Archie's law
Encyclopedia
In petrophysics, Archie's law relates the in-situ electrical conductivity of a sedimentary rock to its porosity
and brine
saturation:
Here,
denotes the porosity, 
the electrical conductivity of the fluid saturated rock, 
represents the electrical conductivity of the brine, 
is the brine saturation, 
is the cementation exponent of the rock (usually in the range 1.8–2.0), 
is the saturation exponent (usually close to 2) and 
is the tortuosity
factor.
Reformulated for electrical resistivity, the equation reads
with
for the fluid saturated rock resistivity, and 
for the brine resistivity.
The factor
is also called the formation factor, where
is the resistivity of the rock filled with only water (
).
The factor
is also called the resistivity index.
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.
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 log
interpretation as it relates borehole electrical conductivity measurements to hydrocarbon
saturations (which, for fluid saturated rock, equals
).
Cementation exponent,
The 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 of the rock, increasing permeability decreases the cementation exponent.
The exponent
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 < 
< 2.0.
The cementation exponent is usually assumed not to be dependent on temperature
.
Saturation exponent,
The saturation exponent 
usually is fixed to values close to 2. The saturation
exponent models the dependency on the presence of non-conductive fluid (hydrocarbons) in the pore-space, and is related to the 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.
Tortuosity factor,
The constant 
, called the tortuosity factor, cementation intercept, lithology factor or, lithology
coefficient is sometimes used. It is meant to correct for variation in compaction
, pore structure and grain size.
) Archie's law can be written
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
and intercept equal to the logarithm of the in-situ brine conductivity.
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 is one model that tries to correct for this.
Porosity
Porosity or void fraction is a measure of the void spaces in a material, and is a fraction of the volume of voids over the total volume, between 0–1, or as a percentage between 0–100%...
and brine
Brine
Brine is water, saturated or nearly saturated with salt .Brine is used to preserve vegetables, fruit, fish, and meat, in a process known as brining . Brine is also commonly used to age Halloumi and Feta cheeses, or for pickling foodstuffs, as a means of preserving them...
saturation:
Here,
denotes the porosity, the electrical conductivity of the fluid saturated rock, represents the electrical conductivity of the brine, is the brine saturation, is the cementation exponent of the rock (usually in the range 1.8–2.0), is the saturation exponent (usually close to 2) and is the tortuosityTortuosity
Tortuosity is a property of curve being tortuous . There have been several attempts to quantify this property. Tortuosity is commonly used to describe diffusion in porous media...
factor.
Reformulated for electrical resistivity, the equation reads
with
for the fluid saturated rock resistivity, and for the brine resistivity.The factor
is also called the formation factor, where
is the resistivity of the rock filled with only water ().The factor
is also called the resistivity index.
It is a purely empirical law attempting to describe ion
Ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...
flow (mostly sodium
Sodium
Sodium is a chemical element with the symbol Na and atomic number 11. It is a soft, silvery-white, highly reactive metal and is a member of the alkali metals; its only stable isotope is 23Na. It is an abundant element that exists in numerous minerals, most commonly as sodium chloride...
and chloride
Chloride
The chloride ion is formed when the element chlorine, a halogen, picks up one electron to form an anion Cl−. The salts of hydrochloric acid HCl contain chloride ions and can also be called chlorides. The chloride ion, and its salts such as sodium chloride, are very soluble in water...
) 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.
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 log
Well logging
Well logging, also known as borehole logging is the practice of making a detailed record of the geologic formations penetrated by a borehole. The log may be based either on visual inspection of samples brought to the surface or on physical measurements made by instruments lowered into the hole...
interpretation as it relates borehole electrical conductivity measurements to hydrocarbon
Hydrocarbon
In organic chemistry, a hydrocarbon is an organic compound consisting entirely of hydrogen and carbon. Hydrocarbons from which one hydrogen atom has been removed are functional groups, called hydrocarbyls....
saturations (which, for fluid saturated rock, equals
).Cementation exponent, 
The cementationCementation (geology)
Cementation involves ions carried in groundwater chemically precipitating to form new crystalline material within sediment pores; this is how "sediment" becomes "rock". The new pore-filling minerals form "bridges" between original sediment grains, thereby binding them together. So sand becomes...
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
Tortuosity
Tortuosity is a property of curve being tortuous . There have been several attempts to quantify this property. Tortuosity is commonly used to describe diffusion in porous media...
of the rock increases this to a higher number than 1. This relates the cementation exponent to the permeability of the rock, increasing permeability decreases the cementation exponent.
The exponent
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 < < 2.0.The cementation exponent is usually assumed not to be dependent on temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
.
Saturation exponent, 
The saturation exponent usually is fixed to values close to 2. The saturationWater content
Water content or moisture content is the quantity of water contained in a material, such as soil , rock, ceramics, fruit, or wood. Water content is used in a wide range of scientific and technical areas, and is expressed as a ratio, which can range from 0 to the value of the materials' porosity at...
exponent models the dependency on the presence of non-conductive fluid (hydrocarbons) in the pore-space, and is related to the wettability
Wetting
Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. The degree of wetting is determined by a force balance between adhesive and cohesive forces.Wetting is important in the bonding or adherence of...
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.
Tortuosity factor, 
The constant , called the tortuosity factor, cementation intercept, lithology factor or, lithologyLithology
The lithology of a rock unit is a description of its physical characteristics visible at outcrop, in hand or core samples or with low magnification microscopy, such as colour, texture, grain size, or composition. It may be either a detailed description of these characteristics or be a summary of...
coefficient is sometimes used. It is meant to correct for variation in compaction
Compaction (geology)
Compaction refers to the process by which a sediment progressively loses its porosity due to the effects of loading. This forms part of the process of lithification. When a layer of sediment is originally deposited, it contains an open framework of particles with the pore space being usually...
, pore structure and grain size.
Measuring the exponents
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 () Archie's law can be writtenHence, 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
and intercept equal to the logarithm of the in-situ brine conductivity.Sands with clay/shaly sands
Archie's law postulates that the rock matrixMatrix (geology)
The matrix or groundmass of rock is the finer grained mass of material in which larger grains, crystals or clasts are embedded.The matrix of an igneous rock consists of finer grained, often microscopic, crystals in which larger crystals are embedded. This porphyritic texture is indicative of...
is non-conductive. For sandstone with clay minerals
Clay minerals
Clay minerals are hydrous aluminium phyllosilicates, sometimes with variable amounts of iron, magnesium, alkali metals, alkaline earths, and other cations. Clays have structures similar to the micas and therefore form flat hexagonal sheets. Clay minerals are common weathering products and low...
, this assumption is no longer true in general, due to the clay's structure and cation exchange capacity
Cation exchange capacity
In soil science, cation-exchange capacity is the maximum quantity of total cations, of any class, that a soil is capable of holding, at a given pH value, for exchanging with the soil solution. CEC is used as a measure of fertility, nutrient retention capacity, and the capacity to protect...
. The Waxman–Smits equation is one model that tries to correct for this.