Shales are the most abundant class of sedimentary rocks, distinguished by being very fine-grained, clayey, and compressible. Their physical and chemical properties are important in widely different enterprises such as civil engineering, ceramics, and petroleum exploration. One characteristic, which is studied here, is a systematic reduction of porosity with depth of burial. This is due increases in grain-to-grain stress and temperature. Vertical stress in sediments is given by the overburden less the pore fluid pressure, σ, divided by the fraction of the horizontal area which is the supporting matrix, (1−φ), where φ is the porosity. It is proposed that the fractional reduction of this ratio, Λ, with time is given by the product of φ4m/3, (1−φ)4n/3, and one or more Arrhenius functions Aexp(−E/RT) with m and n close to 1. This proposal is tested for shale sections in six wells from around the world for which porosity-depth data are available. Good agreement is obtained above 30–40 ∘C and fractional porosities less than 0.5. Single activation energies for each well are obtained in the range 15–33 kJ/mole, close to the approximate pressure solution of quartz, 24 kJ/mol. Values of m and n are in the range 1 to 0.8, indicating nearly fractal water-wet pore-to-matrix interfaces at pressure solution locations. Results are independent of over- or under-pressure of pore water. This model attempts to explain shale compaction quantitatively. For the petoleum industry, given porosity-depth data for uneroded sections and accurate activation energy, E, paleo-geothermal-gradient can be inferred and from that organic maturity, indicating better drilling prospects.

Weathering and erosion transport minerals and organics toward the sea or lake bottoms over geologic time. The finest solids are deposited in lower, less turbulent areas, such as lake bottoms and continental shelves. They sometimes stack up to thicknesses of kilometers, and begin compacting. These sediment sections are called shales, and as initially deposited in water, shales can have porosities up to 50-80{\%} water, As they are buried, many alteration products from oil to slate are produced due to overburden and temperature increasess, making them important to study. A preceeding paper \cite{1} showed that pressure solution is the primary mechanism for porosity reduction, with possible mechanical compaction at shallow depths. Without naming the mineral(s) involved, it postulated that the greater the product of the water and pore interfaces, the faster the reaction would proceed. This term is $ \varphi^{4m/3}(1-\varphi)^{4n/3}$ , where $\varphi$ is porosity and m and n are numbers close to unity. The large exponents, {\em 4/3}, recognize that the reaction occurs at the molecular scale at which the surfaces are rough. A second term, $\exp^{(-E/RT)}$ , indicates that the reaction is impeded by a quantum energy barrier, E, with diminished impeding power as increased available thermal energy, represented by the absolute temperature, T, becomes available at greater depths in the Earth. These two factors combine to allow porosity $\varphi$ to reduce with time, or equivalently for the fraction of solids, $(1-\varphi)$, to increase with time,$\left. \frac{\partial(ln(1-\varphi) ) }{\partial t } \right |_{\sigma} = (\varphi )^{4m/3}( 1-\varphi )^{4n/3}Ae^{-E/(RT)}$. { \em Here it is shown that this equation can cover quantitatively any actual deposition rates which may have been experienced by the six sections studied, the actual deposition rates being unknown for these cases. Hence a time-depth depositional history for any new shale section, known in detail, would allow determination of the parameters m, n, E and a lumped proportionality constant A. This was accomplished by showing that, for a wide range of depositional rates, r, the range of E for any of the studied sections is small compared to laboratory measurements of quartz solution\cite{2}, 24+-15kJ/mol. Results were obtained over this range of r's using the previously determined m and n, and porosity and temperature profiles. The presently existing porosity profiles necessarily incorporate any overpressure or underpressure conditions that may have existed in the past or currently, as the net difference between overburden and pore pressure is a primary driving force for pressure solution. $SiO_2$ usually comprises 20-50{\%} of shales. In conclusion, pressure solution of quartz can account for vertical compaction of shales quantitatively in the studied examples. Separately, a quasi-eqivalent method for discussing reduced porosities in Macron 2 versus Macron 1 as increased vertical stress, rather than additional horizontal stress, is illustrated. Experiments are suggested to clarify the pressure solution mechanism. The role of horizontal forces is discussed.

The grain-to-grain stress vertically in sediments is given by the overburden less the pore fluid pressure, σ, divided by the fraction of the horizontal area which is the supporting matrix , (1 − φ), where φ is the porosity. It is proposed that the fractional reduction of this ratio, Λ, with time is given by the product of φ 4m/3 , (1 − φ) 4n/3 , and one or more Arrhenius functions A exp(−E/RT ) with m and n close to 1. This proposal is tested for shale sections in six wells from around the world for which porosity-depth data are available. Good agreement is obtained above 30-40 C and porosities less than 0.5. Single activation energies for each well are obtained in the range 15-33 kJ/mole, close to pressure solution of quartz, 24 kJ/mol. Values of m and n are in the range 1 to 0.8, indicating nearly fractal pore-matrix spaces and water-wet interfaces. Results are independent of over- or under-pressure of pore water. This model explains shale compaction quantitatively. Given porosity-depth data and accurate activation energy, E, one can infer paleo-geothermal-gradient and from that organic maturity, thus avoiding unnecessary drilling.

Over geologic time wind, rain and snow combine to disintegrate,dissolve or react with land and vegetation, and to move altered solids lower as required by gravitational forces. The finest solids are deposited in low, less turbulent areas, such as lake bottoms and continental shelves. They sometimes stack up to thicknesses of kilometers, and begin compacting. These sediment sections are called shales, and as initially deposited in water, shales can have porosities up to 50-80{\%} water, As they are buried, many alteration products from oil to slate are produced due to overburden and temperature increasess, making them important to study. Besides initial mechanical compaction, other mechanisms can contribute to reduction of porosity. A preceeding paper showed that an important process is pressure solution of some part of the shale minerals. Without naming the mineral(s) involved, it postulated that the greater the product of the water and pore interfaces, the faster the reaction would proceed. This term is $ \varphi^{4m/3}(1-\varphi)^{4n/3}$ , where $\varphi$ is porosity and m and n are numbers close to unity. The large exponents, 3/4, recognize that the reaction occurs at the molecular scale at which the surfaces are rough. A second term, $\exp^(-E/RT)$ , indicates that the reaction is impeded by a quantum energy barrier, E, with diminished impeding power as increased available thermal energy, represented by the absolute temperature, T, becomes available at greater depths in the Earth. These two factors combine to allow porosity $\varphi$ to reduce with time, or equivalently for the fraction of solids, $(1-\varphi)$, to increase with time,$\left. \frac{\partial(ln(1-\varphi) ) }{\partial t } \right |_{\sigma} = (\varphi )^{4m/3}( 1-\varphi )^{4n/3}Ae^{-E/(RT)}$. The current development recognizes that this equation provides snapshots, at all times during the burial, of the sediments at each depth. A time-depth history for a shale section, known in detail, would allow determination of the parameters m, n, E and a lumped proportionality constant A. Lacking this known history, a constant rate of solids deposition, r, can be assumed and these parameters can then be determined. This has been done here for six shale sections, and for a wide range of deposition rates. Satisfactory results were obtained over this range of r's using the previously determined m and n, and porosity and temperature profiles, presently varying only A and E. The present porosity profiles necessarily incorporate any overpressure or underpressure conditions that may have existed in the past or currently, as the net difference between overburden and pore pressure is a primary driving force for pressure solution. The derived activation energy E is close to that for pressure solution of $SiO_2$, which may comprise 20-50{\%} of shales.Experiments are suggested to clarify the pressure solution mechanism. The roles of horizontal forces and pH are discussed. An approximate method for re-casting horizontal forces as a vertical gravitational equivalent force is illustrated for the Macran 2 section.