Different forms of Gassmann equation and implications for dang nhap 188bet estimation of rock properties

Summary

Three new equivalent forms of Gassmann equation are presented that are useful when dang nhap 188bet unknown parameters are dang nhap 188bet Biot- Willis coefficient, dang nhap 188bet dry bulk modulus, and/or dang nhap 188bet grain matrix bulk modulus. We apply these equations to several sets of laboratory measurements to determine dang nhap 188bet profiles of grain matrix bulk modulus and Biot-Willis coefficient as functions of applied pressure, and perform a Monte Carlo simulation to examine dang nhap 188bet effect of uncertainty and/or measurement errors on dang nhap 188bet calculated grain matrix bulk modulus and Biot-Willis coefficient. dang nhap 188bet results show that dang nhap 188bet calculated grain matrix bulk modulus is relatively constant with applied differential pressure (up to 50MPa). However, it is very sensitive to dang nhap 188bet uncertainty of dry and saturated bulk modulus values. Thus, dang nhap 188bet presented new forms of Gassmann equation can be used to effectively quantify dang nhap 188bet uncertainty of dry and saturated bulk modulus (and subsequently, dang nhap 188bet seismic velocities) in fluid identification, fluid substitution, or reservoir monitoring applications.

Key words: Gassmann equation, bulk moduli, Biot-Willis coefficient, sensitivity analysis.

1.       Introduction

dang nhap 188bet Gassmann equations [1] have been used extensively in dang nhap 188bet oil and gas industry for fluid identification and reservoir monitoring applications, despite its various assumptions [2, 3]. dang nhap 188bet first Gassmann equation provides dang nhap 188bet relationship between  dang nhap 188bet  saturated  bulk  modulus  of a rock and its dry frame bulk modulus, porosity, bulk modulus of dang nhap 188bet mineral matrix,  and  bulk  modulus  of  dang nhap 188bet pore-filling fluid. Whereas, dang nhap 188bet second Gassmann equation simply states that dang nhap 188bet shear modulus of dang nhap 188bet rock is independent of dang nhap 188bet presence of dang nhap 188bet saturating fluid:





Where α is dang nhap 188bet Biot-Willis coefficient [4]:

 

dang nhap 188bet moduli are related to dang nhap 188bet seismic velocities and density by:



Berryman [5] gave a concise derivation of Gassmann equations for an isotropic and homogeneous medium using   dang nhap 188bet   quasi-static   poroelastic   theory.   Other forms of Equation (1) can be found in Mavko et al. [6]. Zimmerman [7] presented an equivalent  form  in  terms of compressibility. However, Equation (1) is probably dang nhap 188bet most intuitive in describing dang nhap 188bet effect of fluid presence on dang nhap 188bet bulk modulus.

White and Castagna [8] argued that, since all input parameters for Gassmann equations carry some degrees of uncertainty, a fluid modulus inversion should be performed using a probabilistic approach. Artola and Alvarado [9] evaluated dang nhap 188bet effect of uncertainty of different input parameters and showed that dang nhap 188bet computed compressional velocity of a saturated rock is most sensitive to uncertainties in dang nhap 188bet rock bulk density and dang nhap 188bet dry bulk and shear moduli, while other parameters (porosity, dang nhap 188bet grain matrix and fluid bulk moduli) have negligible effects.

Note that dang nhap 188bet three parameters: dry frame modulus (Kdry), Biot-Willis coefficient (α), and grain matrix bulk modulus (Km) are related by Equation (3); in many instances they are unknown. dang nhap 188bet fluid saturated bulk modulus (Ksat) and fluid bulk modulus (Kf) can also be unknown (e.g. in fluid substitution problem). Asaresult, ad-hoc and empirical correlations have been proposed to address this problem. There are many instances Biot-Willis coefficient is assumedto be 1 due to dang nhap 188bet lack of a better estimate. For sandstone at high  differential  pressure  (40MPa),  Han and Batzle [10] proposed α to be a polynomial function of porosity:



In this study, we present 3 new equivalent forms of Gassmann equation that are useful for different scenarios of available data. We apply these equations to  several sets of laboratory measurements. A stochastic simulation is performed to examine dang nhap 188bet effect of uncertainty and/    or measurement errors on calculated grain matrix bulk modulus and Biot-Willis coefficient.

2.       dang nhap 188bet new equivalent Gassmann equations

When (K , K , K , and φ) are known

This is generally dang nhap 188bet case for laboratory measurements on dry and wet rock samples (e.g. dry and brine saturated acoustic velocities are measured as functions of differential pressure along with rock porosity). In this case we can rewrite Equation (1) as a function of Biot-Willis coefficient α (see Appendix A for a detailed derivation):



Equation (7) is a quadratic equation A2 + B + C = 0, where all coefficients can be readily calculated.



This simple quadratic equation has two solutions:

(11)

However,  Berryman and Milton(11) showed that α is physically bounded between 0 and 1. Equations (9) and (10) show that B is negative since Kdry < Ksat, and C is also negative since Kf < Kdry for consolidated rocks. This means

∆ = B2 – 4AC> B2 and thus, 1=…. Is dang nhap 188bet only.

  Therefore, instead of having two non-linear equations for two unknowns (α and Km), we reduce dang nhap 188bet problem to one simple quadratic equation, Equation (7), that always gives one physically realistic solution.

This provides an independent methodology to calculate dang nhap 188bet grainmatrixbul kmodulusofarockfrom SCAL laboratory acoustic  measurements of dry and saturated rock samples. Traditionally, dang nhap 188bet grain matrix bulk moduli are estimated from averages of dang nhap 188bet rock mineralogical composition (e.g. Voigt-Reuss-Hill average or Hashin- Shtrikman bounds). These bounds may carry large uncertainties since many minerals, especially clays, have a high variance in their bulk modulus values depending dang nhap 188bet measurement conditions [12, 13]. We can further  postulate that: (a) dang nhap 188bet grain matrix calculated from Gassmann equation (using Equations (8) to (12)) must lie between dang nhap 188bet two bounds obtained from mixture theory, and (b) dang nhap 188bet calculated grain matrix values are insensitive to dang nhap 188bet first order to dang nhap 188bet applied pressure. Equations (7) to (11) can also be used to verify dang nhap 188bet applicability of existing estimate Biot-Willis coefficient) for different rocks.

2.2. When (Ksat1, Ksat2, Kf1k Kf2, and ɸ)are known

This case can be encountered in dang nhap 188bet field. dang nhap 188bet same rock can be fully saturated with brine in one well; or it can have verying saturation in dang nhap 188bet same well. Acousstic logs, and density- porosity logs are available. In this case, K   , K  , and α  are unknown in a system of three non-linear equations (two Equation (1) for two different saturation fluids and  Equation (3)). Starting from Equation (7) instead, we end up with (see Appendix B for detaild derivations)

 (12)

(13)

We can write Equation (13) in a more convenient form for numerical calculations:

 possible solution since α2 is negative.

dang nhap 188bet corresponding grain matrix bulk modulus then can be calculated from Equation (3):



 
Kdry, α, and Km can now be calculated very quickly using a simple iteration using Equation (14) and Equation (7)    as follows:

-        Step 1: Make an initial guess, for example:



-        Step 2: Use guessed Kdry value in Equation (7) to find two Biot-Willis coefficients αf1 and αf2 (for two saturations):



-         Step 3: Take dang nhap 188bet average for a new Biot –Wills coefficient



-        Step 4: Use this new α in Equation (14) to find new Kdry.

-        Step 5: Repeat steps 2 to 4 until Kdry converges:



-        Step 6: Use Equations (7) and (12) to find corresponding α and Km.

Note that we have assumed there are no softening or hardening effects caused by dang nhap 188bet saturating fluids on dang nhap 188bet grain bulk modulus (Km is constant). dang nhap 188bet secondassumption  is  that  dang nhap 188bet  rock  dry  frame  is  stiffer  than both fluids, Kdry > max { Kf 1, Kf 2 }, so that Equation (7) still gives only one positive (physically realistic) root. This assumption is generally valid for consolidated sedimentary rocks.

2.3. When (Km, Ksat, Kf, and φ) are known

In this case Kdry and α are unknown. An instance for this case is that fluid data, fluid saturation, acoustic log (Vp, Vs) and density-porosity log are available while Km is estimated from dang nhap 188bet mineralogical composition of dang nhap 188bet rock (FTIR, XRD, thin section of rock cuttings, or mineralogy log). dang nhap 188bet Biot-Willis coefficient can be estimated directly from dang nhap 188bet following equivalent Gassmann equation (see Appendix C for detailed derivations):

(15)

 
And applying this α to Equation (3) gives Kdry. This is equivalent to dang nhap 188bet Kdry solution of Zhu and McMechan [14].

3.       Numerical applications

          Han and Batzle’s sandstone data

We applied Equation (7) to dang nhap 188bet pressure dependent sandstone sample published by Han and Batzle [10] (Figure using dang nhap 188bet density relaρtionshρip:  

 (16)
dang nhap 188bet calculated Biot-Willis coefficient and grain matrix modulus as functions of pressure are plotted in Figure 1. dang nhap 188bet relatively constant value of dang nhap 188bet grain bulk modulus (39GPa) as a function of pressure is a good indicator that Gassmann equation is applicable for this rock. dang nhap 188bet variation of grain bulk modulus at low confining pressure (< 10MPa) is possibly due to higher uncertainty in input values (i.e. higher noise-to-signal ratio from velocity signals).

dang nhap 188bet Biot-Willis coefficient profile is remarkably similar to dang nhap 188bet result measured on a 26% porosity Boise sandstone sample by Fatt  [15]. Note that Gassmann equation gave   a higher value (0.73 at 40MPa) than Han and Batzle’s Equation (6) (0.63).

          Coyner’s limestone data

We employed dang nhap 188bet iteration procedure using Equations (7) to (14) on water and benzene saturated Bedford limestone sample published by  Coyner  [16]  (Figure  2).  In his experiment at room temperature, dang nhap 188bet fluid pore pressure in both saturation cases was maintained at 10MPa. dang nhap 188bet porosity of dang nhap 188bet rock is 11.9%. dang nhap 188bet shear modulus  profile  is  almost  identical  for  all  vacuum  dry, water saturated, and benzene saturated cases, suggesting

dang nhap 188bet back-calculated dry bulk modulus is also plotted (as a dashed line) against dang nhap 188bet various measured moduli in Figure 2. dang nhap 188bet profile is consistently higher than dang nhap 188bet measured vacuum dry bulk modulus profile by approximately 2.5GPa (or 5 - 9%). This is another evidence supporting dang nhap 188bet argument that dang nhap 188bet vacuum dry measured bulk modulus is too dry and should not be used in Gassmann equation [6, 17]. We could have applied dang nhap 188bet measured vacuum dry and either water- or benzene-saturated moduli values on Equation (7), but  that approach would give unrealistically high grain matrix bulk modulus.

Figure 1. Grain bulk modulus and Biot-Willis coefficient of a sandstone sample as a func- tion of pressure calculated from its dry and brine saturated moduli [10] using Gassmann equation.

Figure 2. Bulk and shear moduli as a function of differential pressure for Bedford lime- stone [6]. dang nhap 188bet dashed line is dang nhap 188bet dry bulk modulus calculated from Gassmann equation, consistently higher than dang nhap 188bet vacuum dry measured data.

Figure 3. Calculated grain matrix bulk modulus and Biot-Willis coefficient of Bedford limestone sample as a function of pressure using Gassmann equation from its water - and benzene - saturated moduli [16].

dang nhap 188bet grain matrix bulk modulus and Biot-Willis coefficient profiles obtained from dang nhap 188bet rock water- and benzene-saturated moduli are shown in Figure 3. While dang nhap 188bet grain matrix bulk modulus is similar to Coyner’s reported value of 65GPa, dang nhap 188bet back-calculated Biot-Willis coefficient profile decreases from 0.6 to 0.53, significantly lower than dang nhap 188bet commonly assumed value of 1 while significantly higher than dang nhap 188bet estimated value of 0.34 obtained from Han and Batzle’s 2004 correlation.

          Effects of input data errors on calculated grain bulk modulus

Measured values always have some associated errors. Velocities, especially shear wave velocities may carry significant uncertainties. We would like to determine dang nhap 188bet effects of uncertainties from porosity, Kdry, Ksat, and Kf to dang nhap 188bet uncertainty of dang nhap 188bet predicted Km. Since dang nhap 188bet relationship in Equation (7) is not linear, a Monte Carlo (stochastic) simulation was used.

Table    1   summarises   dang nhap 188bet   input   parameter  values [18] and their estimated ranges of  uncertainties.  dang nhap 188bet  rock sample is a Berea sandstone sample with Voigt- Reuss-Hill average grain bulk modulus  of  39.6GPa from its mineralogical composition. All parameters were assumed to have a normal distribution with means being dang nhap 188bet measured values and dang nhap 188bet errors represent dang nhap 188bet 95% confidence interval. Thus, dang nhap 188bet relative error (uncertainty) of each parameter is defined as:

 (17)

 
Where s is dang nhap 188bet standard deviation of dang nhap 188bet parameter’s sample.

For each set of perturbed errors, 10,000 sets of (porosity, dry bulk modulus, wet bulk modulus, and fluid modulus) values were generated to compute 10,000 grain bulk moduli, which are then analysed for dang nhap 188bet mean value and standard deviation.

In our base case, porosity is assigned a 1% error,  Kdry and Ksat are each assigned a 3% error,  and Kf carries  a 10% uncertainty. dang nhap 188bet resulting Km is also a Gaussian distribution with a mean of 44.6GPa and a standard deviation of 3.45GPa. dang nhap 188bet 95% confidence interval is, therefore, from 37.7GPa to 51.5GPa (or 16% error). dang nhap 188bet Biot-Willis coefficient α is also a Gaussian distribution with a mean of 0.62 and a standard deviation of 0.03. dang nhap 188bet 95% confidence interval is from 0.56 to 0.68 (or 10% error).

Table 1. Mean (measured) values of a Berea sandstone sample [18] and ranges of uncertainties used in Monte Carlo simulations

Figure 4. dang nhap 188bet uncertainty of dang nhap 188bet computed grain matrix bulk modulus Kmusing Gassmann equation as functions of percent error in one input parameter (Kf, α, Kdry, or Ksat), while dang nhap 188bet remaining input parameters carry dang nhap 188bet same errors as dang nhap 188bet base case. Errors from Ksat and Kdry have dang nhap 188bet largest impacts on dang nhap 188bet uncertainty of calculated Km. Porosity and fluid bulk modulus, on dang nhap 188bet other hand, show negligible effects.

Figure 4 shows dang nhap 188bet uncertainty of dang nhap 188bet computed grain matrix bulk modulus Km as functions of percent error in one input parameter (K , φ, K , or K ), while dang nhap 188bet remaining input parameters carry dang nhap 188bet same uncertainties as of dang nhap 188bet base case. Errors from Ksat and Kdry have dang nhap 188bet largest effects on dang nhap 188bet uncertainty of Km. Minus errors in Kdry and Ksat (even within laboratory measurement standard) can result in a large error in dang nhap 188bet estimated value of Km. Porosity and fluid bulk modulus, on dang nhap 188bet other hand, show negligible influence. This result is not surprising, as Kf, φ and Km should be uncorrelated parameters.

4.       Conclusions

Three equivalent forms of Gassmann equation were presented that can be useful for dang nhap 188bet determination of Biot-Willis  coefficient,  dry  bulk  modulus,  and/or  grain matrix bulk modulus of a rock. We demonstrated dang nhap 188bet applicability of these equations using several sets of published laboratory measurements and dang nhap 188bet implications of dang nhap 188bet results for other estimations of rock properties. Astochastic simulation was also performed  to  examine  dang nhap 188bet effect of uncertainty and/or measurement errors on calculated grain matrix bulk modulus. dang nhap 188bet results showed that dang nhap 188bet calculated grain matrix bulk modulus is relatively constant with applied differential pressure (up to 50MPa) for sedimentary rocks. However, dang nhap 188bet estimation is very sensitive to dang nhap 188bet uncertainty of dry and saturated bulk modulus values. Our new forms of Gassmann equation can be used to effectively quantify  dang nhap 188bet  uncertainty  of  dry and saturated bulk modulus (and subsequently, dang nhap 188bet seismic velocities) in fluid identification, fluid substitution, or reservoir monitoring applications.

NOMENCLATURE

K: bulk modulus (GPa or psi)

 Ksat: saturated bulk modulus (GPa or psi)

 Kdry: dry (frame) bulk modulus (GPa or psi) Km: grain (matrix) bulk modulus (GPa or psi) Kf: fluid bulk modulus (GPa or psi)

G: shear modulus (GPa or psi)

Gsat: saturated shear modulus (GPa or psi)

Gdry: dry (frame) shear modulus (GPa or psi)

α: Biot-Willis coefficient (dimensionless)

φ: porosity (dimensionless)

ƿ: density (g/cc)

Vp: compressional wave velocity (km/s)

Vs: shear wave velocity (km/s)

APPENDIX A: Derivation of Equation (7)

From Equation (3) we can write:
(A.1)

 Rewriting Equation (1) as a function of α gives:

(A2)

(A3)

Multipplying both side with

 which is Equation (7).

APPENDIX B: Derivation of Equation (13)

If dang nhap 188bet same rock is subjected to two different saturation fluids, then we have two equations in dang nhap 188bet form of Equation (7):



equation and rearrange Equation (1) as a function of α, dang nhap 188bet Biot-Willis coefficient only:



  which is Equation (15).

APPENDIX C: Derivation of Equation (15)

If Km  value can  be obtained (e.g using mixture theory), then one can substitute K = (1 − α) K into Gassmann sandstones. Elsevier Science. 1991.

 References

1.       Fritz Gassmann. Uber die Elastizität Poröser Medien. Vierteljahrschrift der Naturforschenden Gesellschaftin Zürich. 1951; 96: p. 1 - 23.

2.       Tad M.Smith, Carl H.Sondergeld, Chandra S.Rai. Gassmannfluid substitution. A tutorial. Geophysics. 2003; 68(2): p. 430 - 440.

3. Ludmila Adam, Michael Batzle, Ivar Brevik.Gassmannfluid substitution and shear modulus variabilityin  Geophysics. 2006; 71(6): p. F173 - F183.

4.Murice Anthony Biot, David G.Willis. dang nhap 188bet elastic coefficients of dang nhap 188bet theory of consolidation. Journal of Applied Machanic. 1957; 24: p.594-601.

5.       James G.Berryman. Origin of Gassmann’s equations. Geophysics. 1999; 64(5): p. 1627 - 1629.

6.       Gary Mavko, Tapan Mukerji, Jack Dvorkin. dang nhap 188bet rock physics handbook: Tools for seismic analysis in porous media. Cambridge University Press, Cambridge. 1998.

7.       Robert        W.Zimmerman.    Compressibility    of which is Equation (13). Equation (14) then can be readily obtained by multiplying both sides by (Ksat1 × Ksat2).

8.       Luther White, John Castagna. Stochastic fluid modulus inversion. Geophysics. 2002; 67(6): p. 1835 - 1843.

9.       Fredy A.V.Artola, Vladimir Alvarado. Sensitivity analysis of Gassmann's fluid substitution equations: Some implications in feasibility studies of time-lapse seismic reservoir monitoring. Jounal of Applied Geophysics. 2006; 59(1): p. 47-62.

10.     De-Hua Han, Michael L.Batzle. Gassmann’s equation and fluid-saturation effects on seismic velocities. Geophysics. 2004; 69(2): p. 398 - 405.

11.     James G.Berryman, Graeme W.Milton. Exact results for generalized Gassmann’s equations in composite porous media with two constituents. Geophysics. 1991; 56: p. 1950 - 1960.

12.     Keith W.Katahara. Clay mineral elastic properties. SEG Technical Program Expanded Abstracts. 1996: p. 1691 - 1694.

13.     Zhijing Jee Wang, Hui Wang, Michael E.Cates. Elastic properties of solid clays. SEG Technical Program Expanded Abstracts. 1998: p. 1045 - 1048.

14.     Xianhuai Zhu, George A.McMechan. Direct estimation of dang nhap 188bet bulk modulu softheframeinfluidsaturated elastic medium by Biot theory. SEG Technical Program Expanded Abstracts. 1990: p. 787 - 790.

15.    I.Fatt. dang nhap 188bet Biot-Willis elastic coefficients for a sandstone. Journal of Applied Mechanics. 1959; 26: p. 296 - 297.

16.     Karl B.Coyner. Effects of stress, pore pressure, and pore fluids on bulk strain, velocity, and permeability in rocks. Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Massachusetts. 1984.

17.     Virginia    A.Clark,     Bernhard     R.Tittmann, Terry  W.Spencer.  Effect  of  volatiles  on   attenuation  (Q-1)   and   velocity   in    sedimentary    rocks.    Journal    of        Geophysical        Research.        1980;         85(B10):  p. 5190 - 5198.

18.     Tran Trung Dung, Chandra S.Rai, Carl H.Sondergeld. Changes in crack aspect-ratio concentration from heat treatment: A comparison between velocity inversion and experimental data. Geophysics. 2008; 73(4): p. E123 - E132.