Analogue modelling (geology)

In 1937 King Hubbert described the key principles for scaling analogue models. He defined three types of similarity between models and the natural world: geometric, kinematic and dynamic.[8][15]

To be geometrically similar, lengths in the model and natural example must be proportional and angles must be equal.[15] When the length of a natural prototype (p) is ${\displaystyle l_{n}^{p}}$ (n=1, 2, 3...) and the angle is ${\displaystyle \alpha _{n}^{p}}$. Correspondingly, the length in the model (m) is ${\displaystyle l_{n}^{m}}$ and the angle is ${\displaystyle \alpha _{n}^{m}}$. They need to conform to the following formulas:[1]

${\displaystyle {\frac {l_{1}^{m}}{l_{1}^{p}}}={\frac {l_{2}^{m}}{l_{2}^{p}}}={\frac {l_{3}^{m}}{l_{3}^{p}}}={\frac {l_{n}^{m}}{l_{n}^{p}}}}$ & ${\displaystyle \alpha _{n}^{m}=\alpha _{n}^{p}}$

For example, 1 centimetre in the model represents 1 kilometre in nature.

To be kinematically similar, they must be geometrically similar and the time needed for changes to occur must be proportional.[9] When the required time for changing is ${\displaystyle t_{n}}$:[1]

${\displaystyle {\frac {t_{1}^{m}}{t_{1}^{p}}}={\frac {t_{2}^{m}}{t_{2}^{p}}}={\frac {t_{3}^{m}}{t_{3}^{p}}}={\frac {t_{n}^{m}}{t_{n}^{p}}}}$

For example, 1 second in the model represents 1 thousand years in the nature.

As is known: ${\displaystyle v={\frac {l}{t}}}$, the velocities (${\displaystyle v}$) can be scaled by the following equation:[1]

${\displaystyle v^{p}=v^{m}{\frac {l^{p}t^{m}}{l^{m}t^{p}}}}$

When the models and the natural world are geometrically and kinematically similar, dynamic similarity additionally requires that the various forces acting on a point in the model are proportional to those at a corresponding point in nature.[15] When the forces (${\displaystyle F_{n}}$) acting on the system are ${\displaystyle F_{g}}$ (gravity), ${\displaystyle F_{v}}$ (viscous force), and ${\displaystyle F_{f}}$ (friction):[15]

${\displaystyle {\frac {F_{g}^{m}}{F_{g}^{p}}}={\frac {F_{v}^{m}}{F_{v}^{p}}}={\frac {F_{f}^{m}}{F_{f}^{p}}}={\frac {F_{n}^{m}}{F_{n}^{p}}}}$

However, since the forces acting in the nature are unmeasurable, it is impossible to scale the forces and stresses directly. Scientists have been using different formulas to convert forces into the parameters that can be measured. Cauchy momentum equation is usually used for showing the relationship between forces and densities (${\displaystyle \rho }$ is density):[1]

${\displaystyle {\frac {F^{m}}{F^{p}}}={\frac {\rho ^{m}(l^{m})^{3}}{\rho ^{p}(l^{p})^{3}}}}$(Generating from Cauchy momentum equation[16])

Stokes' law is usually used for showing the relationship between forces and density contrasts (${\displaystyle \Delta \rho }$ is density constant):[1]

${\displaystyle {\frac {F^{m}}{F^{p}}}={\frac {\Delta \rho ^{m}(l^{m})^{3}}{\Delta \rho ^{p}(l^{p})^{3}}}}$(Generating from Stokes' law[17])

(While the gravitational acceleration ${\displaystyle g^{m}=g^{p}}$)

Since the densities and density contrasts are proportional to forces and stresses, it is easy to scale densities or density contrasts instead of scaling forces and stresses.[1]

However, these two equations can lead to different topography scales.[1]

The simple analogue modelling of the fold-thrust belt system taken in the Nanjing University physical laboratory. The white and blue materials are quartz sands.

Different geodynamic processes are simulated by different experimental apparatus.

For example, lateral compression machines are commonly used in simulating deformations involving lithospheric shortening, such as folding,[2] thrust faulting, collision, and subduction. Longitudinal compression machines are usually used for fracturing.[18] There is a large variety of devices based on the different sources of forces applied to the material. Some devices have multiple forcing systems because nature is not homogeneous.[1]

For experimental systems, the energy can be supplied externally (at the boundary) and internally (buoyancy forces). If the deformation is only caused by internal forces, it is a closed system. Conversely, if the deformations are caused by external forces or a combination of internal and external forces, it is an open system.[1]

For the open system, the extrusion or stretching forces are imposed externally. However, the buoyancy forces can be generated both externally or internally. The materials and thermal energy can be added to or remove from the system. For the closed system, there is no energy and materials added to the system. Thus, all the deformations are caused by internal buoyancy forces. Only buoyancy-driven deformation can be simulated in a closed system.[1]

The simple analogue modelling of a subduction zone. The materials this model uses are sand mixture and silicone putty for the continental crust (left in layered brown) and oceanic crust (right in layered brown), and glucose syrup for the asthenosphere (greenish-blue liquid in the glass tank). There is a heater in the tank for heating the liquid.[2][19][20]

Because the major research object of analogue modelling is Earth, the gravity field that most experiments utilize is ordinarily the Earth's field of gravity. However, many models are carried out using simulated gravity, for example, by use of a centrifuge. These technologies are usually used in studying development of gravity-controlled structures, such as dome formation,[21] and diapirism.[1]

Analogue modelling uses various materials, such as sand, clay, silicone and paraffin wax.[2] Different materials came into use for quantitative analysis of analogue modelling experiments, compared to qualitative analysis.[22] Before Hubbert's scaling theory, scientists used natural materials (e.g. clays, soil, and sand) for analogue modelling.[1] For large-scale simulation, analogue modelling should have geometric, kinematic, and dynamic similarity with nature. If the model has these similarities, results from simulation will be more accurate.[8] All these different materials represent the natural features of Earth (such as crust, mantle, and river).[22] Selection of analogue materials is difficult, because of the largely rheology-dependent deformation and inconstant rheology influenced by the thermal gradient in nature. The rheological characteristic of internal layering was developed by the study of seismology and geochemistry.[1]

To simulate layers with different properties, different materials are chosen:

The first analogue model was built by James Hall for simulating folds. He used a lateral compression machine for the simulation, and this machine is still shown in the Royal Society of Edinburgh.[2] The final result from the model is quite close to observation of the Berwickshire coast.[2] Although the model he used is simpler than current ones, the idea remains in use.

The use of more complex compression machines substantially increases the number of simulations of compressional tectonics, including subduction, collision, lithospheric shortening, fracture formation, thrust and accretionary wedge. If the simulation only focuses on the upper crustal, the model is always built in the glass box (or two lateral glass walls) with a piston and/or wedges to supply forces to layers of granular materials (normally called sandbox). Depending on the different natural features, erosion (removal of top materials at a certain angle), décollement (inserted layers with low cohesion, normally glass microbeads), and any other parameters can put into the model, producing various results.[25]

Simulations of mantle influences vary. Because of the different physical and chemical properties between the asthenosphere and lithosphere, viscous materials and a heater (for mantle convection) are also used.[2]

The simple analogue modelling of the extension tectonics which showing the formation of normal fault and salt dome (diapirism). This model is built in a glass box. The darker greyish layer is silicone which represents salt, and brownish layers are dry quartz sands which represent the brittle sedimentary rocks.[13]

Compression machines can also be used in reverse for simulating extensional tectonics, such as lithospheric extension, the formation of rifts, normal faulting, boudinage and diapirs. These models can also be built in a glass box which is similar to the above, but instead of thrust force, tensile force is applied.[13]

The simplified analogue modelling setting of shear deformation. This model is built on two separate horizontal plates. The brownish layers are dry sand, wet clay, and viscous materials, such as silicone or polydimethylsiloxane.[26]

Strike-slip tectonics differ from the dominantly vertical crust movements associated with shortening and extension, being dominantly horizontal in character (in relative terms sinistral or dextral). This kind of horizontal movement will create a shear zone and several types of fractures and faults. A typical model used for strike-slip tectonics has two (or more) horizontal basal plates moving in opposite directions (or only move one of the plates, other are fixed). The visual results are shown from bird's-eye view. Scientists used CT-analysis to collect the cross-section images for the observation of the most influenced area during the simulation.[26]