experiment to find permeability of free space

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Permeability of free space.

The permeability of free space , μ 0 , is a physical constant used often in electromagnetism . It is defined to have the exact value of 4π x 10 -7 N/A 2 ( newtons per ampere squared). [1] It is connected to the energy stored in a magnetic field , see Hyperphysics for specific equations.

It is related to the speed of light by the equation:

• [math]c[/math] is the speed of light and
• [math]\epsilon_0[/math] is the permittivity of free space .

A magnetic field, [math]B[/math] in a region of space has field energy associated with it. That energy density is defined as: [2]

One particular use is the magnetic force . The constant gives how strong the force is between two electric currents separated by a distance: [1]

• [math]F[/math] is the Magnetic force,
• [math]I_1[/math] and [math]I_2[/math] are two currents, and
• [math]r[/math] is the separation between the wires carrying the currents.

Visit Hyperphysics for more information on this constant.

For Further Reading

For further information please see the related pages below:

• Permittivity of free space
• Magnetic field
• Magnetic flux
• Or explore a random page
• ↑ 1.0 1.1 Hyperphysics. (August 28, 2015). Electric field [Online], Available: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefie.html
• ↑ Jackson, John David (1998). Classical Electrodynamics (3rd ed.). New York: Wiley. p. 213

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Permeability/permittivity of free space.

What actually is the permeability of free space and the permittivity of free space? I know it’s not super important for high school physics but I can’t let it go- I want a proper understanding of what it means rather than just throwing it into my equations because what’s the point? It’d be great if somebody could break it down for me, cheers!

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Permeability of Free Space

In ap physics c: e&m.

A physical constant denoted by μ₀, representing the ability of free space to allow magnetic field lines to pass through it. It has a value of approximately 4π x 10⁻⁷ T·m/A.

congrats on reading the definition of Permeability of Free Space . now let's actually learn it.

Related terms

Magnetic Field Strength : The force experienced by a unit positive test charge moving perpendicular to magnetic field lines.

Magnetic Flux Density (B-field) : A measure of the strength and directionality of a magnetic field at any given point.

Ampere's Law : Relates the circulating magnetic field around a closed loop to the electric currents passing through that loop.

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• College Physics: Thermodynamics, Electricity, and Magnetism
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• Astronomy and Cosmology

Why does the permeability of free space have an exact value?

• Thread starter DeadWolfe
• Start date May 21, 2004
• Tags Free space Permeability Space Value
• May 21, 2004
• New measurements of gravitational anomaly at low acceleration favor modified gravity, researcher claims
• Hubble and Chandra find supermassive black hole duo
• New observations shed more light on the nature of a millisecond pulsar binary

DeadWolfe said: And why is it related to pi?

• May 22, 2004

TALewis said: When you see physical constants listed as exact , rather than given to an approximate number of decimals, in my experience this means that the units the constant is given in are defined in some way or another in terms of the constant itself. For example, my text says that the speed of light in vacuum is exactly : 299792458 meters per second How can anyone say this is the number, exactly? It's not because they've got a really good stopwatch and meterstick. They can do it because they they define the meter in the following way: 1 meter is equal to the distance traveled by light in vacuum during a time of 1/299792458 second.

Edit: Through a little more quick research it seems the exact value of the permeability of free space comes from the definition of the Ampere: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 Newton per meter of length.

It's the same kind of mathematical idealism that you see in something like the Pythagoream theorem. Sure, I say that a^2 + b^2 = c^2, but I could never physically construct a right triangle so perfect as to be able to say that the measurements of the sides satisfy the equation exactly. There is always uncertainty. I find the definition of the meter particularly tickling. 1 meter is the distance light travels in 1/299792458 second. OK, how far is that? Well, no one can say, exactly, other than to define it so. To call it one meter. But if I'm going to make a meterstick, I'm not going to shut myself in a dark room with a flashlight and a wristwatch. If I'm going to make a wristwatch I'm not going to count the vibrations of a cesium atom to tell me how long a second is. So while a physicist offers an intellectually satisfying definition, it's up to an engineer to make it practical.  

TALewis said: It's the same kind of mathematical idealism that you see in something like the Pythagoream theorem. Sure, I say that a^2 + b^2 = c^2, but I could never physically construct a right triangle so perfect as to be able to say that the measurements of the sides satisfy the equation exactly. There is always uncertainty.

A PF Asteroid

The value of [tex]\mu_0 = 4 \pi x10^{-7} \frac H m [/tex] is due to the choice of constants in Maxwell's equations. This system is called the Rationalized MKS system. Do a web search on that term you should be able to find the details.  

TALewis said: For example, my text says that the speed of light in vacuum is exactly : 299792458 meters per second How can anyone say this is the number, exactly? It's not because they've got a really good stopwatch and meterstick. They can do it because they they define the meter in the following way: 1 meter is equal to the distance traveled by light in vacuum during a time of 1/299792458 second.

Basically, the permeability of free space is a geometric constant (like pi), not an empirical constant.  

Some of the newer physics texts offer this explanation: The problem is the standard rod or weight. Most of the standards were held in some bureau in France. For example, the standard for mass, the kilo, was a cylidrical metal weight. I'm not as sure about length but I'm sure there was a physical rod as the standard of length. This worked for awhile until a set of problems arose. First, most materials warp, expand, and contract over time make the standard useless. Second no one can precisely duplicate the standard. Tired of useless, imprecise, standards of measurement. Worry no more, the UNIVERSE provides with the new, unshakeable, standard. Light! :surprise: Yes! Light travels at a constant speed. The meter is now defined in terms of light travel time. The second is also defined as 2.999792E8 meters of light travel time. The vibration of a cesium atom is constant. A cesium atom electons are excited and gives of light. From Q.M. the energy of photon also reveals the frequency. So the second is the inverse of the frequency of cesium atom vibrations.  

A PF Mountain

H-bar None said: The meter is now defined in terms of light travel time. The second is also defined as 2.999792E8 meters of light travel time.

krab One problem I have, which I pointed out, is that the definition of the Ampere relies on something physically impossible (infinitely long wires). In fact, since there will be gravitational attraction between any two wires as well, it seems the definition of the Ampere is flawed in comparison to, say, the definitions of the second (which depends on a constant and something countable) or the meter (which depends on the constant speed of light). Second of all, it is said that any time less than the Planck time is meaningless. We can take this to suggest that, for all meaningful intents and purposes, time is quantized. Charge, too, is quantized. Current measures the rate of flow of charge with respect to time. But if current depends on [itex]\mu_{o}[/itex] which depends on [itex]\pi[/itex] which is irrational, then doesn't that contradict the fact that current is based on quantum measurements (sorry if I'm using the wrong terminology). A second must be a multiple of the Planck time, I would think, since, if, say a second was 100.5 Planck times (I know that's way off) that would mean 0.5 Planck times would have some meaning, but it doesn't, as far as I know. So, if an amp is charge/time, then it must be some_natural/some_natural x some units, giving some rational with units, whereas pi is irrational. Is this not contradictory?  

• May 23, 2004

AKG: Although the Pythagorean Theorem is not a definition, it's practicality is essentially irrelevant since it need only apply to mathematical objects which exist "in" mathematics, and need no correspondence to the real world." Are you saying that there is no real world correspondence between the theory and the reality?  

TillEulenspiegel said: AKG: Although the Pythagorean Theorem is not a definition, it's practicality is essentially irrelevant since it need only apply to mathematical objects which exist "in" mathematics, and need no correspondence to the real world." Are you saying that there is no real world correspondence between the theory and the reality?

AKG said: krab One problem I have, which I pointed out, is that the definition of the Ampere relies on something physically impossible (infinitely long wires). In fact, since there will be gravitational attraction between any two wires as well, it seems the definition of the Ampere is flawed in comparison to, say, the definitions of the second (which depends on a constant and something countable) or the meter (which depends on the constant speed of light).

Second of all, it is said that any time less than the Planck time is meaningless. We can take this to suggest that, for all meaningful intents and purposes, time is quantized. Charge, too, is quantized. Current measures the rate of flow of charge with respect to time. But if current depends on [itex]\mu_{o}[/itex] which depends on [itex]\pi[/itex] which is irrational, then doesn't that contradict the fact that current is based on quantum measurements (sorry if I'm using the wrong terminology). A second must be a multiple of the Planck time, I would think, since, if, say a second was 100.5 Planck times (I know that's way off) that would mean 0.5 Planck times would have some meaning, but it doesn't, as far as I know. So, if an amp is charge/time, then it must be some_natural/some_natural x some units, giving some rational with units, whereas pi is irrational. Is this not contradictory?

• May 30, 2004

AKG said: One problem I have, which I pointed out, is that the definition of the Ampere relies on something physically impossible (infinitely long wires). In fact, since there will be gravitational attraction between any two wires as well, it seems the definition of the Ampere is flawed in comparison to, say, the definitions of the second (which depends on a constant and something countable) or the meter (which depends on the constant speed of light).

A PF Molecule

H-bar None said: The vibration of a cesium atom is constant. A cesium atom electons are excited and gives of light. From Q.M. the energy of photon also reveals the frequency. So the second is the inverse of the frequency of cesium atom vibrations.

• Oct 3, 2004

permeability What if I told you that the permeability of free space was increasing due to the expansion of the universe? That would mean that the speed of light is also increasing. Which would mean that light emitted from distance stars billions of years ago would actually be traveling faster now than when it was first emitted, and therefore appear to be red shifted more to the red than it actually would be just due to the expansion of the universe alone.  

A PF Planet

rogerperkins said: What if I told you that the permeability of free space was increasing due to the expansion of the universe? That would mean that the speed of light is also increasing.

rogerperkins said: What if I told you that the permeability of free space was increasing due to the expansion of the universe? That would mean that the speed of light is also increasing. Which would mean that light emitted from distance stars billions of years ago would actually be traveling faster now than when it was first emitted, and therefore appear to be red shifted more to the red than it actually would be just due to the expansion of the universe alone.

FAQ: Why does the permeability of free space have an exact value?

1. why does the permeability of free space have an exact value.

The permeability of free space, denoted by the symbol μ0, is a fundamental physical constant that describes the ability of a vacuum to support the flow of magnetic fields. This value is exact because it is defined based on the International System of Units (SI), which is a globally accepted system of measurement for physical quantities. The value of μ0 is determined by experiments and is not subject to change.

2. How is the exact value of the permeability of free space determined?

The exact value of μ0 is determined through a variety of experiments, including the measurement of the force between two current-carrying wires, the behavior of a magnet in a coil of wire, and the deflection of a charged particle in a magnetic field. These experiments allow scientists to calculate the value of μ0 with a high degree of accuracy.

3. Why is the permeability of free space considered a constant?

The permeability of free space is considered a constant because it does not vary with any external factors such as temperature, pressure, or magnetic field strength. This means that the value of μ0 remains the same regardless of the conditions in which it is measured. This makes it a reliable and fundamental constant in the study of electromagnetism.

4. How does the exact value of the permeability of free space impact other physical quantities?

The value of μ0 plays a crucial role in many equations and formulas in the field of electromagnetism. It is used to calculate other physical quantities such as the magnetic field strength, the force between two electric charges, and the speed of light in a vacuum. Having an exact value for μ0 allows for precise calculations and predictions in various scientific fields.

5. Can the exact value of the permeability of free space change in the future?

The value of μ0 is considered a fundamental constant, meaning it is an unchanging and immutable quantity. As of now, there is no scientific evidence to suggest that the value of μ0 could change in the future. However, as our understanding of the universe and its fundamental laws continues to evolve, it is possible that new discoveries may lead to a re-evaluation of this constant in the future.

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How the value of permitivity of free space is determined?

when electric charges are placed on the medium then force of interaction between them decreases. We say this factor permitivity of the medium. The force of interaction between the charges is max if the charges are at space or if there is absence of medium. We know the electric permittivity of free space is $8.85 * 10^{-12} F/m$. My question is how this value is determined. Is there any experiment held?

• electrostatics
• experimental-physics

• $\begingroup$ These days it has a defined value. WP describes the history. en.wikipedia.org/wiki/Permittivity_of_free_space#Value $\endgroup$ –  user4552 Commented May 23, 2013 at 14:57
• $\begingroup$ related: physics.stackexchange.com/q/3644/4552 physics.stackexchange.com/q/64841/4552 $\endgroup$ –  user4552 Commented May 23, 2013 at 15:48

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• Open access
• Published: 05 September 2024

Permeability partitioning through the brittle-to-ductile transition and its implications for supercritical geothermal reservoirs

• Gabriel G. Meyer   ORCID: orcid.org/0000-0003-4628-6879 1 ,
• Ghassan Shahin   ORCID: orcid.org/0000-0003-3160-2779 1 ,
• Benoît Cordonnier   ORCID: orcid.org/0000-0001-8632-3844 2 &
• Marie Violay   ORCID: orcid.org/0000-0002-7402-8263 1  

Nature Communications volume  15 , Article number:  7753 ( 2024 ) Cite this article

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• Volcanology

Geothermal projects utilizing supercritical water (≥400 °C) could boost power output tenfold compared to conventional plants. However, these reservoirs commonly occur in crustal areas where rocks are semi-ductile or ductile, impeding large-scale fractures and cracking, and where hydraulic properties are largely unknown. Here, we explore the complex permeability of rocks under supercritical conditions using mechanical data from a gas-based triaxial apparatus, high-resolution synchrotron post-mortem 3D imagery, and finite element modeling. We report a first order control of strain partitioning on permeability. In the brittle regime, strain localizes on permeable faults without necessarily increasing sample apparent permeability. In the semi-ductile regime, distributed strain increases permeability both in deformation bands and the bulk, leading to a more than tenfold permeability increase. This study challenges the belief that the brittle-ductile transition (BDT) marks a cutoff for fluid circulation in the crust, demonstrating that permeability can develop in deforming semi-ductile rocks.

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Introduction.

Geothermal power plants that harness natural, high-enthalpy geothermal systems have emerged as a crucial solution to combat the global climate crisis. The recent discovery of supercritical water resources above magma bodies has sparked immense interest in enhancing geothermal energy output by extracting supercritical fluids characterized by pressures above 22 MPa and temperatures surpassing 375 °C 1 , 2 . This approach, estimated to boost geothermal plant energy output by a factor of 10, has led to ambitious projects worldwide, including the Japan Beyond-Brittle Project (JBBP), the Taupo Volcanic Zone-Deep geothermal drilling project in New Zealand (TVZ-DGDP), and the NDDP project at the Newberry volcano, USA 2 , 3 , 4 . However, realizing this potential is currently hindered by the coexistence of supercritical geothermal fluids and the transition of rocks from brittle to ductile behavior.

The brittle-to-ductile transition (BDT) occurs at depth and marks the boundary between two regions of the crust with distinct rheology 5 , 6 , 7 . As pressure and temperature rise, the manifestation of deformation at the macroscopic level transitions from being concentrated along sharp fractures and faults to being homogeneously distributed throughout the bulk 8 , 9 (localized-ductile transition; LDT). At the microscopic level, the mechanism for accommodating strain transitions from being primarily driven by microcracking to being governed by crystal lattice distortion accommodated by mechanisms such as dislocation glide 9 , 10 (brittle-plastic transition; BPT).

Understanding the transition from brittle to ductile behavior in rocks, where BDT refers to the conjoined effect of both LDT and BPT, is vital for optimizing geothermal energy extraction. Production efficiency and feasibility heavily rely on the permeability of the host rock (i.e, existing macro and micro fractures facilitating fluid circulation) 11 , 12 , 13 , 14 , 15 , 16 . Moreover, ductile reservoirs offer significant advantages over classical brittle reservoirs, such as reduced seismicity due to the absence of large-scale tectonic faults and overall homogeneity of rock properties 17 . However, the reduced cracking activity in favor of crystal plasticity and the absence of large-scale faults raise concerns about permeability. For these reasons, the brittle-to-ductile transition is widely considered a critical threshold for hydrothermal circulation in the crust 18 , 19 , 20 , which not only bears implications for the exploitation of geothermal resources, but also for all crustal processes involving the circulation of fluids.

Unraveling the permeability dynamics of nominally ductile systems is imperative to advance the exploitation of supercritical geothermal reservoirs. Assessing the permeability of rock systems below 2–3 km has been hindered by significant technological limitations in in-situ borehole measurements, especially under supercritical conditions 21 . Experimentalists have embarked on filling this knowledge gap. Extensive studies of low porosity rocks in the brittle regime, characterized by relatively low pressure and temperature, have demonstrated that deformation increases permeability, yielding promising results for enhanced geothermal systems 15 , 22 , 23 , 24 , 25 , 26 . Moreover, the permeability of nominally brittle samples but under supercritical conditions have been tested, showing a clear effect of fluid critical state on permeability 22 . However, these studies were unable to test permeability as a function of deformation in the ductile regime (higher pressure and temperature), as the technical complexity of measuring permeability under such extreme conditions remains critically challenging. Additionally, while dilatancy has historically served as a proxy for permeability at elevated temperature and pressure, its relationships with permeability is all but trivial and prone to estimation errors 20 , 27 .

In this study, we present permeability assessment of deforming crustal rocks in the semi-ductile regime. Continuous permeability measurements have been performed on granite specimens deformed under elevated effective pressure (100 MPa) and a broad range of extreme temperatures (200–800 °C). The macroscopic measurements have been corroborated by local, spatially resolved permeability quantification conducted through X-ray imaging and numerical modeling. We demonstrate that permeability can be created in deforming semi-ductile rocks and that the resulting diffuse strain can lead to greater equivalent permeability than in brittle, localized rocks. Most importantly, we challenge the long established belief that the BDT marks a cutoff for the circulation of fluids in the crust.

Mechanical data and measured permeability

Cylindrical samples of Lanhélin granite with a diameter of 20 mm and height of 42 mm were subjected to elevated confining pressure (150 MPa), argon pore pressure (50 MPa) and a range of temperatures (200–800 °C) while maintaining a constant axial strain rate of 10 −6  s −1 , corresponding to conditions under which granite displays localized to semi-ductile behavior in the laboratory 20 (Fig.  1 A, B). Permeability was recorded continuously by means of the oscillatory method.

A An optical micrograph of intact Lanhélin granite ( B ). Stress as a function of strain at all tested temperatures ( C ). Permeability as a function of strain at all tested temperatures where thicker lines are a moving average of 8 data points of the thinner lines ( D ). Normalized permeability k / k 0 as a function of strain whereby initial permeability k 0 is taken as the permeability right before initiating deformation ( E ).

Within the temperature range of 200–600 °C, Lanhélin granite exhibits a mechanical behavior typical of the brittle, localized deformation, with stress increasing with strain until reaching a maximum value ( \({Q}_{\max }\) ) before failure. In this regime, \({Q}_{\max }\) displays a moderate temperature dependence, decreasing from 624 MPa at 200 °C–588 MPa at 600 °C. Specimens tested at 200 °C and 400 °C exhibit a more compliant anelastic response during initial deformation than the sample deformed at 600 °C, which can be attributed to the closure of multiple pistons interfaces within the loading column. In contrast, the specimen tested at 800 °C demonstrates marked differences in \({Q}_{\max }\) and deformation behavior. Notably, no observable strength peak is present, and stress increases with strain in a non-linear fashion up to a stress plateau. The sample’s strength at 800 °C is more than halved, reaching 272 MPa at 5% strain. This sample shows semi-ductile rheology.

Measured permeability in all experiments ranged between 10 −20 and 10 −18  m 2 , with inception values consistently below 10 −19  m 2 .

The initial permeability of all samples (i.e., after pressurization and thermal loading, but before axial deformation) remains between 2 and 6 × 10 −19  m 2 . There is no trend between initial permeability and applied temperature (Fig.  1 D).

In the 200 °C and 400 °C experiments, we observe fluctuations around the initial permeability throughout deformation. The 600 °C and 800 °C specimens displayed marked increases in permeability. The permeability growth in the 800 °C sample was as much as one order of magnitude by the end of the experiment. The brittle specimen deformed at 600 °C displayed nearly constant permeability after the strength peak was achieved whereas, by contrast, the semi-ductile specimen deformed at 800 °C displayed continuous increases in permeability. Most of these observations are counter-intuitive. On one hand, permeability is expected to increase upon brittle failure. On the other hand, permeability is anticipated to be larger in rocks deformed in the brittle regime compared to the ductile regime. Therefore, a closer look at the microstructure to understand the failure mechanisms and modes and their influence on permeability was necessary to explain these intriguing observations.

Microstructure analysis

We conducted X-ray scans of the post-mortem specimens at the European Synchrotron Radiation Facility, beamline BM18, in Grenoble, France in order to examine the micromechanisms active in the brittle and semi-ductile regimes and their implications on permeability changes.

Vertical (parallel to axial loading) and horizontal (perpendicular to the axial loading) slices obtained from the X-ray tomography images are presented in Fig.  2 . In these images, four distinct phases/gray-scales can be identified: a dark gray phase representing cracks, a silver phase representing the quartz and feldspar, a light gray phase representing the mica, and a nearly white phase representing the copper casing.

Middle column shows vertical sections of the samples while left and right column show radial cross-section taken in places indicated in red and yellow. The blue boxes represent the sub-volumes in which permeability, crack length and aperture was computed and the numbers labeled are linked to the results presented in Table  1 . The bottom inset show the x , y and z coordinates used in the later calculations. Note that z is parallel to the applied deviatoric stress.

All specimens display significant cracking induced by deformation. The cracks vary in intensity and distribution across the specimen. Vertical sections reveal deformation bands (DB) that appear as structures of highly concentrated cracks, inclined between 45 and 60° with the horizontal. The deformation band is narrow in the specimen deformed at 200 °C (<1 mm), while, in the sample deformed at 800 °C, the band is wider (3–4 mm) and contains marked cataclasis (Fig.  2 A, D). All DBs intersect with one of the specimen’s ends and the lateral surface, but never connect both ends.

In the horizontal sections, the DBs extend between the lateral surfaces. In the vertical section, the DBs are curved. Deformation band branching (two or multiple branches) was observed, especially near the edges and especially in the sample deformed at 800 °C (Fig.  2 D). It is important to note here that some fragments were washed out by a confinement leak during unloading, which explains the excessive void inside the deformation band of the semi-ductile specimen’s deformation band (see the puncture in the surrounding copper jacket in the bottom left of the left in Fig.  2 D).

Moreover, the scans reveal arrays of secondary cracks present off-band in the bulk rock. These cracks are smaller and thinner than the DB, generally spanning a few grains. These cracks are inclined parallel to the direction of compression. While these arrays are ubiquitous among samples, their density varies and is greater at higher temperatures. Notably, the secondary crack network of the 800 °C specimen (Fig.  2 D) is denser and more spread across the specimen bulk; individual cracks are wider (in some places are as wide as 50  μ m) and connect the upper and lower specimen ends (Fig.  2 D).

The effect of semi-ductile deformation on in-situ permeability

The mechanical data illustrate a marked transition from localized to semi-ductile deformation in Lanhélin granite with increasing temperature. This transition is characterized by changes in the initial stiffness and stress-strain response (Fig.  1 C).

Surprisingly, the initial permeability of the samples (i.e., before deformation but after pressurization and thermal loading) does not show a trend with applied temperature (Fig.  1 D). Under room or low confining pressure, thermal stressing of rocks, particularly granite, typically results in crack formation due to the mismatched thermal expansion coefficients of the constituent minerals, which in turns, translates to an increase in permeability 28 , 29 , 30 . In our experiments, the samples were pressurized before being heated. This heating appears not to have impacted the samples permeability suggesting that elevated confining pressures could suppress thermally induced cracking. However, a more systematic study is necessary to understand the interplay between thermal cracking and confining pressure.

When compared to the rock samples’ initial permeability, the permeability data clearly showcases the impact of the transition from localized to semi-ductile behavior on the deforming specimens’ hydraulic properties. While permeability remains nearly constant at 200 °C and 400 °C, it evolves at different rates at higher temperatures. At 600 °C, permeability increases five-fold and at 800 °C, it increases thirty-fold (Fig.  1 E).

Intriguingly, these results appear to contradict dilatancy measurements on Westerly granite, where lower dilatancy was observed at 800 °C compared to lower temperatures 20 . To resolve this apparent paradox, we need to extract the permeability of the deformation band and that of the bulk to better understand the controls on apparent sample permeability during deformation.

Post-mortem crack analysis and permeability modeling

To understand the spatial complexity of the microstructure and permeability, we extracted three cubic, 1500 voxels edge length (3.9 mm), sub-volumes from the 3D reconstructions of the two end member specimens deformed at 200 °C and 800 °C: a sub-volume centered on the deformation band (DB), a sub-volume excluding the deformation band, but located in its close proximity (hence representative of the damage zone), and a sub-volume away from the DB (representative of the bulk; see Fig.  2 for locations). In these sub-volumes, we computed the total crack length and the average crack aperture (Table  1 ).

Additionally, permeability simulations were conducted in x , y , and z direction (see coordinates in Fig.  2 ) on these sub-volumes and the results in the z direction (parallel to the direction of compression) are presented in Table  1 . More details about the computation method are provided in the Methods section. The computed permeabilities for all three directions can be found in  supplementary material (Table  S1 )

In the 200 °C specimen, the sub-volume centered on the DB contains around three times more cracks than the sub-volumes in the damage zone and bulk with a cumulative crack length of 73801  μ m. With an average width of ≈17  μ m, the cracks in the DB are wider than in the damage zone and bulk rock ( ≈16  μ m and ≈14  μ m, respectively).

In the 800 °C specimen, the DB is the most intensely cracked sub-volume with a cumulative crack length of 210795 μ m, followed by the damage zone (162134  μ m) and the bulk (75782  μ m). Interestingly, the bulk rock of the 800 °C appears as cracked as the deformation band of the 200 °C specimen. Average crack aperture ranges from ≈20  μ m in the DB to ≈18  μ m in the bulk.

Computed ex-situ permeability values are nearly seven orders of magnitude greater than the measured in-situ values. The scans were carried out at atmospheric pressure and temperature, and therefore do not account for the actual aperture of the crack network when subjected to the extreme confining pressures applied during the experiments. In fact, cracks are very sensitive to effective confining pressure and it is expected that, under experimental conditions, the aperture of the fracture network would be greatly reduced 31 . Nevertheless, this discrepancy does not prevent relative comparisons between the post-mortem samples since the error introduced is expected to be similar in all samples.

In the 200 °C specimen, the permeability is similar between the DB and the damage zone (≈5.5 × 10 −12  m 2 ). However, the bulk rock away from the DB features a markedly lower permeability of  ≈0.2 × 10 −12  m 2 .

In the 800 °C specimen, the permeability of the DB is twice that of the DB in the 200 °C specimen, while the permeability of the damage zone remains comparable in both specimens. On the other hand, the permeability in the bulk in the 800 °C specimen is comparable to that of its DB and its damage zone, and is an order of magnitude greater than the permeability in the bulk in the 200 °C specimen.

Partitioning of strain as a main control on the partitioning of permeability

Our microstructural analysis highlights the first order role of strain partitioning in influencing sample permeability.

Generally, the transition from localized to ductile behavior is marked by a gradual change in the distribution of strain between macroscopic localized shear fractures and diffuse bulk damage 8 , 9 , 32 . As depth and temperature increase, a greater portion of strain is accommodated by the bulk rock, and hence, a greater amount of damage is expected to occur away from the DB. This is supported by our microstructural observation.

In fact, the brittle sample deformed at 200 °C displays a narrow fracture that accumulated most of the strain applied to the sample. It led to the formation of a permeable fracture embedded in an otherwise less permeable bulk rock.

Conversely, the semi-ductile sample deformed at 800 °C displays a wide deformation band characterized by intensely damaged rock and which proved more permeable than the brittle, sharp fracture. However, in this case, a significant amount of strain was accommodated in the bulk of the sample. It resulted that the damage-induced permeability increase was not limited to the deformation band but is spread-out in the sample and therefore, the permeable semi-ductile DB exists in a relatively damaged and permeable bulk rock.

In the light of these microstructural findings, we can interpret our in-situ apparent permeability data readings.

In our experimental setup, permeability measurements are confined to the sample vertical axis, where permeability depends on both the matrix and fracture permeability. Because these two systems are acting in series, the lowest permeability will control the apparent permeability of the sample as a whole. It resulted that in the brittle sample, given all strain was accommodated on the fault, permeability in the bulk (and therefore the apparent permeability) did not increase during deformation (Fig.  3 A–B1). Conversely, in the semi-ductile regime, the deformation was distributed between the matrix and the large-scale deformation band. Hence, continued deformation of the specimen lead to an increase in crack density in the bulk rock which ultimately resulted in high apparent permeability (Fig.  3 A–B2).

From ( A – B1 ) in the brittle regime, strain is chiefly accommodated on a fault that does not connect the two ends of the samples and hence the apparent permeability k apparent remains constant with increasing deformation. Conversely, from ( A – B2 ) in the semi-ductile regime, strain is diffuse in the sample and k apparent increases with strain. In ( C ) is a diagram opposing a shallow brittle reservoir where permeability is controlled by a main fault to a supercritical ductile reservoir where both bulk and fault control the overall permeability.

In summary, conducting in-situ permeability measurements provided us with a unique scalar representing the apparent, unidirectional permeability of the sample as a whole. This permeability value can be seen as a lower bound value for permeability in granite under conditions representative of the BDT, which does not fully capture the possible heterogeneity (e.g., permeability in a fracture) or anisotropy (e.g., preferential crack orientation) in the sample. Conversely, post-mortem microstructural analysis enabled us to grasp the spatial complexity of permeability within the samples. It could be leveraged to calculate the permeability within particular sub-volumes of the samples and appreciate potential differences and anisotropy. However, this method can only be applied on post-mortem samples, and as such, under pressure and temperature conditions that are not representative of the subsurface. Overall, it was only through the combined application of both methods that we could attain a comprehensive understanding of permeability, a multifaceted physical property more accurately characterized as a tensor rather than a scalar.

Implications for supercritical geo-reservoirs

We saw that a greater degree of strain partitioning translates to a greater degree of permeability partitioning. In the localized, brittle regime, permeability of the sample was controlled almost uniquely by a sharp fault and it resulted that the apparent permeability of the sample remained low. On the other end of the spectrum, in our semi-ductile samples, since strain was partitioned between bulk and fault, permeability was controlled by the two and the apparent permeability of the sample increased during deformation.

This transition in how strain and permeability is distributed macroscopically forms the fundamental distinction between brittle and semi-ductile reservoirs.

In shallower, cooler reservoirs, it is anticipated that macroscopic, localized faults will predominantly occur within a generally intact, less permeable bulk rock. Such features must be the focal points of drilling in order to achieve economically viable permeabilities (Fig.  3 C). However, this pursuit comes with the potential risk of reactivating faults and inducing potentially dangerous seismic activity.

Conversely, in ductile reservoirs, both macroscopic fractures and microscopic cracks within the bulk could be effectively utilized to establish a permeable system, especially considering the very low viscosity of supercritical water. Specifically, we illustrated that the permeability of macroscopic deformation bands remains comparable to their brittle counterparts. Furthermore, we show that due to the partitioning of strain towards the bulk, the permeability of the rock surrounding these deformation bands can be significantly enhanced. This suggests that heat exchange processes could occur at greater distances from the targeted fractures and penetrate into larger volumes of the reservoir rock (Fig.  3 C). Similarly, the presence of a dense array of micro fractures could help mitigate the risk of creating shortcuts between injection and production wells.

Another way to extrapolate our results to the reservoir scale would be the equivalent permeability approach 33 whereby the equivalent permeability k e of a rock mass containing flow parallel permeable fractures is given by:

where k bulk and k frac are the permeability of the bulk rock and fracture respectively, w bulk and w frac are the width of the host rock and total fracture width respectively and W is the total width ( w bulk  +  w frac ). In this context, for a brittle, localized reservoir with k bulk at least an order of magnitude lower than k frac , the most economical way to improve k e would be to target as many fractures as possible (i.e., increase w frac ). Conversely, in a semi-ductile reservoir where k bulk is an order of magnitude greater than in a brittle reservoir and comparable to k frac , the equivalent permeability would be controlled in equal parts by the volume of bulk rock and the number of fractures crossing the borehole. If a borehole was extended to intersect more fractures (i.e., increase w frac ) it would naturally result that a greater volume of bulk rock would be sampled (i.e., increase w bulk ) resulting in a sizeable increase in k e .

While our study demonstrates that permeability can be created in strained semi-ductile samples, the up-scaling of such processes to the field scale remains unclear. It has been proposed that permeability in supercritical reservoirs can be stimulated hydraulically 24 and the exploitation of underground resources could create significant stress changes, but how these will compare to our results is yet to be determined.

Similarly, the time dependent nature of our results is yet to be investigated. In our experiments, the damaging of the samples was the result of a transient process and how the increased permeability would evolve over time is unaccounted for. This aspect is crucial for the long-term exploitation of supercritical reservoirs and has thus to be studied in the future.

Overall, the exploitation of supercritical fluids at depth has yet to overcome many challenges. In this context, our study only offers an initial perspective on their feasibility and questions the previously prevailing notion that the BDT marks a cutoff limit for hydrothermal circulation in the crust 21 .

Mechanical testing at varied temperature

Our investigation focuses on Lanhélin granite, a coarse-grained granodiorite sourced from Brittany, France, valued for its homogeneous and isotropic texture, low porosity (<1%), and large average grain size of 2 mm 34 . This rock is of particular interest due to its characteristics representative of deep geothermal reservoir lithologies.

Cylindrical samples with a diameter of 20 mm and height of 42 mm were cored out of a single block of Lanhélin granite. The specimens were rectified to remove irregularities and ensure parallelism of the loading surfaces (±100  μ m). Prior to testing, the samples were oven-dried at 60 °C for a minimum of 24 h. Each specimen was then jacketed in two layers of copper. The first layer, a thin foil (0.025 mm thick), serves to enhance contact between the specimen and the jacket and prevent any pore fluid flow between sample and jacket. The second layer consists of a thicker copper jacket (0.2 mm) that effectively isolates the sample from the confining medium.

The experimental apparatus, named TARGET, is a gas medium deformation apparatus based on the Paterson-type design, and recently installed at the EPFL Laboratory of Experimental Rock Mechanics, Switzerland 35 . TARGET is capable of withstanding confining pressures up to 400 MPa and temperatures up to 1000 °C. Confining pressure ( P c ) is applied by an external servo-controlled intensifier. Upon starting an experiment, P c was first increased to 100 MPa. Subsequently, the pore pressure system was flushed with argon to remove oxygen before the pore pressure ( P p ) was adjusted to its 50 MPa target value through two external low-volume (70 mL) syringe pumps. Once P p was stable, P c was increased further to its target value of 150 MPa, resulting in an effective pressure ( P eff ) of 100 MPa. Both confining and pore pressure were applied using argon gas, an inert gas which allow us to isolate the mechanical effects of deformation on permeability by minimizing fluid-rock chemical interactions during the experiments. Once the effective pressure was stable (≈10 min) thermal loading was gradually increased at a rate of ~25 °C/min via an internal furnace. Axial loading was applied last, once sample temperature was stable (≈30 min), by the mean of a hydraulic piston. All experimental conditions are gathered in Table  2 .

Upon reaching the target temperature and achieving thermal stability, pore pressure oscillations (5 MPa amplitude around the nominal pore pressure with 20 min period) were applied to the upstream side of the sample and consistently maintained throughout the entire experiment, while recording the pressure response on the downstream side. The harmonic calculation method was then employed to infer permeability during the experiments 36 . For each data point, a moving window of three oscillations was used to compute the permeability, with storativity being neglected. Furthermore, before initiating axial loading, an equilibration period of three oscillations was observed to minimize the influence of the initial pressure transient 36 , 37 .

Axial force was applied to the sample using a hydraulic piston driving the loading column. The axial load was recorded by two load cells, one external and one internal. Shortening of the sample was measured using two linear variable differential transformers (LVDTs) and then converted to strain by normalizing it with the initial sample length. Loading was carried out at a nominal strain rate of 10 −6  s −1 until either catastrophic failure occurred or the jacket punctured under extreme straining. The experiments lasted on average 15 h.

After the experiments, the samples were meticulously retrieved within their jackets to preserve the microstructure.

Image acquisition

X-ray tomography is a non-invasive material characterization technique that enables the quantification of micro-scale properties and mechanisms.

All X-ray scans were conducted at the European Synchrotron Radiation Facility beamline BM18. Samples were scanned at an average energy of 130 KeV with a propagation distance of 2 meters. Scans were acquired on an IRSI 15 teledyne camera with a 5056 × 2960 4.25  μ m resolution sensor. The field of view was 25.50 × 7.78 mm under half acquisition with a rotation axis off centered of 2200 pixels (87% of the sensor width) and a resolution of 2.63  μ m pixel size. The exposure time was 0.075 sec with a signal accumulation of 2 leading to a total exposure time of 0.15s (latency time included). 15000 projections were acquired over 360 degrees. Each specimen was scanned through a vertical series of six subsequent stages. The x-ray synchrotron white beam was filtered by 1.3 of Molybdenum and 20 mm of SiO 2 .

3D X-ray tomography images were constructed from the radiographs using in-house built software. The resulting images have a resolution of around 2.6  μ m 3 /voxel.

Image processing

X-ray images were processed to isolate the crack network and analyze permeability.

The resulting images were processed in ImageJ (Windows OS version 5.2) and Matlab (R2017b). Phase segmentation to isolate cracks was processed as follows.

Starting from the raw image (Fig.  4 A, B), a median filter (radius = 3 pixels) was applied to reduce the image noise. The ImageJ Mexican Hat Filter was applied with a radius of 5 pixels (Fig.  4 C). This inverted Gaussian-Laplace filters smoothed the data and amplifies gray-scale peaks depending on the data gradient. The image is then partitioned into a foreground (crack) and a background (Fig.  4 D).

A Raw, grayscale image. B Focused visualization with gray-scale adjustment showing intense cracks. C Outcomes of the Mexican Hat Filter. D Thresholding after applying the Mexican Hat Filter.

Further processing was conducted using Matlab. After trimming off-specimen areas, the artifacts resulting from the image-segmentation stage were removed using the Matlab Bwareaopen function, which removes objects smaller than a given volume. Our cut-off object size was 500 voxels (Fig.  4 D). Objects are recognized based on voxel connectivity, defined through the following criteria.

Voxels are connected if their faces, edges, or corners are in touch. Furthermore, two adjoining pixels are part of the same object if they are both on and are connected in (1) either in, out, left, right, up, and/or down direction, (2) a combination of two directions, such as a right-down or in-up, (3) combination of three directions, such as in-right-up or in-left-down.

Absolute permeability analysis

Absolute permeability simulations in x , y , and z directions were carried out using the Avizo XLab extension. The software computes the absolute permeability (hereafter referred to as permeability) from Darcy’s law 38 :

where μ is the dynamic fluid viscosity, L is the flow length, Δ P is the differential pressure applied on the specimen ends in the flow direction, and A is the area perpendicular to the flow direction.

Q is the volumetric flow rate and is defined based on finite element velocity computations of the Stokes flow equations 38 :

where u is the fluid velocity vector, p is the fluid pressure. The finite element mesh is generated based on the X-ray image voxel grid. The method has been validated in several geomechanics studies 39 , 40 . Other methods for computing permeability based on X-ray images are present, and a recent study 41 comparing predictions based on these methods suggests that all methods have a reasonable agreement with experimental measurements (the error is <20%).

Data availability

The mechanical data supporting the findings of this study are available from the Zenodo repository at https://doi.org/10.5281/zenodo.13319443 . The reconstructed 3D volumes are available on the ESRF repository at https://data.esrf.fr/doi/10.15151/ESRF-DC-1859045611 .

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Acknowledgements

The authors wish to extend our gratitude to Paul Tafforeau for his assistance during image acquisition at the ESRF. This project received fundings from the European Research Council Starting Grant under Project No. 757290-BEFINE and from the Swiss National Science Foundation (SNF) under Project No. 200021_204943. G.S. has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 101034260. G.S. is also funded by the Swiss Federal Office of Energy (SFOE) as part of the EPFLRock4CCS project. This project was also funded by Quaise and Alta Rock Energy. All X-ray data were acquired at ESRF, beamline BM18. The authors are solely responsible for the conclusions and findings.

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G.M. and M.V. designed the study and the experiments. G.M. conducted the experiments and processed the mechanical and permeability data. B.C. produced the XCT images. G.S. conducted the permeability modeling. All authors contributed to the redaction of the manuscript.

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Meyer, G.G., Shahin, G., Cordonnier, B. et al. Permeability partitioning through the brittle-to-ductile transition and its implications for supercritical geothermal reservoirs. Nat Commun 15 , 7753 (2024). https://doi.org/10.1038/s41467-024-52092-0

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DOI : https://doi.org/10.1038/s41467-024-52092-0

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