The Equations Archie Forgot: Anisotropy of the Rocks
- Carlos F. Haro (Occidental Oil and Gas)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- October 2010
- Document Type
- Journal Paper
- 823 - 836
- 2010. Society of Petroleum Engineers
- 5.4.1 Waterflooding, 1.6.9 Coring, Fishing, 4.3.4 Scale
- tortuosity, Laplace equation, pore geometry, Archie equation
- 1 in the last 30 days
- 1,400 since 2007
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Archie's empirical equation is used extensively to estimate hydrocarbons in place. This power-laws combination has stood the test of time with few changes. However, it is still poorly understood and considered an ad hoc relation. Our original analysis will prove these laws rigorously, show how they must be amended, and introduce additional accompanying equations. This comprehensive model, which represents the electrical flow through the intricate conductive paths of the rock, is confirmed with Archie's and Hamada's core data sets. It corrects for Archie's inaccuracies.
A thorough appreciation of the pore-scale physics behind the modified version of Archie's equation is presented. The principles can be applied in clean and complex formations (shaly sands, thin beds, and vuggy or fractured carbonates) to obtain enhanced values of water saturation. The theory sheds light on the role and quantification of anisotropy.
Solving for the elaborate pore geometry, we use the Laplace differential equation (not Ohm's law), appropriate in the analysis of electrostatic fields in charge-free regions. Rock morphology dictates its boundary conditions (Jin 2007; Ghous 2005), characterized as corner angles. The corresponding particular solution (flow around a corner) and modeling tactic delineate the streamlines throughout the pores. The angles establish strong mathematical links among the exponents of Archie's equation, the geometry of the rock frame, and the spatial fluid distribution. This quantitative method is lacking in previous saturation models.
The solution constitutes the basis to solve more-complicated rock layouts. It enables the calculation of equivalent resistivities (normalized resistances) to take advantage of well-established electrical relationships. The extra equations compute the variable exponents and coefficients of Archie's equation at every depth. They obtain the saturation exponent in clean rocks as a function of water saturation, crucial to the quality control of core electrical data and to the quantification of reservoirs under changing saturation (waterflooding). Therefore, improved calculations of original and remaining hydrocarbons are achieved.
|File Size||1 MB||Number of Pages||14|
Adisoemarta, P.S., Anderson, G.A., Frailey, S.M., and Asquith, G.B. 2000. Historical Use of m and a in Well LogInterpretation: Is Conventional Wisdom Backwards? Paper SPE 59699 presentedat the SPE Permian Basin Oil and Gas Recovery Conference, Midland, Texas, USA,21-23 March. doi: 10.2118/59699-MS.
Aguilera, R. 1995. Naturally Fractured Reservoirs, second edition,521. Tulsa: Pennwell Books.
Amaefule, J.O., Altunbay, M., Tiab, D., Kersey, D.G., and Keelan, D.K. 1993.Enhanced Reservoir Description:Using Core and Log Data to Identify Hydraulic (Flow) Units and PredictPermeability in Uncored Intervals/Wells. Paper SPE 26436 presented at theSPE Annual Technical Conference and Exhibition, Houston, 3-6 October. doi:10.2118/26436-MS.
Ara, T.S., Talabani, S., and Vaziri, H.H. 2001. In-Depth Investigation of theValidity of the Archie Equation in Carbonate Rocks. Paper SPE 67204presented at the SPE Production and Operations Symposium, Oklahoma City,Oklahoma, USA, 24-27 March. doi: 10.2118/67204-MS.
Archie, G.E. 1942. The electrical resistivity log as an aid in determiningsome reservoir characteristics. SPE-942054-G. Trans., AIME, 146:54-62.
Belhaj, H.A., Agha, K.R., Nouri, A.M., Butt, S.D., Vaziri, H.F., and Islam,M.R. 2003. Numerical Simulation ofNon-Darcy Flow Utilizing the New Forchheimer's Diffusivity Equation. PaperSPE 81499 presented at the Middle East Oil Show, Bahrain, 9-12 June. doi:10.2118/81499-MS.
Bussian, A.E. 1982. A Generalized Archie Equation. Paper E presented at theSPWLA 23rd Annual Logging Symposium, Corpus Christi, Texas, USA, 6-9 July.
Gómez-Rivero, O. 1977. Some Considerations About the Possible Use of theParameters a and m as a Formation Evaluation Tool Through Well Logs. Paperpresented at the SPWLA Annual Logging Symposium, Houston, 5-8 June.
Halliday, D., Resnick, R., and Walker, J. 1997. Fundamentals of Physics:Part 2, fifth edition, 654. New York City: John Wiley & Sons.
Haro, C.F. 2006. PermeabilityModeling in Porous Media: Setting Archie and Carman-Kozeny Right. Paper SPE100200 presented at the SPE Europec/EAGE Annual Conference and Exhibition,Vienna, Austria, 12-15 June. doi: 10.2118/100200-MA.
Haro, C.F. 2008. Advanced Petrophysical Modeling: Correcting Shaly-SandTheory. Poster II presented at the SPWLA Annual Logging Symposium, Houston,25-28 May.
Herrick, D.C. and Kennedy, W.D. 1993. Electrical Efficiency: A PoreGeometric Model For The Electrical Properties of Rocks. Paper HH presented atthe SPWLA Annual Logging Symposium, Calgary, 13-16 June.
Holtz, M.H., Jackson, J.A., Jackson, K.G., and Major, R.P. 2002. Petrophysical Characterization ofPermian Shallow-Water Dolostone. Paper SPE 75214 presented at the SPE/DOEImproved Oil Recovery Symposium, Tulsa, 13-17 April. doi: 10.2118/75214-MS.
Jennings, J.W. Jr. and Lucia, F.J. 2001. Predicting Permeability From WellLogs in Carbonates With a Link to Geology for Interwell PermeabilityMapping. Paper SPE 71336 presented at the SPE Annual Technical Conferenceand Exhibition, New Orleans, 30 September-3 October. doi: 10.2118/71336-MS.
Jin, G., Torres-Verdín, C., Devarajan, S., Toumelin, E., and Thomas, E.C.2007. Pore Scale Analysis of the Waxman-Smits Shaly-Sand Conductivity Model.Petrophysics 48 (2): 104-120.
Jordan, E.C. and Balmain, K.G. 1968. Electromagnetic Waves and RadiatingSystems, second edition, 46. Englewood Cliffs, New Jersey, USA:Prentice-Hall.
Kennedy, W.D. 2007. The Porosity-Water Saturation Conductivity Relationship:An Alternative to Archie's Model. Petrophysics 48 (5):335.
Kennedy, W.D. and Herrick, D.C. 2004. Conductivity Anisotropy in Shale-FreeSandstone. Petrophysics 45 (1): 38.
Kimminau, S. and Schwartz, L. 2003. A Review of Pore and Grain GeometricModels. Paper AA presented at the SPWLA 44th Annual Logging Symposium,Galveston, Texas, USA, 22-25 June.
Knackstedt, M.A., Ghous, A., Bauget, F., and Beck, G.F. 2005. Resistivityand Permeability Anisotropy Measured in Laminated Sands Via Digital CoreAnalysis. Paper VVV presented at the SPWLA Annual Logging Symposium, NewOrleans, 26-29 June.
Lucia, F.J. 1999. Carbonate Reservoir Characterization, 68. Berlin:Springer-Verlag.
Mahood, B.C. and Boyd, D.A. 1993. Formation Factor Relationships of WesternCanada. Paper FF presented at the SPWLA Annual Logging Symposium, Calgary,13-16 June.
Mavko, G., Mukerji, T., and Dvorkin, J. 1998. The Rock Physics Handbook:Tools for Seismic Analysis of Porous Media, 260. Cambridge, UK: CambridgeUniversity Press.
Schlumberger. 1984. Log Interpretation Charts, 1984 edition, 15.Sugar Land, Texas, USA: Schlumberger.
Sen, P.N. 1980. The Dielectricand Conductivity Response of Sedimentary Rocks. Paper SPE 9379 presented atthe SPE Annual Technical Conference and Exhibition, Dallas, 21-24 September.doi: 10.2118/9379-MS.
Serra, O. 1986. Advanced Interpretation of Wireline Logs, 110. SugarLand, Texas, USA: Schlumberger.
Shang, B.Z., Hamman, J.G., and Caldwell, D.H. 2003. A Physical Model to Explain the FirstArchie Relationship and Beyond. Paper SPE 84300 presented at the SPE AnnualTechnical Conference and Exhibition, Denver, 5-8 October. doi:10.2118/84300-MS.
Shang, B.Z., Hamman, J.G., and Caldwell, D.H. 2004. Water Saturation Estimation UsingEquivalent Rock Element Model. Paper SPE 90143 presented at the SPE AnnualTechnical Conference and Exhibition, Houston, 26-29 September. doi:10.2118/90143-MS.
Sharma, M.M., Garrouch, A., and Dunlap, H.F. 1991. Effects of Wettability,Pore Geometry, and Stress on Electrical Conduction in Fluid-Saturated Rocks.The Log Analyst 32 (5): 511-526.
Toumelin, E. and Torres-Verdin, C. 2005. Influence of oil saturation andwettability on rock resistivity measurements: a uniform pore-scale approach.Paper PPP presented at the SPWLA Annual Logging Symposium, New Orleans, 26-29June.
Warren, J.E. and Root, P.J. 1963. The Behavior of Naturally FracturedReservoirs. SPE J. 3 (3): 245-255; Trans., AIME,228. SPE-426-PA. doi: 10.2118/426-PA.
White, F.M. 2003. Fluid Mechanics, fifth edition, 250. New York City:McGraw-Hill Higher Education.
Worthington, P.F. 1981. The Influence of Formation Anisotropy uponResistivity--Porosity Relationships. Paper AA presented at the SPWLA AnnualLogging Symposium, Houston, 23-26 June.
Wyllie, M.R. and Spangler, M.B. 1952. Application of electrical resistivitymeasurements to problem of fluid flow in porous media. Bulletin of theAmerican Association of Petroleum Geologists 36 (February):359-403.