress is very low in comparison to moduli of solid phase, Hertz-Mindlin
K G
1
0
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moduli tend to zero and we obtain:
(21)
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Expression (21) shows that with no loading, the ratio of EPM’s porosity and that of the initial
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material equals inverse of the critical porosity. The theoretical value of critical porosity is the porosity of a random pack of identical spherical grains that is 0.36 (Mavko et al., 2009). However, in reality this parameter depends on the microstructure of material. In fact, the
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critical porosity of sandstone is about 0.40 (Mavko et al., 2009). Using this value of
0,
the
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ratio of EPM’s porosity and that of the initial material is found equal to 2.5 which is in
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agreement with equation (14) (for free stress condition) obtained above from the calibration on experimental data of Han et al. (1986). Equations (19) and (20) also allow confirming a K
and
G
with the mean effective stress.
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decrease of
CONCLUSION
In this paper, the concept of EPM is proposed to model elastic properties and sonic velocities of uncemented sandstones. A scaling factor of 2 is multiplied to the initial rock porosity to obtain EMP porosity in the case of clean sandstone. When extending to shaly sandstone, the model parameters must comprise of the porosity, the mineralogy and the stress effects. In this case, EPM is the porous medium of a supported solid matrix and soft phase inclusions. The solid matrix is the combination of the grains that support the effective stress. The soft inclusions are formed by fluids, gas, unsupported clay and organic constituents. A factor of 2 is again used for defining the volume of EPM soft-phase. However, to take the stress dependency into consideration, this factor can varies between 1.5 and 2.5. The results were 12
ACCEPTED MANUSCRIPT verified against the experimental measures of a sandstone rock (Han et al.1986). This approach shows a strong agreement with the heuristic modified Hashin-Strikman model
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proposed by Walls et al. (1998).
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ACKNOWLEDGEMENTS
The authors wish to express their special thanks to the reviewers for their contribution to
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improve the quality of this paper.
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Nguyen, S. T., Dormieux, L., Yann, L. E., & Sanahuja, J. (2011). A Burger model for the effective behavior of a microcracked viscoelastic solid. International Journal of Damage
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Norris, A. N., & Johnson, D. L. (1997). Nonlinear elasticity of granular media. Journal of
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Sayers, C. M. (2002). Stress‐dependent elastic anisotropy of sandstones. Geophysical
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prospecting, 50(1), 85-95.
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Sayers, C. M., & Kachanov, M. (1995). Microcrack‐induced elastic wave anisotropy of brittle rocks. Journal of Geophysical Research: Solid Earth (1978–2012), 100(B3), 4149-4156. Schöpfer, M. P., Abe, S., Childs, C., & Walsh, J. J. (2009). The impact of porosity and crack
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density on the elasticity, strength and friction of cohesive granular materials: insights from DEM modelling. International Journal of Rock Mechanics and Mining Sciences, 46(2),
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des équations intégrales singulières. PhD Thesis. Ecole des Ponts ParisTech 2012.
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Walls, J.D., Dvorkin J., & Smith B.A (1998). Modeling Seismic Velocity In Ekofisk Chalk. SEG
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Annual Meeting, 13-18 September, New Orleans, Louisiana. Walsh, J. B. (1965). The effect of cracks on the uniaxial elastic compression of rocks. Journal of Geophysical research, 70(2), 399-411.
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Willis, J. R. (1981). Variational and related methods for the overall properties of composites. Advances in applied mechanics, 21, 1-78. Wyllie, M. R. J., Gregory, A. R., & Gardner, L. W. (1956). Elastic wave velocities in heterogeneous and porous media. Geophysics, 21(1), 41-70.
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Solid mineral
Solid matrix
EPM
s
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Pore s
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Interface
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Figure 1. The concept of Equivalent Porous Medium (EPM)
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Equivalent porosity
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5.0
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Computed velocity [km/s]
6.0
2.0
Vs
3.0 4.0 5.0 Measured velocity [km/s]
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2.0
Vp
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Figure 2. EPM concept for water saturated clean sandstone with eq 2 . Comparison with
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the data measured by Han et al. (1986)
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50
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EPM model
35 30 25 Quartz
20 15 10
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Numerical simulation
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45 40
35
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Effective shear modulus [GPa]
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Feldspar
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0.1
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0.3 0.4 Porosity
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Feldspar
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Numerical simulation
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Effective bulk modulus [GPa]
EPM model
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Figure 3. Comparison between EPM concept for dry clean sandstone with
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eq=2
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Shear mdulus (GPa)
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Volumetric fraction of clay in soft phase
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Figure 4. Hashin-Shtrikman under bound for shear modulus of a mixture of floating clay particles in water. The elastic moduli used for the simulation are: clay bulk modulus is 21
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T SC R
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Computed velocity [km/s]
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Figure 5. Comparison between EPM concept for water saturated shaly sandstones with
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3.0
eq Vsoft
eq Vsoft
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Figure 6. Stress sensitivity of EPM
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Vp Vs
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10 20 30 40 Mean effective stress [MPa]
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Equivalent porous media is defined for modeling of sonic properties of sandstones. The simulation is compared with laboratory measurements The effect of clay content is accounted The stress sensitivity is also discussed.
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