SPWLA 46 th Annual Logging Symposium, June 26-29, 2005

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1 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 PETROPHYSICAL ROC PHYSICS ODELING: A COPARISON OF THE RIEF AND GASSANN EQUATIONS, AND APPLICATIONS TO VERIFYING AND ESTIATING COPRESSIONAL AND SHEAR VELOCITIES ichael Hlmes and Antny Hlmes Cpyright 005, held jintly by the Sciety f Petrphysicists and Well Lg Analysts (SPWLA) and the submitting authrs. This paper was prepared fr presentatin at the SPWLA 46 th Annual Lgging Sympsium held in New Orleans, Luisiana, United States, June 6-9, 005. ASTRACT The Gassnn Equatin, first published in the 1950 s, relates cmpressinal acustic velcity t bulk mduli f the empty rck frame, slids and fluids, density and prsity. When cmbined with it s analysis (1961), cmpressinal and shear velcities can be determined. In rder t slve the Gassnn/it mdel, it is necessary t determine bulk mdulus f the rck. Several empirical equatins have been published t achieve this. rief (1987) appraches the same rck physics mdel smewhat differently. In additin t elastic mduli f slids and fluids, the mdel incrprates shear mduli f the slids and the it cmpressibility cnstants. Thus, bth cmpressinal and shear velcities are available frm rief. th mdels allw fr predictins f velcity variatins as fluids change (gas vs. water). An initial rck mdel is frmulated, cnsisting f rck trix vlume, shale vlume, and prsity. est surce data fr prsity is a density/neutrn cmbinatin, because calculatins are relatively independent f fluid cntent and trix lithlgy. th mdels, adapted t petrphysical applicatins, invlve average bulk mduli fr fluid (water and gas) as a functin f pressure. Zne input includes average values ver the zne depth f: Cmpressinal travel time trix Cmpressinal travel time shale Shear travel time trix Shear travel time shale th mdels are a cmprehensive slutin t velcity prperties f rcks ver the cmplete range f lithlgies, shale cntent and prsity. In high prsity sft rcks (prsity values f 40% r greater) bth mdels give cmparable results cmpressinal velcity slwing in the presence f gas. Hwever, at lwer prsities, and particularly fr rcks with less than 0% prsity, the Gassnn mdel predicts a much larger velcity slwing effect than des rief, depending upn the rck bulk mdulus mdel that is applied. When applying the mdels, fr calibratin f any ne reservir, it is best t have bth cmpressinal and shear measurements. Then, by suitable adjustment f the zne parameters, a tch between the pseud cmpressinal and shear data (mdeled lgs) and real measurements can be achieved. In ny wells, and particularly fr shear data, ccasinal depth intervals shw emphatic differences between the pseud and measured data. In such cases assuming that prsity calculatins are reliable (n bad hle prblems), data prcessing t generate the shear curve is suspect. Once a reservir sequence has been crrectly zned, the pseud shear curve is reliable, even if n shear measurements have been cllected. Fr key wells where n acustic measurements f any kind have been cllected, pseud cmpressinal and shear velcity data can be estited, and used fr gephysical calibratin. Examples frm a number f clastic reservirs are presented. INTRODUCTION Acustic prperties f rck (bth cmpressinal and shear respnses) are influenced by the mechanical prperties f the slid cmpnents as well as the cmpressibility f the cntained fluids il, gas, free 1

2 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 water, bund water. Standard petrphysical interpretatin f cmpressinal acustic data invlves empirical relatins such as the Wyllie (1956) time series equatin and the Raymer-Hunt-Gardner (1980) transfrm. It is well knwn that these equatins can be suspect particularly if gas is present. Petrphysical interpretatin f shear data is even mre apprxite - usually invlving average ratis f cmpressinal t shear travel times fr different trix/shale cmpnents. ecause cmpressinal and shear acustic prperties are s imprtant in gephysical seismic interpretatin, a great deal f effrt has been dented t understanding velcity behavir fr bth cmpressinal and shear energy. A number f relatinships have been develped based n fundamental rck and fluid prperties. st f these analyses have nt been expressed in ways suitable fr petrphysical applicatin. In additin there is ften incnsistency in terminlgy and mnemnics, which cmplicates cmparisns amng the varius mdels prpsed. In this paper, we analyzed sme f these relatinships, and present equatins that can be used fr the interpretatins f standard wireline lg data. DEFINATION OF ROC AND FLUID PROPERTIES The fllwing definitins are used in subsequent analyses (synnyms frm the literature are indicated) = ulk dulus = ρ ( 4 Vp V ) s = Elastic dulus = V ρ 4 p = + μ μ = Shear dulus = Vs ρ β = it Cmpressibility Cnstant = = Rck & Fluid it Cefficient 1 β φ φ = + F φ 1 φ V p = V c = Cmpressinal velcity V s = Shear velcity ρ b = ulk density φ = Prsity = ulk mdulus f rck, fluid urated = ulk mdulus f rck = = ulk mdulus f the trix F = Fluid = ulk mdulus f the fluid μ = Shear mdulus f the trix μ = Shear mdulus f the urated rck μ = Shear mdulus f the rck = Elastic mdulus f urated rck = Elastic mdulus f rck = Elastic mdulus f rck mineral F = Elastic mdulus f the fluid DTC = DTP = Cmpressinal Travel Time DTS = Shear Travel Time SOURCE EQUATIONS FRO GEOPHYSICAL LITERATURE A. Gassnn/it = The mdel assumes + φ μ = μ ( ) Fluid Fluid. Simplified Gassnn published by Crain (1986) Gives nly V p data = 1 + φ fl 1 φ + C. Simplified Gassnn published by avk (1988) + φ ( ) Fluid F

3 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 D. rief V p = μ = μ ( 1 β ) ( 1 β ) = ρ ρ 4 V p V 1 β φ φ = + f 1 β = + 4 μ = ρ + ( 1 φ) 1 φ + β S EQUATIONS USED IN THIS ANALYSIS A. Gassnn (Crain) t determine V p nly = 1 + φ fl 1 φ + recgnizes the range culd be frm 1.5 t 5 * dynes per cm. Several empirical relatinships have been suggested t predict frm and prsity: Geerts (1961) 1 = 1 ( φ ) Nur (1995) 1 φ φ = c Where φ c is critical prsity aterial Sandstne Limestne Dlmite Chalk Igneus φ % = 1 it ( ) φ c 1 φ. Gassnn/it t determine V p and V s The prcedures t slve the equatins are: 1. Determine fr the specific frtin: Sandstne 7.9 * dynes/cm fl + φ fl Limestne 67 * dynes/cm Dlmite 8 * dynes/cm. Determine Fluid fr any assumed fluid uratin. Cmmn values are: Water.05 * dynes/cm Oil 0.4 * dynes/cm Gas 0.04 * dynes/cm T extract V p and V s frm it is necessary t determine μ μ = μ 1 ( β ) ; mechanics data μ is available frm rck Sandstne 44 * dynes/cm Limestne * dynes/cm Dlmite 54 * dynes/cm Values fr gas are pressure and cmpnent dependent. Determine. Crain suggests an average value f abut * dynes per cm, but

4 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 C. rief t determine V P and V s V p = μ = μ ( 1 β ) ( 1 β ) = ρ ρ 4 V p V 1 β φ φ = + f 1 β = + 4 μ = ρ + ( 1 φ) 1 φ + β S Input parameters are similar t the Gassnn/it equatin except there is n requirement t slve fr. ETHODOLOGY All f the equatins require determinatin f the fllwing parameters: = = ulk mdulus f the trix ρ = ulk density Fluid = F = ulk mdulus f the fluid cmbinatin μ = Shear mdulus f the rcks Gassnn/it and rief mdels = ulk mdulus f rcks Gassnn (Crain and Gassnn/it) φ = Prsity In ur apprach we incrprate the influence f shale by invlving bulk mduli f the shale cmpnent. The methdlgy cnsists f: 1. Determine, at each level, a rck mdel f: atrix vlume Shale vlume Prsity This is best achieved using density/neutrn cmbinatin, because calculatins are relatively independent f fluid cntent and trix lithlgy.. Assuming different fluid cmbinatin (gas vs. water) calculate: ρ V p V s Neutrn lg respnse. Frm V p and V s, determine DTC and DTS 4. Cmpare calculated values with lg data. The cmparisn, level-by-level, will indicate what fluid cmbinatin is seen by each lg. The methdlgy allws the calculatin f acustic prperties (bth cmpressinal and shear) even if n acustic lg data has been run n the well. dels t Establish in the Gassnn Equatin The fur mdels which can be used t estite frm are: 1. Crain Infers a cnstant value f f 1.86 * dynes/cm. Geerts. it 4. Nur dels,, and 4 give quite different results as expressed in the fllwing table ( in dynes/cm * ) 4

5 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 Prsity Geerts it Nur The implicatins f the differences are very significant. Velcity slwing due t the presence f gas are predicted t be much greater if the Crain r Geerts mdels are used as cmpared with the it r Nur mdels. When cmpared with rief (which des nt invlve ) a tch wuld be achieved if a mdel clse t Nur is applied. EXAPLES Data frm the fllwing reservirs are presented: 1. High prsity shallw sandstne frm SW Wyming. High prsity sandstne frm the Gulf f exic. Lw prsity tight gas sandstne frm NW Clrad 4. Lw prsity tight gas sandstne frm Central Wyming Fr each example, the fllwing plts are shwn: Recnstructed prsity lgs (including cmpressinal and shear acustic data), cmpared with actual wireline lgs fr the fllwing mdels: Gassnn using Geerts mdel t calculate Gassnn using it mdel t calculate rief DISCUSSION OF RESULTS 1. High Prsity Gas Sandstne, SW Wyming Fig 1: Gassnn (Geerts del) Difference between theretical wet DTC and Gas filled DTC is abut 0 0 micrsecnds per ft. Actual DTC meanders between theretical wet and theretical gas. Generally, when gas is 5

6 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 indicated frm density/neutrn crss ver, the actual DTC is clse t the gas theretical curve. Recnstructin f the theretical DTS agrees well with measured DTS. Fig : Gassnn (it del) There is very little difference between theretical DTC gas and theretical DTC wet and the theretical curves d nt tch measured DTC. Theretical DTS is a clse tch t measured DTS. Fig : rief Theretical DTC gas is abut 0 micrsecnds per ft higher than theretical DTC wet. Actual DTC meanders between theretical DTC gas and theretical DTC wet in a nner cnsistent with gas indicatins frm density/neutrn respnse. Recnstructed DTS agrees well with measured DTS. 6

7 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005. High Prsity Sandstne frm the Gulf f exic This example is frm an inter-bedded sequence f wet sandstnes and shales. Fig 4: Gassnn (Geerts del) Separatin between theretical DTC gas and DTC wet is abut 40 micrsecnds per ft. The actual DTC tracks the theretical wet DTC (as it shuld). There is fair agreement between actual and theretical DTS, and the theretical DTS rati (wet) tracks the actual rati quite well. 7

8 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 Fig 5: Gassnn (it del) Separatin between theretical DTC gas and DTC wet is abut 0 micrsecnds per ft. The actual DTC tracks the theretical wet DTC (as it shuld). There is fair agreement between actual and theretical DTS, and the theretical DTS rati (wet) mstly tracks the actual DTS rati quite well. Fig 6: rief Separatin between theretical DTC gas and DTC wet is abut 0 micrsecnds per ft. Actual DTC tracks theretical DTC wet mre clsely then fr the Gassnn mdels. Agreement between theretical and actual DTS is a little better than fr the Gassnn mdels. There is gd agreement between actual rati and theretical wet rati. 8

9 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005. Lw Prsity, Tight Gas Sandstne frm NW Clrad Fig 7: Gassnn (Geerts del) Separatin between theretical DTC gas and DTC wet is abut 10 micrsecnds per ft. Theretical DTS agrees well with measured DTS, but the ratis f the tw theretical curves d nt tch the measured rati. 9

10 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 Fig 8: Gassnn (it del) There is n separatin between theretical DTC gas and DTC wet, but the tch with measured DTC is excellent. Theretical DTS agrees quite well with measured DTS, but the theretical rati curve des nt tch measured rati. Fig 9: rief There is n separatin between theretical DTC gas and DTC wet. Hwever, all theretical curves (including ratis) agree well with measured data. 10

11 SPWLA 46 th Annual Lgging Sympsium, June 6-9, Tight Gas Sandstne, Pwder River asin, Wyming Fig 10: Gassnn (Geerts del) Separatin between theretical DTC gas and DTC wet is abut 10 micrsecnds per ft. Theretical DTS agrees well with actual DTS. Theretical wet rati f DTS t DTC agrees well with measured DTS t DTC rati. Fig 11: Gassnn (it del) There is n separatin between theretical DTC gas and DTC wet. All theretical curves agree well with measured curves. 11

12 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 Fig 1: rief There is n separatin between theretical DTC gas and DTC wet. All theretical curves agree well with measured data. VELOCITY SLOWING DUE TO THE PRESENCE OF GAS The three different Gassn Dry mdels presented shw different predictins with respect t the values f Dry as prsity dependent functins ( Dry is in dynes per cm x ). As a cnsequence, each mdel gives different results with respect t velcity slwing in the presence f gas. Fig 1: Dry Geerts vs. Prsity 1

13 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 Fig 14: Dry avk/it vs. Prsity Fig 16: Cmparisns f the mdels Fig 15: Dry Nur vs. Prsity Figure 16 sumrizes the results; data are based n an assumed gas uratin f 80%. The cnclusin is reached that the Geerts rck bulk mdulus mdel leads t significant verestites f the effect f gas, particularly in prsities less than abut 0% (DT f 85 micrsecnds per ft). 1

14 SPWLA 46 th Annual Lgging Sympsium, June 6-9, 005 CONCLUSIONS 1. Petrphysical adaptatin f the Gassnn and rief acustic mdels has been successfully applied t the analysis f acustic cmpressinal and shear data. Gd cmparisns exist with measured data.. Fr the Gassnn mdel, it is necessary t estite rck bulk mdulus. A number f published empirical equatins were used. It appears that the it and Nur mdels give better and mre cnsistent results ver the entire prsity range then des the Geerts mdel; the Geerts mdel leads t significant verestitin as t the degree f velcity slwing (increase in DT) in the presence f gas.. The rief mdel, which des nt invlve rck bulk mdulus, appears t be stable fr the entire range f prsity. It is relatively clse t the Gassnn (it and Nur) mdels, but smewhat mre cnsistent. 4. Frm either the Gassnn/it r rief mdels, it appears that velcity slwing due t the presence f gas is n mre than three micrsecnds per ft in rcks with less than abut 15% prsity. 5. The techniques presented allw fr the predictin f acustic prperties (bth cmpressinal and shear) frm ther standard wireline lg measurements. Thus, pseud cmpressinal and shear data can be reliably estited curve when such measurements have nt been de. It is necessary t calibrate frm wells in the area, since accurate values f trix and shale prperties are required. REFERENCES it,.a., 1956 Thery f prpagatin f elastic waves in a fluid urated prus slid I. Lw frequency range and II. Higher- frequency range. J. Acust. Sc. Am., 8, yer, S., ari, J.L., Seismic Surveying and Well Lgging (1997) Editins Technip, Paris. Crain, E.R., The Lg Analysis Handbk (1986) Penn Well ks. Gassnn, F., Über Die Elastizität pröser edien Vier. der Natur. Gesellschaft in Zürich, Geerts, J., 1961 Velcity-lg interpretatin: The effect f rck bulk cmpressibility Sc. Pet. Eng. J., 1, rief,., Garar, J., Stellingwerff, J., and Ventre, J., 1990 A petrphysical interpretatin using the velcities f P and S waves (full-wavefrm snic) The Lg Analyst, 1, Nvember, avk, G., ukerji, T., Dvrkin, J., The Rck Physics Handbk (1998) Cambridge University Press. Nur, A., avk, G., Dvrkn, J., and Gas, D., Critical prsity: The key t relating physical prperties t prsity in rcks, in Prc., 65 th Ann. Int. eeting, Sc. Expl. Gephys., 878. Raymer, L.L., Hunt, E.R., and Gardner, J.S. An Imprved Snic Transit Time-t-Prsity Transfrm, SPWLA Lgging Sympsium Transactins (July 1980). Wyllie,.R.J., Gregry, A.R., and Gardner, G.H.F., Elastic Wave Velcities in Hetergeneus and Prus edia, Gephysics, Vl 1, N. 1 (January 1956), pp AOUT THE AUTHORS ichael Hlmes has a Ph.D. frm the University f Lndn in gelgy and a Sc. frm the Clrad Schl f ines in Petrleum Engineering. His prfessinal career has invlved emplyment with ritish Petrleum, Shell Canada, arathn Oil Cmpany and H.. van Pllen and Assciates. Fr the past 15 years he has wrked n petrphysical analyses fr reservirs wrldwide under the auspices f Digital Frtin, Inc. Antny. Hlmes has a S in Cmputer Science frm the University f Clrad. He has been invlved with the develpment f petrphysical sftware fr 15 years with Digital Frtin, Inc., particularly with regards t the implementatin f petrphysical analyses. 14