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Отправлено: 28.02.13 03:26. Заголовок: A simple geometric model of sedimentary rock to connect transfer and acoustic properties
Arabian Journal of Geosciences February 2013 A simple geometric model of sedimentary rock to connect transfer and acoustic properties Gabor Korvin, Klavdia Oleschko, Abdulazeez Abdulraheem Abstract A simple rock model is presented which reproduces the measured hydraulic and electric transport properties of sedimentary rocks and connects these properties with each other, as well as with the acoustic propagation velocities and elastic moduli. The model has four geometric parameters (average coordination number Z of the pores, average pore radius r, average distance between nearest pores d, and average throat radius δ) which can be directly determined from the measured porosity Φ, hydraulic permeability k, and cementation exponent m of the rock via simple analytic expressions. Inversion examples are presented for published sandstone data, and for cores taken from Saudi Arabian, Upper Jurassic and Permian carbonate reservoirs. For sandstone, the inversion works perfectly; for carbonates, the derived rock model shows order-of-magnitude agreement with the structure seen in thin sections. Inverting the equations, we express the transfer properties Φ, k, and m as functions of r, d, δ, and Z. Formulae are derived for the bulk density D b, formation factor F, and P-wave velocity in terms of the proposed geometrical parameters. References (81) Abdassah D, Ershaghi I (1986) Triple-porosity systems for representing naturally fractured reservoirs, SPE Formation Evaluation, April 1986, 113–127 Ahr WA (2008) Geology of carbonate reservoirs. The identification, description, and characterization of hydrocarbon reservoirs in carbonate rocks. Wiley, Hoboken Akbar NA (1993) Seismic signatures of reservoir transport properties and pore fluid distribution, Ph. D. Thesis, Stanford University, Stanford, CA Akbar N, Mavko G, Nur A, Dvorkin J (1994) Seismic signatures of reservoir properties and pore fluid distribution. Geophysics 59(8):1222–1236 » CrossRef Andrä H, Combaret N, Dvorkin J, Glatt E, Han J, Kabel M, Keehm Y, Krzikalla F, Lee M, Madonna C, Marsh M, Mukerji T, Saenger EH, Sain R, Saxena N, Ricker S, Wiegman A, Zhan X (2012) Digital rock physics benchmarks—Part 1. Imaging and segmentation; Pt. 2. Computing effective properties. Comput Geosci 50:25–32. doi:10.1016/j.cageo.2012.09.005 Archie GE (1942) The electric resistivity log as an aid in determination of some reservoir characteristics. Trans AIME 146:54–62 Arns C, Knackstedt MA, Val Pinczewski W, Martys NS (2004) Virtual permeametry on microtomographic images. J Petrol Sci Eng 45(1–2):41–46 » CrossRef Benson P, Meredith P, Schubnel A (2006) Examining the role of void space fabric in the permeability development of crustal rock with pressure. J Geophys Res 111: Art. No. B12203 Benson A, Schubnel A, Vinciguerra S, Trovato C, Meredith P, Young RP (2006) Modelling the permeability evolution of micro-cracked rocks from elastic wave velocity inversion at elevated hydrostatic pressure. J Geophys Res 111: Art. No. B04202 Bernabe Y, Brace WF, Evans B (1982) Permeability, porosity, and pore geometry of hot-pressed calcite. Mech Mater 1:173–183 » CrossRef Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. J Acoust Soc Am 28:168–178 » CrossRef Biswal B, Mauwart C, Hilfer R, Bakke S, Øren PE (1999) Quantitative analysis of experimental and synthetic microstructures for sedimentary rock. Physica A 273:452–475 » CrossRef Biswal B, Øren P-E, Held RJ, Bakke S, Hilfer R (2009) Modeling of multiscale porous media. Image Anal Stereol 28(1):23–34 » CrossRef Boylan AL, Waltham DA, Bosence DWJ, Badenas B, Aurell M (2002) Digital rocks linking forward modeling to carbonate facies. Basin Res 14(3):401–415 » CrossRef Brandt H (1955) A study of the speed of sound in solid granular media. Trans ASME 22:479–486 Carman P (1937) Fluid flow through a granular bed. Trans Inst Chem Eng 15:150–167 Clerke EA (2003) Beyond porosity-permeability relationships—determining pore network parameters for the Ghawar Arab-D using the Thomeer method. Geofrontier (Dhahran, Saudi Arabia) 1(3):12–17 Cole KS, Cole RH (1941) Dispersion and absorption in dielectrics. Part 1. Alternating current fields. J Chem Phys 9:341 » CrossRef Dong H (2007) Micro CT imaging and pore network extraction, Ph. D. Dissertation, Imperial College, London Dong H, Fjeldstad S, Alberts L, Roth S, Bakke S, Øren P-E (2008) Pore network modeling on carbonate: a comparative study of different micro-CT network extraction methods. Soc. Core Anal. Int’l. Symp. UAE, Oct. 29–2 Nov., 2008. Paper SCA2008-31. Doyen PE (1987) Crack geometry of igneous rocks: a maximum entropy inversion of elastic and transport properties. J Geophys Res 92(B8):8169–8181 » CrossRef Duda A, Koza Z, Matyka M (2011) Hydraulic tortuosity in arbitrary porous media flow. Phys Rev E 84:036319 » CrossRef Dvorkin J, Derzhi N, Fang Q, Nur A, Nur B, Grader A, Baldwin C, Tono H, Diaz E (2009) From micro to reservoir scale: permeability from digital experiments. Lead Edge 28:1446 » CrossRef Dvorkin J, Derzhi N, Diaz E, Fang Q (2011) Relevance of computational rock physics. Geophysics 76(5):E141–E153 » CrossRef Fortin J, Schnubnel A, Guéguen Y (2005) Elastic wave velocities and permeability evolution during compaction of Bleuswiller sandstone. Int J Rock Mech Min Sci 42:873–889 » CrossRef Glover PW (2009) What is the cementation exponent? A new interpretation. Lead Edge 28:82–85 » CrossRef Grosse-Kunstleve RW, Brunner GO, Sloane NJA (1996) Algebraic description of coordination sequences and exact topological densities for zeolites. Acta Crystallogr A52:879–889 Guéguen Y, Dienes J (1989) Transport properties of rocks from statistics and percolation. Math Geol 21:1–13 » CrossRef Halawani MA (2000) Stratigraphic column for the phanerozoic rocks of Saudi Arabia, Techn. Rept. BRGM-TR-2000-3, 95 pp Hassanzadeh H, Pooladi-Darvish M (2006) Effects of fracture boundary conditions on matrix-fracture transfer shape factor. Transp Porous Media 64:51–71 » CrossRef Jorgensen DA (1988) Using geophysical logs to estimate porosity, water resistivity, and intrinsic permeability, USGS Water-Supply Paper # 2321, 24 pp Kachanov M (1994) Elastic solids with many cracks and related problems. Adv Appl Mech 30:259–345 » CrossRef Kalam Z, Al Dayyani T, Grader A, Sisk C (2011) Digital rock physics analysis in complex carbonates, World Oil, 232, May 2011 Kayser A, Ziauddin M (2006) A closer look at pore geometry. Oilfield Review, Spring, 2006, 4-14 Keehm Y (2003) Computational rock physics transport properties in porous media and applications. Ph. D. Dissertation, Stanford Kirkpatrick S (1973) Percolation and conduction. Rev Mod Phys 45(4):574–588 » CrossRef Knackstedt M, Arns C, Sheppard A et al (2007) Archie’s exponents in complex lithologies derived from 3D digital core analysis. Annual Logging Symp. of SPWLA, SPWLA Paper UU:1-1C Knackstedt M, Madadi M, Arns Ch, Baechle G, Eberli G, Weger R (2009a) Carbonate petrophysical parameters derived from 3d images. Search & Discovery article #40393, posted March 20, 2009 Knackstedt M, Latham S, Sheppard A, Vaslet T, Arns C (2009b) Digital rock physics: 3D imaging of core material and correlations to acoustic and flow properties. Lead Edge Jan. 2009, 28–33 Korvin G (1984) Shale compaction and statistical physics. Geophys J R Astron Soc 78:35–50 » CrossRef Korvin G (1992a) A percolation model for the permeability of kaolinite-bearing sandstones. Geophys Trans 37(2–3):177–209 Korvin G (1992b) Fractal models in the earth sciences. Elsevier, Amsterdam Kozeny J (1927) Über kapillare Leitung der Wasser in Boden, Sitzungs-Ber. Akad Wiss Wien 136:271–306 Kuster GT, Toksöz MN (1974) Velocity and attenuation of seismic waves in two-phase media. Pt. 1, Theoretical formulations. Geophysics 39(5):587–618 » CrossRef Lamm PK (1997) Solution of ill-posed Volterra equations via variable smoothing Tikhonov regularization. In: Engl EW et al (eds) Inverse problems in geophysical applications. SIAM, 92-108 Lifshitz EM, Pitaevskii LP (1980) Statistical physics, Part 1. Pergamon, New York Lim KT, Aziz K (1995) Matrix-fracture transfer shape factors for dual-porosity simulators. Petrol Sci Eng 13:169–178 » CrossRef Lucia FJ (1998) Carbonate reservoir characterization. Springer, Berlin Mavko G, Mukerji T, Dvorkin J (1998) The rock physics handbook. Tools for seismic analysis in porous media. Cambridge University Press, Cambridge, UK Meyer FO, Price RC, Al-Raimi SM (2000) Stratigraphic and petrophysical characteristics of core Arab-D super-k intervals, Hawiyah area, Ghawar field, Saudi Arabia. GeoArabia 5(3):355–384 Pal M (2012) A unified approach to simulation and upscaling of single-phase flow through vuggy carbonates. Int J Numer Methods Fluids 69:1096–1123 » CrossRef Pelton WH, Ward SH, Halloff PG, Still WR, Nelson PH (1978) Mineral discrimination and removal of inductive coupling with multifrequency induced polarization. Geophysics 43:588–609 » CrossRef Peng S, Hu Q, Dultz S, Zhang M (2012) Using X-ray computed tomography in pore structure characterization for a Berea sandstone: resolution effect. J Hydrol 472–473(23):254–261 » CrossRef Perez-Rozales C (1982) On the relationship between formation resistivity factor and porosity. SPE J (Aug., 1982): 531-536 Pulido H, Samaniego FV, Cinco-Ley H, Rivera J, Guadalupe G (2007) Triple porosity mode—double permeability with transient hydraulic diffusivity in naturally fractured reservoirs, Proceedings of 32nd Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California, January 22–24, 2007, SGP-TR-183 Pulido H, Galicia-Munoz G, Valdés-Pérez AR, Diaz-Garcia F Improve Reserves estimation using interporosity skin in naturally fractured reservoirs. Proceedings of 32nd Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California, January 31–February 2, 2011, SGP-TR-191 Rassenfoss S (2011) Digital rocks out to become a core technology. J Petrol Technol May, 2011, 36–41 Reuss A (1929) Berechnung der Fliessgrenzen von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z Angew Math Mech 9:49–58 » CrossRef Sayers CM, Kachanov M (1995) Microcrack induced elastic wave anisotropy of brittle rocks. J Geophys Res 100:4149–4156 » CrossRef Schlumberger (1991) Log interpretation principles/applications. Schlumberger Educational Services, Houston, Texas Schubnel A, Guéguen Y (2003) Dispersion and anisotropy in cracked rocks. J Geophys Res 108:2001. doi:10.1029/2002JB001824 » CrossRef Sok RM, Knackstedt MA, Sheppard AP, Pinczewski WV, Lindquist WB, Venkatarangan A, Paterson L (2002) Direct and stochastic generation of network models from tomographic images; effect of topology on residual saturations. Transp Porous Media 46:345–372 » CrossRef Sok R, Varslot T, Ghous A, Latnam S, Sheppard AP, Knackstedt MA (2009) Pore scale characterization of carbonates at multiple scales: integration of MCT, BSEM and FIBSEM, International Symp. of Soc. Core Anal Sorbie KS, Skauge A (2011) Can network modeling predict two-phase flow functions? International Symp. Soc. Core Anal., Austin, TX, USA, 18–21 Sept., 2011. Paper SCA2011-29 Stenger B, Pham T, Al-Afaleg N, Lawrence P (2003) Tilted original oil/water contact in the Arab-D reservoir, Ghawar field, Saudi Arabia. GeoArabia 8(1):9–39 Telford WM, Geldart LP, Sheriff RE (1990) Applied geophysics. Cambridge University Press, Cambridge » CrossRef Thomeer JHM (1960) Introduction of a pore geometrical factor defined by a capillary pressure curve. Petr Trans AIME 219(TN 2057):354–358 Thomeer JHM (1983) Air permeability as a function of three pore-network parameters. J Petr Technol April, pp. 809-814 Touati M, Suicmez S, Funk J, Cinar Y, Knacksted M (2009) Pore network modeling of Saudi Aramco rocks: a comparative study. SPE Saudi Arabia Section Techn. Symp., 9–11 May, 2009, Al Khobar, SA Turcotte DL, Schubert G (1982) Geodynamics, applications of continuum physics to geological problems. Wiley, New York Walsh JB, Brace WF (1984) The effect of pressure on porosity and the transport property of rock. J Geophys Res 89B(11):9425–9431 » CrossRef Walton K (1987) The effective moduli of a random packing of spheres. J Mech Phys Solids 35:213–226 » CrossRef Warren JE, Root PJ (1963) The behavior of naturally fractured reservoirs. SPE J 426:245–255 Widjajakusuma J, Biswal B, Hilfer R (1999) Quantitative prediction of effective material properties of heterogeneous media. Comput Mater Sci 16(70) Worthington PF (1993) The uses and abuses of the Archie equations, 1: the formation factor-porosity relationship. J Appl Geophys 30(3):215–228 » CrossRef Worthington PF (2011) The petrophysics of problematic reservoirs. J Petrol Technol December 2011, 88–97 Wyllie MRJ, Gregory AR, Gardner GHF (1956) Elastic wave velocities in heterogeneous and porous media. Geophysics 21:41–70 » CrossRef Wyllie MRJ, Gregory AR, Gardner GHF (1958) An experimental investigation of factors affecting elastic wave velocities in porous media. Geophysics 23:459–493 » CrossRef Yonezawa F, Cohen MH (1983) Granular effective medium approximation. J Appl Phys 54:2895–2899 » CrossRef Zhang X, Knackstedt MA (1995) Direct simulation of electrical and hydraulic tortuosity in porous solids. Geophys Res Lett 22(17):2333–2336 » CrossRef Zimmerman RW (1991) Compressibility of sandstones. Elsevier, Amsterdam
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