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| In [[Reservoir Petrophysics|petrophysics]], '''Archie's law''' relates the in-situ [[electrical conductivity]] of a sedimentary rock to its [[porosity]] and [[brine]] saturation:
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| :<math>C_t = \frac{1}{a} C_w \phi^m S_w^n</math>
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| Here, <math>\phi\,\!</math> denotes the porosity, <math>C_t</math> the electrical conductivity of the fluid saturated rock, <math>C_w</math> represents the electrical conductivity of the brine, <math>S_w</math> is the brine [[water saturation|saturation]], <math>m</math> is the cementation exponent of the rock (usually in the range 1.8–2.0 for sandstones), <math>n</math> is the saturation exponent (usually close to 2) and <math>a</math> is the [[tortuosity]] factor.
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| Reformulated for [[electrical resistivity]], the equation reads
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| :<math>R_t = a \phi^{-m} S_w^{-n} R_w </math>
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| with <math>R_t</math> for the fluid saturated rock resistivity, and <math>R_w</math> for the brine resistivity.
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| The factor
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| :<math>F = \frac{a}{\phi^m} = \frac{R_o}{R_w}</math>
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| is also called the '''formation factor''', where <math>R_o</math> is the resistivity of the rock filled with only water (<math>S_w=1</math>).
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| The factor
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| :<math>I = \frac{R_t}{R_o} = S_w^{-n}</math>
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| is also called the '''resistivity index'''.
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| It is a purely [[empirical law]] attempting to describe [[ion]] flow (mostly [[sodium]] and [[chloride]]) in clean, consolidated sands, with varying intergranular porosity. Electrical conduction is assumed not to be present within the rock grains or in fluids other than water.
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| Archie's law is named after [[Gus Archie]] (1907–1978) who developed this empirical quantitative relationship between porosity, electrical conductivity, and brine saturation of rocks. Archie's law laid the foundation for modern [[Well logging|well log]] interpretation as it relates borehole electrical conductivity measurements to [[hydrocarbon]] saturations (which, for fluid saturated rock, equals <math>1 - S_w</math>).
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| ==Parameters==
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| ===Cementation exponent, <math>m</math>===
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| The [[cementation (geology)|cementation]] exponent models how much the pore network increases the resistivity, as the rock itself is assumed to be non-conductive. If the pore network were to be modelled as a set of parallel capillary tubes, a cross-section area average of the rock's resistivity would yield porosity dependence equivalent to a cementation exponent of 1. However, the [[tortuosity]] of the rock increases this to a higher number than 1. This relates the cementation exponent to the [[Permeability (earth sciences)|permeability]] of the rock, increasing permeability decreases the cementation exponent.
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| The exponent <math>m</math> has been observed near 1.3 for unconsolidated sands, and is believed to increase with cementation. Common values for this cementation exponent for consolidated sandstones are 1.8 < <math>m</math> < 2.0.
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| In carbonate rocks, the cementation exponent shows higher variance due to strong diagenetic affinity and complex pore structures. Values between 1.7 and 4.1 have been observed. <ref>Verwer, K., Eberli, G.P. and Weger, R.J., 2011, Effect of pore structure on electrical resistivity in carbonates: AAPG Bulletin, no. 20, v. 94, p. 1-16</ref>
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| The cementation exponent is usually assumed not to be dependent on [[temperature]].
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| ===Saturation exponent, <math>n</math>===
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| The saturation exponent <math>n</math> usually is fixed to values close to 2. The [[Water content|saturation]] exponent models the dependency on the presence of non-conductive fluid (hydrocarbons) in the pore-space, and is related to the [[Wetting|wettability]] of the rock. Water-wet rocks will, for low water saturation values, maintain a continuous film along the pore walls making the rock conductive. Oil-wet rocks will have discontinuous droplets of water within the pore space, making the rock less conductive.
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| ===Tortuosity factor, <math>a</math>===
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| The constant <math>a</math>, called the ''tortuosity factor'', ''cementation intercept'', ''lithology factor'' or, ''[[lithology]] coefficient'' is sometimes used. It is meant to correct for variation in [[Compaction (geology)|compaction]], pore structure and grain size.<ref>{{cite journal |last=Winsauer |first=W.O. |coauthors=Shearing, H.M., Jr., Masson, P.H., and Williams, M. |year=1952 |title=Resistivity of brine saturated sands in relation to pore geometry |journal=AAPG Bulletin |volume=36 |issue=2 |pages=253–277}}</ref>
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| ===Measuring the exponents===
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| <!-- note that "Pickett plot" redirects directly to this section header -->
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| In petrophysics, the only reliable source for the numerical value of both exponents is experiments on sand plugs from cored wells. The brine conductivity can be measured directly on produced water samples. Alternatively, the brine conductivity and the cementation exponent can also be inferred from downhole electrical conductivity measurements across brine-saturated intervals. For brine-saturated intervals (<math>S_w=1</math>) Archie's law can be written
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| :<math>\log{C_t} = \log{C_w} + m \log{\phi}\,\!</math>
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| Hence, plotting the logarithm of the measured in-situ electrical conductivity against the logarithm of the measured in-situ porosity (a so-called '''Pickett plot'''), according to Archie's law a straight-line relationship is expected with slope equal to the cementation exponent <math>m</math> and intercept equal to the logarithm of the in-situ brine conductivity.
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| ==Sands with clay/shaly sands==
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| Archie's law postulates that the rock [[matrix (geology)|matrix]] is non-conductive. For sandstone with [[clay minerals]], this assumption is no longer true in general, due to the clay's structure and [[cation exchange capacity]]. The [[Waxman–Smits equation]]<ref>{{cite journal |first1=M.H. |last1=Waxman |first2=L.J.M. |last2=Smits |title=Electrical conductivities in oil-bearing shaly sands |journal=SPE Journal |volume=8 |issue=2 |pages=107–122 |year=1968 |doi=10.2118/1863-A}}</ref> is one model that tries to correct for this.
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| This Law forms the basis of studying and interpreting WELL LOGS.
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| ==See also==
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| *[[Birch's law]]
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| *[[Byerlee's law]]
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| *[[Jason's law]]
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| ==References==
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| *{{cite journal |first=G.E. |last=Archie |title=The electrical resistivity log as an aid in determining some reservoir characteristics |journal=Petroleum Transactions of AIME |year=1942 |volume=146 |pages=54–62}}
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| *{{cite journal |first=G.E. |last=Archie |title=Electrical resistivity an aid in core-analysis interpretation |journal=American Association of Petroleum Geologists Bulletin |year=1947 |volume=31 |issue=2 |pages=350–366}}
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| *{{cite journal |first=G.E. |last=Archie |title=Introduction to petrophysics of reservoir rocks |journal=American Association of Petroleum Geologists Bulletin |year=1950 |volume=34 |issue=5 |pages=943–961}}
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| *{{cite journal |first=G.E. |last=Archie |title=Classification of carbonate reservoir rocks and petrophysical considerations |journal=American Association of Petroleum Geologists Bulletin |year=1952 |volume=36 |issue=2 |pages=278–298}}
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| * {{cite book |first=Malcolm H. |last=Rider |title=The Geological Interpretation of Well Logs |publisher=Whittles Publishing Services |edition=Second |year=1999 |isbn=0-9541906-0-2 |pages=288}}
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| * {{cite book |first=Darwin V. |last=Ellis |title=Well Logging for Earth Scientists |publisher=Elsevier |isbn=0-444-01180-3 |year=1987}}
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| * {{cite book |first1=Darwin V. |last1=Ellis |first2=Julian M. |last2=Singer |title=Well Logging for Earth Scientists |edition=Second |publisher=Springer |isbn=1-4020-3738-4 |year=2008 |pages=692}}
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| <references/>
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| {{DEFAULTSORT:Archie's Law}}
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| [[Category:Geophysics]]
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| [[Category:Equations]]
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| [[Category:Well logging]]
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The writer's name is Christy Brookins. To climb is some thing she would by no means give up. For years she's been living in Kentucky but her spouse wants them to transfer. Since I was 18 I've been working as a bookkeeper but soon my wife and I will start our own business.
Here is my blog post: best psychics