A historical review on porous‐media research

At the end of the 18th century, serious problems in dyke constructions in Northern Germany and the need to understand coupled solid‐water problems initiated first attempts to describe porous media. Many attempts followed until a sound Theory of Porous Media (TPM) was born on the basis of continuum mechanics of multi‐component materials with multi‐physical properties. The present article roughly describes the development of the TPM from its origins to contemporary applications, thus presenting a short historical review of porous‐media research.

F I G U R E 1 Detail of a railroad dam, shear zone development (left) as a result of a heavy rainfall event (right).
To set an example of a geomechanically based porous-media problem, consider a heavy rainfall event acting on a railway dam, compare of Figure 1 (right).The water content varies from red (empty) directly beneath a nearly impermeable layer and fully saturated (blue) on top of this layer and at the bottom of the displayed zone which is understood as the groundwater level.As a result of the rainfall, the water accumulates on top of an impermeable layer and leaks at the airside of the slope, while the soil is subjected to buoyancy forces.As is seen from Figure 1 (left), this leads to an accumulation of plastic soil deformations in a thin shearing zone with a shape of a double-curved shell embedded in the three-dimensional (3-d) soil.This example has been computed in parallel around 10 years ago by use of a combination of M++ 1 and PANDAS 2with around 11 million degrees of freedom (DOF) on 88 processors by a time consumption of more than 1070 h.As modern machines would be faster, computer power is also an important factor of the computation of complex problems and has therefore also been playing an important role in the history of porous-media research.Problems like that of Figure 1 can only be handled since the late 90th of the last century, although these problems have been observed in nature during all times.

THE EARLY DAYS
In its roots, the TPM dates back to the end of the 18th century, when Reinhard Woltman (*28 December 1757; † 20 April 1837, left portrait), the director of hydraulic engineering (Direktor der Strom-und Uferwerke und Leiter des gesamten Wasserbaus) of the city of Hamburg, discovered the concept of volume fractions as the ratio of the volumetric portions of the soil and the pore water compared to the volume of the overall dyke as significant components of any dyke construction [9].Based on this concept, compare Figure 2, he was also able to conclude to the partial densities of mud as a mixture of soil and water.
Comparable ideas have been presented by Achille Ernest Oscar Joseph Delesse (*3 February 1817; † 24 March 1881, right portrait).In his early career as a mining engineer, he had the problem to distinguish between the portions of the minerals in a mine.From a seam, he could observe the area fraction of the minerals, but was that equivalent to the volume fractions?
By intensive studies, Delesse [10] found out by statistical investigation of various slices of mineral conglomerates that area fractions and volume fractions are equivalent.In a modern setting, this leads to The concept of volume fractions and the weightiness of mud components, taken from Woltmann [9].stating that the volume fraction   is obtained by relating the local volume or the local area element of the th component to the overall volume element d or the overall area element d.After his period as a mining engineer, Delesse became a renown scientist, when he was appointed professor for geology and mineralogy at Besançon (1845-1850) and later as professor for geology at the Sorbonne in Paris (1850-1864).In 1864, he became professor for agriculture at the École des Mines, where he finally was appointed as the inspector-general of mines in 1878.In the same area of time, Henry Philibert Gaspard Darcy (*10 June 1803; † 3 January 1858) was working as a hydraulic engineer at the city of Dijon (1834-1840), where he became the chief engineer for the Cote d'Or in 1840.As a result of health problems, he asked for an early retirement in 1850.In 1856, Darcy [10] published an extended treatise on the public water supply for the city of Dijon, compare Figure 3.
In this treatise, he also included the results of various experiments that guided him to formulate his famous filter law yielding "Il paraît donc que, pour un sable de même nature, on peut admettre que le volume débité est proportionnel à la charge et en raison inverse de l'épaisseur de la couche traversé ".This led to the equation where grad ( ⋅ ) =  ( ⋅ )∕ with  as the location vector.Darcy's empirical equation given here in its basic one-dimensional (1-d) and in a modern three-dimensional (3-d) notation expresses the filter velocity   of the pore fluid as a linear function of the gradient of the hydraulic head h.While   is given in m/s, ℎ is measured in m and the hydraulic conductivity   in m 3 /(m 2 s), often reduced to m/s, respectively.From a modern point of view, Darcy's law can be found by combination of a constitutive equation for the direct momentum production term and the momentum balance of the liquid component included in a binary system of solid and fluid.Furthermore, Darcy's law only proves to hold in case of quasi-static situations, when a creeping hydraulic flow motivates a neglect of acceleration terms and when, furthermore, the frictional fluid stresses can be neglected by arguments of dimensional analysis in comparison with the friction included in the constitutive assumption for the momentum production, compare [11].However, Darcy's law is widely used today in hydraulic engineering as a given constitutive equation without the consideration of its domain of validity.

THE PERIOD OF GEOMECHANICS
At the beginning of the 20th century, geotechnical constructions had to be considered that needed a precise analysis of soil systems composed of soil particles and pore water.Especially, dam-construction and foundation problems such as the computation of settlements of tall buildings had to be tackled.In the framework of these engineering-dominated problems, geotechnical experts like Terzaghi and Biot came into play.Karl von Terzaghi (Karl Anton Terzaghi Edler von Pontenuovo) (*2 October 1883; † 15 October 1963, left portrait) was a civil engineer with a deep interest in geotechnical problems.In 1912, he visited various dam construction sites in the USA.Obviously, he was aware of the complexity of soil as a binary medium of solid grains and water, but he was completely unaware of any theoretical description of porous-media problems.Therefore, he tried himself to solve the problem, when he published on soil mechanics, for example, in his book "Erdbaumechanik auf bodenphysikalischer Grundlage" [13] of 1925, where he firstly discovered the principle of effective stresses yielding +  with  = 1, 2, 3 for the three dimensions in space. (3) Therein,   are the main principle stresses (pressure positive) acting in the whole fully saturated medium of porous solid and pore liquid, are the effective stresses acting only in the solid, while u is the pore pressure acting everywhere.However, Terzaghi was not a mathematician nor was he very much interested in struggling with theories such that he did not find continuum-mechanically based approaches satisfying the standard that has earlier been found, for example, by Stefan and Jaumann.As an engineer, Terzaghi always tried to combine theory and practice.
Thus, it was not astonishing that Terzaghi's work led to oppositions.Especially, when Terzaghi was a professor for hydraulic engineering at the Technical University of Vienna (1929)(1930)(1931)(1932)(1933)(1934)(1935)(1936)(1937)(1938), Paul Fillunger (*25 June 1883; † 7 March 1937, right portrait), who was a professor for applied mechanics at the same university since 1923, became his major scientific opponent.In 1913, Fillunger published a first paper on buoyancy forces in gravity dams [14].With this work, he presented a masterpiece, when he considered the problem as a binary medium of two interacting continua, soil and water.From this point of view, Fillunger can be regarded as the pioneer of the modern TPM, a conclusion that has been drawn by de Boer in his book on porous media [4].However, and this is the tragedy of Fillunger's work, his buoyancy equation included a mistake by presenting the buoyancy force as a linear function of the difference between the volume and the surface porosity, compare [4].This mistake was recognized by Terzaghi.Fillunger tried to justify his result, the onset of a severe conflict between the opponents that has been carried out both personally and scientifically.When both worked on capillary forces in porous media, it came to a final conflict.In his book "The engineer and the scandal" [15], de Boer portrayed the whole story that ended with Fillunger's suicide in 1937, after a scientific commission of the Technical University of Vienna came to the conclusion that Fillunger was wrong.
Fillunger's and Terzaghi's basic work on geotechnically based porous media has been continued by a variety of scientists, in the early days by the Belgium Maurice Anthony Biot (*25 May 1905; † 12 September 1985) and by the Austrian Gerhard Heinrich (*18 April 1902, † 18 March 1983) who later also published with Kurt Desoyer.While Biot, who worked as a research associate with Theodore von Kármán at Caltech, was a follower of Terzaghi, Heinrich exclusively used the ideas of Fillunger when he started to publish on the settlement of clay layers in 1938 [16].This separation of the porousmedia society is, by the way, still present.While the procedure of Terzaghi is, from a modern point of view, more or less unsatisfactory, Fillunger's approach is still modern, because he started with the balance equations of two overlaying components, soil and water, and treated this aggregate in the sense of a mixture with immiscible but interacting components and not, like Terzaghi, from an intuitive basis as a conglomerate of solid and fluid.Biot followed Terzaghi's basic ideas in his first papers of 1935 and 1941 [17,18] and published intuitively based treatises.Nevertheless, Biot's work is still highly cited and in use whenever young researchers are looking for basic material with the goal to solve porous-media problems.Unfortunately, Fillunger's work got nearly lost.Also Heinrich and Desoyer's work including the first numerical solution of the 3-d consolidation problem [19] vanished in the scientific jungle.

THE MODERN ERA
The modern era began with the recovery of continuum mechanics at the beginning of the 1950th, when the US-American scientist Clifford Ambrose Truesdell III (*18 February 1919; † 14 January 2000) entered the stage.
Truesdell's work originated the modern view on continuum mechanics and thermodynamics [20,21] including mixture theories.However, in Truesdell and Toupin's description of mixtures, there was no consideration of a balance of angular momentum for the mixture components [20], Section 215.Furthermore, an entropy inequality was also missing, although the entropy principle of Clausius was part of the description of standard single-phasic materials for a long time, compare [20], Sections 245-258.Without raising their hypothesis to a principle, Truesdell and Toupin surmised that the entropy inequality of heterogeneous media would basically be the same as that of a single-component medium.This assumption, however, turned out to be wrong.In the 1960th, the continuum-mechanical mixture theory has been brought to a certain perfection including a sound angular-momentum balance for the individual components of the mixture and the conclusion that an entropy inequality had to be formulated for the whole mixture instead as for each constituent separately, compare Bowen [6].However, some confusion was left.Based on Truedell's three metaphysical principles [5], page 221, the angular-momentum balance of the individual components allowed non-symmetric Cauchy stresses, although the overall Cauchy stress had to be symmetric.This would allow couple stresses at the component level that would have to vanish by summing them up.On the other hand, consider a liquid-saturated porous solid, where a homogenisation over the solid and fluid microstructures yields the component balance equations of mixture theories.If both the solid and the fluid are Cauchy continua (without couple stresses) on their microstructures, the homogenisation will result in partial continua with symmetric stresses, as the mean of symmetric tensors cannot produce non-symmetricity.

POROUS-MEDIA RESEARCH TODAY
Nowadays and as it has been said before, the porous-media society is still split into the Biot-theory-based and into the TPMbased groups.However, on the basis of the rigorous TPM approach, the following numerical examples have been generated by use of PANDAS.The first example stems from geomechanical engineering and exhibits a hydraulic fracturing problem carried out by use of the phase-field approach to fracture (phase field   ) for a ternary model of a brittle porous solid saturated by pore water and gas, compare [22].While the solid and the pore liquid are considered materially incompressible meaning that their intrinsic densities do not vary under constant temperature, the gas follows the ideal gas law. Figure 4 displays the numerical results as a reaction of pressing a fracking fluid from the left into the notch.
The second example stems from medical biomechanics and represents a chemotherapeutic treatment of a brain tumour that has its origin in metastatic lung-cancer cells that have been able to circumvent the blood-brain barrier (BBB), compare [23].For the surgery, four infusion doses have been planned by use of 2.5 × 10 −4 mol∕m3 of TRAIL 3 with a time interval of 5 days in between.Figure 5 characterises the result.
Prior to the first treatment, the tumour in black has got a critical size, such that a surgery has been found necessary.For the treatment, a convection-enhanced method has been chosen, compare Bobo et al. [24], where therapeutic agents enter the brain via an infusion needle that has to be drilled into the skull, thus bypassing the BBB that typically would hinder the passage of therapeutics with high molecular weights.At day 1 of the surgery, the first treatment is applied, and the tumour shrinks heavily (figures from left to right).At day 6, the tumour has regrown but exhibits after the second treatment a nearly full apoptosis.Five days later at day 11, the tumour has regrown again.However, the third surgery is successful to a certain extend meaning that the tumour cannot be seen afterwards.Nevertheless, a final treatment at day 16 is planned.After this treatment, the tumour seems to have undergone a full apoptosis, as nothing of the tumour can be detected anymore (not displayed).However, the right figure of the forth treatment (bottom right) shows the situation 15 days after the final treatment, where the tumour unfortunately started again to regrow.
The above computational results reflect the clinical reality of cancer therapy in a certain sense.However, the surgeon has finally to decide, whether or not a further treatment is advisable.In reality, this means that the question has to be answered against the background of the problem, whether the patient can be rescued or injured by further treatments.

CONCLUSION
The present article does not only exhibit the difficult way of finding sound conclusions on how to describe complex coupled problems of porous solids that are saturated by liquids and/or gases, such as the consolidation problem in porous-media research.It also elucidates the rapid growth of knowledge in the field of porous-media research during the last decades.

F I G U R E 4
First line: development of the fracture from a notch (left) via a vertical precrack (middle) towards a wing-like crack; second and third lines: streamlines of the pore liquid and the pore gas.

F I G U R E 5
First treatment (top left), second treatment (top right), third treatment (bottom left), fourth treatment (bottom right).
This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project Number 327154368 -SFB 1313.Open access funding enabled and organized by Projekt DEAL.O R C I D Wolfgang Ehlers https://orcid.org/0009-0008-1107-5151R E F E R E N C E S