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

Summary

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

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

1.       Introduction

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





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

 

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



Berryman [5] gave a concise derivation of Gassmann equations for an isotropic and homogeneous medium using   188bet vipz   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 188bet vipz most intuitive in describing 188bet vipz effect of fluid presence on 188bet vipz 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 188bet vipz effect of uncertainty of different input parameters and showed that 188bet vipz computed compressional velocity of a saturated rock is most sensitive to uncertainties in 188bet vipz rock bulk density and 188bet vipz dry bulk and shear moduli, while other parameters (porosity, 188bet vipz grain matrix and fluid bulk moduli) have negligible effects.

Note that 188bet vipz 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. 188bet vipz 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 188bet vipz 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 188bet vipz effect of uncertainty and/    or measurement errors on calculated grain matrix bulk modulus and Biot-Willis coefficient.

2.       188bet vipz new equivalent Gassmann equations

When (K , K , K , and φ) are known

This is generally 188bet vipz 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 188bet vipz only.

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

This provides an independent methodology to calculate 188bet vipz grainmatrixbul kmodulusofarockfrom SCAL laboratory acoustic  measurements of dry and saturated rock samples. Traditionally, 188bet vipz grain matrix bulk moduli are estimated from averages of 188bet vipz 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 188bet vipz measurement conditions [12, 13]. We can further  postulate that: (a) 188bet vipz grain matrix calculated from Gassmann equation (using Equations (8) to (12)) must lie between 188bet vipz two bounds obtained from mixture theory, and (b) 188bet vipz calculated grain matrix values are insensitive to 188bet vipz first order to 188bet vipz applied pressure. Equations (7) to (11) can also be used to verify 188bet vipz 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 188bet vipz field. 188bet vipz same rock can be fully saturated with brine in one well; or it can have verying saturation in 188bet vipz 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.

188bet vipz 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 188bet vipz 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 188bet vipz saturating fluids on 188bet vipz grain bulk modulus (Km is constant). 188bet vipz secondassumption  is  that  188bet vipz  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 188bet vipz mineralogical composition of 188bet vipz rock (FTIR, XRD, thin section of rock cuttings, or mineralogy log). 188bet vipz Biot-Willis coefficient can be estimated directly from 188bet vipz following equivalent Gassmann equation (see Appendix C for detailed derivations):

(15)

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

3.       Numerical applications

          Han and Batzle’s sandstone data

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

 (16)
188bet vipz calculated Biot-Willis coefficient and grain matrix modulus as functions of pressure are plotted in Figure 1. 188bet vipz relatively constant value of 188bet vipz grain bulk modulus (39GPa) as a function of pressure is a good indicator that Gassmann equation is applicable for this rock. 188bet vipz 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).

188bet vipz Biot-Willis coefficient profile is remarkably similar to 188bet vipz 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 188bet vipz 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, 188bet vipz fluid pore pressure in both saturation cases was maintained at 10MPa. 188bet vipz porosity of 188bet vipz rock is 11.9%. 188bet vipz shear modulus  profile  is  almost  identical  for  all  vacuum  dry, water saturated, and benzene saturated cases, suggesting

188bet vipz back-calculated dry bulk modulus is also plotted (as a dashed line) against 188bet vipz various measured moduli in Figure 2. 188bet vipz profile is consistently higher than 188bet vipz measured vacuum dry bulk modulus profile by approximately 2.5GPa (or 5 - 9%). This is another evidence supporting 188bet vipz argument that 188bet vipz vacuum dry measured bulk modulus is too dry and should not be used in Gassmann equation [6, 17]. We could have applied 188bet vipz 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]. 188bet vipz dashed line is 188bet vipz dry bulk modulus calculated from Gassmann equation, consistently higher than 188bet vipz 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].

188bet vipz grain matrix bulk modulus and Biot-Willis coefficient profiles obtained from 188bet vipz rock water- and benzene-saturated moduli are shown in Figure 3. While 188bet vipz grain matrix bulk modulus is similar to Coyner’s reported value of 65GPa, 188bet vipz back-calculated Biot-Willis coefficient profile decreases from 0.6 to 0.53, significantly lower than 188bet vipz commonly assumed value of 1 while significantly higher than 188bet vipz 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 188bet vipz effects of uncertainties from porosity, Kdry, Ksat, and Kf to 188bet vipz uncertainty of 188bet vipz predicted Km. Since 188bet vipz relationship in Equation (7) is not linear, a Monte Carlo (stochastic) simulation was used.

Table    1   summarises   188bet vipz   input   parameter  values [18] and their estimated ranges of  uncertainties.  188bet vipz  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 188bet vipz measured values and 188bet vipz errors represent 188bet vipz 95% confidence interval. Thus, 188bet vipz relative error (uncertainty) of each parameter is defined as:

 (17)

 
Where s is 188bet vipz standard deviation of 188bet vipz 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 188bet vipz 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. 188bet vipz resulting Km is also a Gaussian distribution with a mean of 44.6GPa and a standard deviation of 3.45GPa. 188bet vipz 95% confidence interval is, therefore, from 37.7GPa to 51.5GPa (or 16% error). 188bet vipz Biot-Willis coefficient α is also a Gaussian distribution with a mean of 0.62 and a standard deviation of 0.03. 188bet vipz 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. 188bet vipz uncertainty of 188bet vipz computed grain matrix bulk modulus Kmusing Gassmann equation as functions of percent error in one input parameter (Kf, α, Kdry, or Ksat), while 188bet vipz remaining input parameters carry 188bet vipz same errors as 188bet vipz base case. Errors from Ksat and Kdry have 188bet vipz largest impacts on 188bet vipz uncertainty of calculated Km. Porosity and fluid bulk modulus, on 188bet vipz other hand, show negligible effects.

Figure 4 shows 188bet vipz uncertainty of 188bet vipz computed grain matrix bulk modulus Km as functions of percent error in one input parameter (K , φ, K , or K ), while 188bet vipz remaining input parameters carry 188bet vipz same uncertainties as of 188bet vipz base case. Errors from Ksat and Kdry have 188bet vipz largest effects on 188bet vipz uncertainty of Km. Minus errors in Kdry and Ksat (even within laboratory measurement standard) can result in a large error in 188bet vipz estimated value of Km. Porosity and fluid bulk modulus, on 188bet vipz 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 188bet vipz determination of Biot-Willis  coefficient,  dry  bulk  modulus,  and/or  grain matrix bulk modulus of a rock. We demonstrated 188bet vipz applicability of these equations using several sets of published laboratory measurements and 188bet vipz implications of 188bet vipz results for other estimations of rock properties. Astochastic simulation was also performed  to  examine  188bet vipz effect of uncertainty and/or measurement errors on calculated grain matrix bulk modulus. 188bet vipz results showed that 188bet vipz calculated grain matrix bulk modulus is relatively constant with applied differential pressure (up to 50MPa) for sedimentary rocks. However, 188bet vipz estimation is very sensitive to 188bet vipz uncertainty of dry and saturated bulk modulus values. Our new forms of Gassmann equation can be used to effectively quantify  188bet vipz  uncertainty  of  dry and saturated bulk modulus (and subsequently, 188bet vipz 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 188bet vipz same rock is subjected to two different saturation fluids, then we have two equations in 188bet vipz form of Equation (7):



equation and rearrange Equation (1) as a function of α, 188bet vipz 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. 188bet vipz elastic coefficients of 188bet vipz 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. 188bet vipz 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 188bet vipz bulk modulu softheframeinfluidsaturated elastic medium by Biot theory. SEG Technical Program Expanded Abstracts. 1990: p. 787 - 790.

15.    I.Fatt. 188bet vipz 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.