Petrophjysics to Well Log

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GEOPHYSICS, VOL. 64 , NO. 6 (NOVEM BER-DECE MBER 1999); P . 1756-1759,6 FIGS.

Short NoteIdentifying patch y saturation from w ell logs

Jack Dvorkin *, D a nie l Mo o s , J a mes L. Packwood**and A mos M. Nur*INTRODUCTION

Ga ssmann's (1951 ) equations relate the elastic bulk (Ksat)and shear (Gsat) moduli of a fully saturated rock to the elasticbulk and shear moduli of the dry-rock frame (K d , y and G dry ,respectively), porosity , and the bulk m oduli of the mineralphase (K s ) and pore fluid (K f ):IKdry (1 + q5 )Kf Kdry /KS +Kf(1 4)K f + 4K K fKdry/K,s

(1)Gdry=Gsat=G.

These equations are valid at low frequencies w hich, in mostpractical cases, include the seismic and sonic data ranges (thebounds for the low -frequency approximation are d iscussed inMa vko et al. , 1998). Gassmann's equations are used for porefluid identification from seismic and sonic because the elasticmoduli are directly related to V P and Vs :V P =/(Ksat + 4G/3)/p,S = G/p,2)where p is bulk density.When using Gassmann's equations at partial saturation, onemust calculate the effective bulk mod ulus of a fluid mixture inthe pore space. One physically meaningful assumption is thatwave-induced increments of pore pressure in each phase of themixture equilibrate during a seismic period. T his assumptionleads to the isostress (Reuss) avera ge for the effec tive bulk

modulus of the mixture (e.g., gaswater):Kf1 =SWKw'+(1 Sw)K g 1 ,3)where S w is water sa turat ion and where K w and K 9 are thebulk moduli of water and gas, respectively. The condition of

pore-pressure equilibrium is satisfied if imm iscible gas and w a-ter coexist at the pore sc ale, i.e., every po re contains S,, volu-metric fraction of water and 1 S W fraction of gas. It can bealso satisfied if gas and water occ upy fully dry and fully satu-rated patches, respectively. However, the length scale of suchpatches m ust be smaller than K K w / f,u, where K is permeabil-ity, f is seismic frequency, and is dynamic viscosity of water(Mavk o and M ukerji, 1998). This upper bound applies to anymultiphase mixture, with K W and being the bulk mod ulusand viscosity of the most viscous phase. I t can be on the orderof a few millimeters at sonic frequencies and on the order ofa meter at seismic frequencies. This state of partial saturationwhere equation (3), tog ether with Gassmann's equations, is ap-plicable for calculating the rock's elastic moduli is uniform orhomogeneous saturation (Ma vko and M ukerji, 1998).

Given that K 9 < < K W , the value of K f as calculated fromequation (3) remains much sm aller than K W at all saturationsexcept those very close to 100% . Therefore, Vp remains practi-cally insensitive to water saturation in the do minant part of itsrange (Figure 1). Domenico (19 76, 1977 ) seems to be the firstto reproduce this effect experimentally. He also notes that ve-locity (in sand and glass beads) may significantly deviate fromthe equation (3) curve if saturation is heterogeneous.To ca lculate the elastic moduli of rock w ith patchy saturation,consider a volume of rock at partial saturation S W . The dryframe of the rock is elastically uniform within the volume. Letus assume that saturation is perfectly patchy, i.e. , all water isconcentrated in fully saturated patches and ga s is concentratedin patches with S W = 0 . This situation is perceived as a limitingcase that may not necessarily be realized in rocks (since thereis always residual water saturation in dry patches).At patchy sa turation, Ga ssmann's equations can be appliedseparately to fully saturated and dry patches. The volume fra c-tion of the former in the volume is S.. The volume under

Ksat = Ks

Manuscript received by the Editor July 7, 1997; revised manuscript received February 26, 1999.*Department of Geophysics, Stanford University, Mitchell Building, Stanford, California 94305-2215. E-mail: [email protected]; [email protected].$Geomechanics International, 2250 Park B oulevard, Palo Alto, California 94306. E-mail: moos@ geomi.com.**BP Am oco, 501 W estlake Park Bo ulevard, Houston, Texas 77079. E-mail: packw ood@ alumni.stanford.org., 1999 Soc iety of Exploration Geophysicists. All rights reserved.1756

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Patchy Saturation from W ell Logs75 7exam ination includes regions with different bulk mo duli butthe same shear mod ulus. The effective bulk mod ulus Ksatp ofthe volume is (Hill, 1963)(KsatP +4G/3)-1= Sw (K1 +4G/3) '

+ (1 Sw)(K& + 4G/3) I , (4 )where K l and K o are the bulk mo duli of the rock a t S. = 1 andS W = 0, respectively. At patchy saturation, V, is very sensitiveto S W (Figure 1).At patchy saturation, equation (1) ca nnot be applied to theentire rock volume bec ause there is no such physical entityas uniform pore fluid. Still, mathematically a pore-fluid bulkmodulus exists that, if substituted into Gassmann's equations,will yield any desired KSatp. Dom enico (1976) proposes calcu-lating this modulus of fictitious homogeneous pore fluid fromthe isostrain (Voigt) avera ge:

Kf = S w K,,, + (1 S,,,)K g. (5These values are close to the (fictitious) pore-fluid modulusback-calculated from the d ata using Ga ssmann's equations.Brie et a l . (1995) observe a d eviat ion of the V p/Vs ratiof rom typical hom ogeneous saturat ion va lues and a t t ributeit to patchy saturation. To model this deviation, they useGassmann's equations with an additional, ad hoc expressionfor the bulk modulus of the fictitious uniform pore fluid:

Kf = (K u, Kg )S + Kg, (6where e is a calibration parameter (Figure 1). By adjusting e,Brie et al. match the in-situ data.In this paper, we o ffer a first-principle-based (as op posedto ad ho c) m ethod for identifying patchy sa turation in situ.First we assume that saturation is homog eneous and use equa-t ions (1), (2), and (3 ) to calculate the dry-fram e bulk andshear moduli from V, and V. T hen we ca lculate the dry-frame

Poisson's ratio asV d ry = 0.5(3Kdry/G 2 )/(3Kdy/G +1 ).7 )

We use v a r y as the criterion for patchy saturation: if its values,as calculated from in-situ data and the homogeneous satura-tion equations, exceed a rea sonable experimentally establishedthreshold, saturation is patchy. In this case one must furtherverify the existence of patchiness by obtaining reasonable va ryfrom the patchy saturation equations. Once sa turation type isestablished, saturation values can be c alculated from so nic logdata using the appropria te (homog eneous or patchy) set ofequations. This routine eliminates the need for adjusting non-physical free parameters.We a pply this criterion to sonic log d ata from a soft formationand identify a patchy saturation interval.

P OIS S ON'S RAT IO AS A D IS CRIM INAT ORF O R S A T U R A T I O N P A T T E R ND ry-fra me v a l ues

Spencer et al. (199 4) report that at in-situ pressure many un-consolidated sands from the G ulf of Mexico have d ry-framePoisson's ratios near 0.18, while other Gulf Co ast reservoirshave these values as low as 0.115. Only occasionally (among alarge number of rock samples) does v ar y exceed 0.2, w ith themaximum value 0.237. To further support this fact, we presentVd ry data at 20 MP a effective pressure reported by H an (1986)for consolidated sandstones, Jizba (1991) for tight gas sand-stones, and Strandenes (1991) and B langy (1992 ) for North Seasandstones (Figure 2). All data sets include rocks w ith varyingamounts of clay.M ost of the data points lie below 0.2. Only a few of themreach 0.25. In Figure 2a we also show a data po int relevant tothe case study below. This datum (Y in, 1993) is used in Figure 1(see description there). The sample is an artificial mixture ofOttawa sand and k aol ini te wi th 10% weight fract ion of thelatter. Its v ar y is 0.23.Based o n these observations, we chose 0.25 as an upperbound for Vd ry in sands.

2.5 Ottawa+ Clay at 20 MPaPorosity = 28.4%dentifying patchy saturationIn Figure 3a w e plot the Poisson's ratio of the Ottaw a/ kao li-nite sample versus saturation. The curves are computed fromthe dry-rock data a t 20 M Pa using homogeneous and patchy

B13

1.5o.5Sw at 20 MPa.. . .an._........lzba DRockat 20 MPa . . . . . . . . . . . . . . . Ottawa+- . . . . . .Clay1 2.0FIG. 1. Velocity versus saturation for a sample that is an arti-ficial mixture of Ottawa sand and kaolinite with 10% weightfraction of the latter (Yin,1993). Porosity is 28.4% . At 2 0 M Paeffective pressure, the dry-frame mo duli of the rock are 2.637and 1.74 GPa for bulk and shear, respectively. The bulk mod-uli of gas and w ater are 0.033 and 2 .9 GPa, respectively. Thecurves are H for homo geneous saturation, P for patchy satu-ration, V for Voigt fluid mixture as in equation (5), B2 for theBrie mixture with e = 2 as in equation (6) , B3 for e = 3, andB13 for e = 13. A ll curves are calculated from the dry-roc kmoduli using equations (1) (6). B.trandeneslangy0.1.2.3.4.1.2.3.4yyFIG. 2. (a) D ry-rock Poisson's ratio versus porosity for consol-idated low -to-medium-porosity sandstones. (b) H igh-porosityfast (S. Strandenes, personal communication, 1991) and slow(Blangy, 1992 ) sands. The filled square in (b) is for Yin's (1993)data.Downloaded 07 Jun 2011 to 202.46.129.17. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/


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1758 Dvorkin et al.saturation equat ions. The dif ference between the homog e-neous and patchy Poisson's ratio is drastic.W e take ad vantage of this difference in identifying the sat-uration pattern from measurements. Let us assume that thesample has patchy saturation and that the measured V, andV, are 1.8 and 0.934 km/s, respectively, at 40% saturation (Fig-ure 1). The bulk density is 2.03 g /cm 3 . Also, the measurementsat other saturations are not ava ilable; neither are the dry-fram eelastic moduli of the rock. Given these measured va lues andthe properties of the mineral and pore-fluid phases, we calcu-late the dry-frame elastic properties using the inverse of equa-tion (1):

_ 1 (1 QI)Ksat/Ks 4Ksat/Kj (8 )dry KS 1 + OKa/Kf Ksat/Kswith K f given by equation (3). The resulting Vd ry is 0.316.The mistake we m ade by c hoosing the homog eneous in-version becomes apparent because 0.316 significantly exceeds

FIG. 3. Po isson's ra t io versus saturat ion. (a) Soft Ottaw a-kaolinite sample (Yin, 1993). (b) Fast No rth Sea sample. Solidblack curves are for the values calculated from the dry-framemod uli. H is for homo geneous, and P is for patchy. The graysolid lines are for v ar y calculated from patchy data by hom oge-neous inversion. The gray d otted line is for the upper bound(0.25) of Vd ry .

the upper bound of 0.25. The p atchy inversion gives a reason-able value of 0.23. The homog eneous inversion, if applied to thepatchy data, gives unreasonable vary values over practically theentire S,. range (Figure 3a). Therefore, we use Vd ry calculatedfrom data by homogeneous or patchy inversion as an indicatorof a saturation pattern: if its values are unreasonable, the ap-plied inversion is wrong and the saturation pattern is oppositeto that assumed.This method may fail for fast rock s. Consider a No rth Seasample of 28.9% po rosity with the dry-frame bulk and shearmod uli 8.75 and 7.69 GP a, respectively, at 20 M Pa effectivepressure (Strandenes, 1991). The c alculated Poisson's ratio atpartial saturation is given in Figure 3 b. The ho mogeneous inver-sion applied to the patchy data produces wrong but reasonable

var y values in the entire saturation range.C A S E S T U D Y

W e examine open-hole log curves from a vertical well thatpenetrates unconsolidated and very soft ga s sands. The fre-quency of the sonic tool is such that both P- and S -wa vessample the formation at about 1 m away from the well . Theshale content in the gas-saturated interval, as calculated fromthe gamma-ray curve, varies between 0.1 and 0.4. However, theclay content measured on cores is as low as 0.1 (Figure 4a). Thetotal porosity and saturation w ere calculated by simultaneouslysolving the bulk-density equation and A rchie's equation w iththe Humble formation factor formula for sands (Schlumberger,1989). It is very close to the co re porosity (Figure 4b). Watersaturation may be as low as 0.2 (Figure 4c). The grain densitywas estimated as 2.64 g/cm 3 based on 10% clay content ; thecorresponding solid-phase bulk modulus is 34 G Pa. The bulkmoduli and densities of water and gas are 2.9 and 0.033 GPaand 1.055 and 0.112 g/cm 3 , respectively.A question remains w hether Archie's equation is applica-ble to estimating saturation from resistivitynot only for the

FIG. 4. (a) Shale and clay content versus depth. The curve is calculated from the gamma-ray data; symbols indicate core-measuredvalues. Depth, as shown, is measured depth m inus a constant. (b) Neutron porosity (NP NI), porosity calculated from bulk density(PhiRHO ), total porosity (solid black curve), and core-measured porosity. (c) W ater saturation.Downloaded 07 Jun 2011 to 202.46.129.17. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/


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Patchy Saturation from W ell Logs75 9To illustrate the importance of identifying patchy saturationin situ, we ca lculate the dry-fram e bulk mo dulus at the well . Ifsaturation is assumed to be homogeneous (common fluid sub-stitution approach), the errors in Kd ry may exceed 50% (F ig -ure 6b). Such errors must be taken into a ccount because theymay strongly affect seismic interpretation for d irect hydrocar-bon indication.D I S C U S S IO N A N D C O N C L U S I O N

FIG. 5. Velocity versus total porosity. G ray symbo ls are for thiscase study.This case study show s that patchy saturation can be d etected

in situ using the dry-frame Poisson's ratio (calculated fromdata) as a discriminator. We em phasize that both homo geneousand pa tchy saturation mod els must be treated as idealized endmembers (or bounds) for a realistic situation. Where one ofthese two fluid substitution methods fails to give reasonableelastic mo duli values for the dry frame, the other may do so. Wesuggest that in unconsolidated partially saturated sands, twof luid subs t itut ion techniquesho mogeneous a nd pa tchy beused to ca lculate the dry-rock elastic m oduli from w ell logs. Ifthe homog eneous inversion gives dry-rock Poisson's rat ios thatexceed 0.2-0.25 and the patchy inversion gives lower and morereasonable values, the interval is l ikely to be patchy sa turated.Then the dry-rock bulk mo duli must be calculated accordingly.We recommend using both techniques for bounding the possi-ble dry-rock Po isson ratio and bulk m odulus values.The solution given in this paper show s that there is no needfor introducing ad hoc nonphysical mod els with adjustable freeparameters . Patchy saturat ion can b e ident if ied and qua nt ita-tively interpreted using rigoro us equations.

ACK NOWL ED G M ENTFIG. 6. (a) Po isson's ratio versus depth (measured d epth minusa co nstant as in Figure 4). The bold gray curve is for measuredPoisson's ratio. Black curves are for homogeneous and patchyvdty values. Vertical bars indicate likely intervals with patchysaturat ion. (b) D ry-f rame bulk mo dulus corrected for patchysaturat ion (sol id black curve) . Dot ted curves are for hom oge-neous (H) and patchy (P) invers ion.homog eneous, but also for patchy saturation. W e are not aw areof any experimental data to refute or support our statement.The measured V p and V, values are plot ted versus poros i tyin Figure 5. We a lso plot the velocit ies for the North Sea high-porosi ty samples and the Ottaw a and kaol inite sample, a l l a t20 M Pa effective pressure (the approximate effective pressurein the well) and 25% saturation for com parison. The North Seavelocit ies significantly exceed those in the w ell . How ever, themixture of Ot tawa sand and 10% kao l ini te is e las t ically veryclose to the gas-saturated formation.To calculate vd ry (Figure 6a) , we separa tely use the homo-geneous and patchy inversion [the latter directly follow s fromequation (4) and is described in Dvorkin and Nur, 1998]. An in-direct indication of patchy saturation between 1495 a nd 1520 mis the unreasonably high dry-frame homogeneous Vd ry . Thepatchy vd nly sl ightly exceeds the 0.25 upper bound and re-mains w ithin reasona ble l imits.The o ther possible interval with patchy saturation is between1480 and 1490 m. Although the homogeneous inversion givesa vdry that is below 0.25, the patchy vd ,y has lower and morereasonable values.

This work was supported by the Stanford Rock Physics Lab-oratory.R E F E R E N C E S

Blangy, J. P., 1992, Integrated seismic lithologic interpretation: Thepetrophysical basis: P h.D. thesis, Stanford U niv.Brie, A., Pampuri, F., Marsala, A. F., and M eazza, 0., 1995, Shear sonicinterpretation in gas-bearing sands: SPE paper 30595, 701-710.Domenico, S. N., 1976, Effect of brine-gas mixture on velocity in anunconsolidated gas reservoir: Geo physics, 41, 882-894.1977 , Elastic properties of unconsolidated porous sand reser-voirs: Geophysics, 42, 1339-1368.Dvorkin, J., and Nur, A., 1998, Acoustic signatures of patchy satu-ration: Internat. J. Solids and Structures, 35, 4803-4810.Gassmann, F., 1951, Elasticity of porous media: Uber die elastiz-itat poroser medien: Vierteljahrsschrift der Naturforschenden Ges-selschaft, 96,1 -23.Han, D.-H., 1986, Effects of porosity and clay content on acousticproperties of sand stones and unco nsolidated sediments: Ph.D. thesis,Stanford Univ.Hill, R., 1963, Elastic properties of reinforced solids: Some theoreticalprinciples: J. Mech. Phys. Solids, 11, 357-372.Jizba, D., 1991, Mechanica l and acoustical properties of sandstones andshales: Ph.D. thesis, Stanford Univ.Mavko , G., and Mukerji, T., 1998, Bounds on low-frequency seismicvelocities in pa rtially saturated rock s: G eophysics, 63, 918-924.Mavko, G ., Mukerji, T., and Dvorkin, J. , 1998, The rock physics hand-book: C ambridge Univ. Press.Schlumberger, 1989, Log interpretation principles/applications:Schlumberger Wireline & Testing, H ouston.Spencer, J. W., Cates, M. E., and Thompson, D. D., 1994 , Frame mod-uli of unconsolidated sands and sandstones: Geophysics, 59, 1352-1361 .Yin, H., 1993, Acoustic velocity and attenuation of rocks: Isotropy,intrinsic anisotropy, and stress induced anisotropy: Ph.D . thesis,Stanford Univ.

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Petrophjysics to Well Log