Least-action variational continuum description of Darcian permeation of liquids and gases through stone materials

Watertightness of natural or artificial stone materials, like trap rock or concrete, is a basic property for assessing durability of the built heritage to weathering exposure. Fundamental research of the past century has shown that a reliable experimental determination of intrinsic permeability in stone materials can be achieved by uniaxial water-permeation tests under open-flow conditions, together with the opportunity of employing air permeation tests in place of water-tests to obtain important additional information on a given stone material. Correlations of practical relevance in such laboratory applications, among inlet pressure, outlet pressure and flow rate, stem all from solutions to the general continuum modelling problem of the permeation through a non-deformable homogeneous filter of an incompressible or compressible fluid under stationary uniaxial conditions. The present study addresses such class of mechanical problems on a theoretical variational basis to infer general correlations of practical relevance in permeability testing with a specific consideration of applications in which air-permeability testing combined with use of thinner filters makes the hypothesis of isothermal process in the permeating gas not straightforwardly applicable.


Introduction
The unusual complexity of the problem arising from the determination of stone permeability to liquids and gases was early pointed out by Madgwick [16]. The Author remarked that several of the main difficulties are originated by the wide variation in the constitution and structure of different stone materials and by the many physical and chemical properties of a given stone involved in permeation.
Notwithstanding the reliability of Darcy's linear equation when the flow is laminar, in the formulation of stone permeability modelling, with validation of theoretical predictions against experimental data, it is well known that taking proper account of secondary phenomena, other than Darcian flow, possibly affecting permeation during test run time, is essential. For instance, in concrete and concrete-like materials the most important of these secondary phenomena is known to be the decrease of permeability to water during the test, due to the possible presence of reactive substances, like unhydrated cement fractions in concretes [18,35]. An opposite phenomenon of permeability gain can be instead originated by leaching, if significant fractions of soluble components, like gypsum [17,33] or other soluble admixtures [9], are present in the stone matrix, as well as by other possible associated phenomena like swelling and osmosis.
Fundamental research of the past century has shown that in many natural and artificial stones in which the effect of the above mentioned secondary phenomena is negligible -or can be safely bounded -like standard Portland cement pastes which have reached complete maturation and in which swelling and soluble components are controlled by standard limits [22,23], a reliable experimental determination of intrinsic permeability can be achieved by uniaxial water-permeation tests under open-flow conditions [11,16,18,22,25,26]. The possibilities offered by air permeation tests in place of water-tests to obtain important additional information on a given stone material have been shown early on by Madgwick [16]. A review with a historical and technical retrospective of lessons learned from the past can be found in a recent publication by Monaco et al. [19], as well as in a study by Serpieri and Monaco [28], where the convenience, for the sake of minimal invasiveness and preservation of the historical concrete built heritage [4], of testing samples with low thickness extracted from the concrete cover is shown, and more recent provisions, like EN 12390 [10], are critically discussed.
For the laboratory characterization of a given stone permeability by uniaxial flow tests, correlations of practical relevance, among inlet pressure, outlet pressure and flow rate stem all from solutions of the general continuum modelling problem relative to the permeation through a non-deformable homogeneous filter of an incompressible or compressible fluid under stationary uniaxial conditions [16,20].
From the theoretical standpoint, a fairly general statement of the problem of uniaxial permeation of gases through porous stone materials is known to require a thermomechanical description. While for thicker filters an isothermal process can be generally taken for granted [1,20], for thinner stone filters of higher permeability the approach to an isentropic process in the gas cannot be excluded on principle.
The present study addresses such problem of uniaxial permeation of liquids and gases through inert stone filters in general terms on a theoretical variational basis, with the main objective of inferring correlations of general practical relevance in permeability testing. A specific consideration is taken, in particular, of applications in which air-permeability testing combined with use of thinner filters makes the hypothesis of isothermal process in the permeating gas not necessarily satisfied.
The choice of a theory suitable for describing permeation is not straightforward. While fundamental studies by Darcy belong to the middle of the nineteenth century [7], and quite comprehensive treatises dedicated to the continuum description of flow of homogeneous fluids through porous media were already available in the first half of the past century [20], further fundamental advancements have been contributed by Biot in the middle of the past century (see, e.g. [2,3]). However, still quite recently, different authors have emphasized the necessity of tackling the problem of a consistent general physical foundation to the poroelasticity theory in continuum mechanics [8,15]. The interested reader can find in a book by Coussy [6] a formulation emphasizing the importance of the concept of representative elementary volume and can find a retrospective on multiphase continuum theory of poroelasticity, with a specific focus on variational theories, in the first chapter of the book by Serpieri and Travascio [30].
Herein the resort to a macroscopic variational continuum theory is motivated by the capability of variational approaches to provide from the single unitary least-Action principle most general classes of conservative models, thus reducing, compared to other proposed statements of interacting continua (e.g. [12,34]), the number of introduced postulates. As this study will show, the description of intrinsically nonconservative phenomena, like Darcy's friction, can be built off, as a second step, from the conservative variational model adding minimal heuristic positions.
On the basis of canonical variational arguments and proceeding from the consideration of a minimum possible number of kinematic descriptors, Serpieri and Travascio [30] have derived a continuum mechanics theory of permeation of a fluid phase in a porous medium which can be sufficiently general to address the case of a compressible fluid. Such theory, which had been previously proposed in a less mature form by Serpieri et al. [31] and by Travascio et al. [32], is purely-mechanical, purely variational and purely macroscopic (in that it does not require a detailed knowledge or detailed assumptions on the small-scale features of the porous medium and of the solid-fluid interaction). In this study, to describe gas permeation, Serpieri and Travascio's theory is accordingly recalled and combined with the theory of ideal gases.
The document is organized as follows: in Sect. 2, a concise review of the permeability of concrete and stones to incompressible fluids is reported, whereas in Sect. 3 a general variational framework for uniaxial permeation is discussed, developing the general analytical solution to the thermomechanical problem and special solutions for isothermal and adiabatic processes. In Sect. 4, considerations are drawn from the preceding sections about the role of the temperature and a brief examination is reported on the relevance of the developed solutions for the interpretation of air-permeation tests in stone specimens. Conclusions follow in Sect. 5.
A final note concerning the notation is given. Subscripts and superscripts s and f are reserved to specify whether a given mechanical quantity is associated with the solid or the fluid phases. Lower-case and upper-case indices are associated to current and reference configurations, respectively. Bold symbols indicate in implicit format, if lowercase, vector quantities and tensor ones if uppercase. According to the adopted notational conventions, for instance, the deformation gradient of the fluid phase is indicated, in intrinsic notation, as F (f) while its components are denoted both in the format F (f ) iK and in the format The notation indicates the element of spatial index i and reference index K belonging to the 3 × 3 inverse matrix of the deformation gradient matrix of the fluid.

Permeability to water and incompressible fluids
As reviewed in Muskat's treatise [20], the permeation of an incompressible liquid through a cylindrical filter made of a macroscopically homogeneous porous medium saturated by a Newtonian fluidunder the usual hypotheses of stationary conditions, macroscopically uniform, uniaxial and nonturbulent flow-is well described by Darcy's linear relation [7] reading: where Q is the volume of fluid passing through the stone filter, A and L are the cross-sectional area and the length of the filter, and Δh = h 1 − h 2 is the drop of fluid head from the inlet value h 1 , to the outlet value h 2 . Since in the applications at hand, concerned with permeability of natural or artificial concretelike stone, Reynold's number is sufficiently below a threshold that can be safely set to a unit value, laminar viscous flow conditions are given for granted [20] and K P is a proportionality constant (in [m/s]) which is characteristic of the porous medium and also depends on the fluid properties. For incompressible liquids, the pressure gradient and the velocity are uniform through the filter length and relation (1) can be conveniently expressed in the form corresponding to: where v f is the seepage velocity, i.e., the volume of fluid entering an ideal square meter of a control cross section of the porous medium in the filter per second. In Eq. (2) K Pi is the intrinsic permeability, having the dimension of an area (in [m 2 ]). Such quantity is, in first reasonable approximation, a constant independent of the nature of the liquid and determined only by the structure of the porous medium, while p 1 and p 2 are the inlet and outlet fluid pressures, respectively, and is the viscosity of the permeating fluid. Denoting by ̂f and g the fluid mass density and the gravitational acceleration, respectively, so that Δp =̂f gΔh , the relation between the two permeability constants is:

A general variational framework
The governing PDEs of the variational two-phase continuum theory of permeation of Serpieri and Travascio [30] are hereby recalled and specialized to uniaxial permeation through an undeformable filter, employing the original notations therein and, for the representation of Eulerian balances, the notation used, among others, by Serpieri and Rosati [29]. In brief, the hat notation for the fluid density, ̂f , is reserved to denote the true fluid density (i.e., the density averaged inside the pore void space) while a bar notation, f , indicates the macroscopic fluid density (i.e., the average of the fluid density taken over the entire RVE domain), where the well-known relation f = f̂f holds. The notation v f , already used in scalar form in Eq. (2), is employed in the following to indicate the vector seepage velocity, defined componentwise as above. In analogy with the hat and bar notation for ̂f and f , symbol v f indicates the vector quantity v f = v f ∕ f , subsidiary to v f and corresponding to the ideal velocity that fluid particles would have under the assumptions of pore space fully interconnected and uniform velocity field within the pores.
The problem of a non-deformable filter permeated by a fluid under stationary conditions is considerably simpler since the momentum balance of the solid phase can be ruled out, being trivially satisfied by the identity placement function. Herein, such resulting simplified theory is for brevity directly addressed. Proceeding from the consideration of a macroscopic uniaxial fluid flow in the filter, scalar descriptors can be employed for the velocities v f and v f , where the relation v f = fvf holds. The reference and spatial coordinates are denoted by X and x , respectively, while x = x f (X, t) is the placement function of the fluid phase.
Concerning the kinematics of two generally deforming continua under finite deformations, under the employment of a minimal set of kinematic descriptors related to volume fraction fields for both the solid and the fluid phases and for both the reference Lagrangian setting ( s0 (X) , f 0 (X) ) and the spatial Eulerian setting ( s (X) , f (X) ), a consistent definition of volumetric balances capable of describing non-isochoric deformations in both the solid phase and the fluid phase requires two volumetric strain descriptors for the solid phase ( J s , Ĵ s ) and two for the fluid phase ( J f , Ĵ f ) [30].
The macroscopic volumetric strains J s = det x s ∕ X and J f = det x f ∕ X are defined as the Jacobian of their respective deformation gradients, while the intrinsic volumetric strains ( Ĵ s , Ĵ f ) describe in the general case the occurrence of non-isochoric deformations in the solid phase (when Ĵ s ≠ 1 ) and in the fluid phase (when Ĵ f ≠ 1 ) or permit to introduce, under the conditions Ĵ s = 1 and/or Ĵ f = 1 , possible constraints of isochoric deformations.
The relations between volumetric fractions and volumetric strains are: For the much simpler problem at hand of 1-dimensional stone filter where the solid is assumed undeformable and homogeneous, the conditions J s (X) = 1 , Ĵ s (X) = 1 and s0 (X) = s0 hold. The hypothesis of complete void space saturation is introduced, corresponding to coincidence of void and fluid volume fractions, v = f . Complete space saturation implies that in both reference and spatial settings the two constraints must further hold: The first of Eq. (3.1), under the condition for undeformable solids J s (X) =Ĵ s (X) = 1 , implies that all along the filter s x s (X) = s0 (X) = s0 , so that from (3.2) it also results that all through the filter , porosity is uniform in both space and time. This last property makes the second of Eq. (3.1) trivially simplified to the coincidence J f =Ĵ f .
Conservation of the fluid mass implies the respect of the two algebraic equations: where f 0 and ̂f 0 are the macroscopic fluid density and the true fluid density in the reference configuration, respectively. Also, seeking a minimal equation and considering that the weight of the fluid contained in the filter is small, the specific weight of the fluid is neglected in the momentum balance equation. The Lagrangian form for stationarity of Action with respect to translational equilibrium of the fluid phase, which has been derived on account of the least-Action Principle [30], reads under the current simplifying hypotheses: The newly introduced symbols entering (5) denote the following quantities: • b fs drag volume force exerted by the solid phase over the fluid one; • � Π f scalar stress measure of the fluid phase standardly defined in terms of work conjugation as: where ̂f 0 is the strain energy density of the fluid. Although the stationarity condition in Eq. (5) is not a balance, in a strict sense, it is hereafter referred to as linear momentum balance of the fluid phase. The relation of � Π f with the ordinary macroscopic fluid pressure p is recovered recalling that the Cauchy stress tensor of the fluid phase (f ) ij = −p ij ( ij being Kronecker's delta), which is a spatial stress measure assumed spherical in the continuum model underlying balance (5), is related to the first Piola-Kirchhoff stress measure, iK , by the standard push-forward transformation: where F (f ) iK are the components of the two-point deformation gradient of the fluid F (f ) . The strain energy function which returns a spherical stress tensor is iK must account, in the general case, for the saturation constraint (3.2). However, under the current very special condition of porosity uniform in both time and space, as discussed above, the condition of maximum simplicity J f =Ĵ f is inferred from (3.1) and (3.2), and function as the internal function in the energy function ̂f 0 .
Accordingly, the expression giving P (f ) iK is: Recalling definition (6) and the property that the derivative of the determinant of a deformation with respect to the deformation gradient is then expression (8) turns out to be equal to: Substitution of the last equation in (7) returns: By comparing (11) with the spatial tensor repre- The Eulerian spatial form of the linear momentum balance of the fluid phase can be obtained from the Lagrangian form (5) by standardly applying the Piola transform to convert the reference gradient ∕ X into a spatial gradient ∕ x . Application of the Piola transform for scalar functions to the gradient in the first term on the left-hand side (LHS) of (5) yields: Concerning the drag interaction term b fs in (5), a heuristic position has to be necessarily invoked at this stage for extending the variational formulation-which is intrinsically conservative-in order to make it suitably embrace the dissipative behavior of Darcy's viscous interaction. The representation of b fs recovering for incompressible fluids Darcy's equation (that in the current problem is directly written for a non-deformable solid filter) is: The resulting linear momentum balance of the fluid phase in the Eulerian setting is obtained by substitution of (13) and (14) into the LHS of (5), and by expressing the right-hand side (RHS) as the convective derivative of the velocity field Upon simplifying the common factor J f f over all terms of the resulting expression, the linear momentum balance of the fluid phase in the Eulerian setting is written: where the last equality trivially stems from the condition of stationary flow.
The Eulerian form of the mass balance (4) is obtained by standard push-forward transforms of (4) and by use of the convective derivative of the velocity fields (see, e.g., among many, Serpieri and Rosati [29]). As well known, such balance reads in the Eulerian setting: The set of equations governing the purely mechanical stationary problem of 1D permeation is composed by: • the stationary form of Eq. (16) constituting the fluid mass balance in the Eulerian representation as specialized by the condition of complete space saturation:  (15): A second heuristic amendment has to be introduced at this stage into the variational formulation to (14) address the fundamental intrinsically-dissipative constitutive behavior of gases described by the empirical laws by Boyle, Charles and Gay-Lussac, resulting into a full thermomechanical coupled description. In the stationary thermomechanical statement of the 1D gas permeation problem, the set of purely mechanical governing equations in the bullet list above has to be supplemented by: • the ideal gas law pV = nRT , with R the universal gas constant, T the absolute temperature, n the number of moles and V the volume of a control amount of gas at thermodynamic equilibrium; • the first principle of thermodynamics ΔE = q − w , with ΔE , q and w being the variation of internal energy, the absorbed heat and the work done by a control volume of gas permeating the void space of the solid medium.
The employed representation of the ideal gas law is obtained, as usual, attributing local validity of such law in a continuum sense. Accordingly, both sides of the pV = nRT equation are multiplied by the ratio M∕V (with M being the molar mass of the gas), obtaining pM =̂f RT since, by definition, ̂f = nM∕V . Finally, substitution of the Lagrangian mass fluid balance ̂f =̂f 0 ∕Ĵ f , yields p = ̂f 0 RT ∕(MĴ f ) , an equation which extends the set of purely mechanical governing equations as a constitutive constraint having the form of an algebraic field equation.
A similar elaboration into the continuum mechanics format is carried out for the first principle of thermodynamics. In rate form, the change of internal energy can be written as Ė = nC vṪ = ̂f 0 ∕M C vṪ where C v is the constant-volume heat capacity. Similarly, the heat inflow by fluid conduction is written in the usual form as ̇q = − q x ∕ x = −q � x , with q x being the x component of the local heat flux density vector related to temperature by the conduction law q x = −k T T∕ x = −k T T � , where k T is the thermal conductivity. The power of the work spent towards the external environment, also treated as a local variable, is ẇ = ṗĴ f . The specializations of the rates of absolute temperature and fluid volumetric strain under stationary conditions, Ṫ = T � v f and ̇Ĵ f =Ĵ f �v f , combined with the last four equations summarized in the bullet list above gives the continuum mechanics format of the first principle which is presented, altogether with the full set of governing equations, in the following equations (17) A more compact form of the system of equations (17)-(21) above, desirable for approaching its integration, is obtained by substitution of (17) into (19) and by solving for Ĵ f the algebraic Eqs. (17) and (18) to obtain Ĵ f = v f ∕v f 0 , a relation which, employed in turn into (21), yields (17) f v f =̂ f 0 v f 0 Complete space saturation + stationary Eulerian mass balance

Linear momentum balance under stationary conditions
Energy balance under stationary conditions Use of the last two explicit expressions in (20) provides: The last substitution allows one to remain with the purely differential system of (22) and (23) in the only two unknown functions p and v f , supplemented by boundary conditions at the inlet and outlet points x = 0 and x = L.

General solution to the permeation problem
A further considerable simplification of the problem (22)-(23) is achieved by recognizing the presence of negligible terms among the coefficients of this system. It can be quite easily verified that account of the typical ranges which the variables entering the coefficients in (22) experiment during an ordinary air-permeation test in a stone filter make the inertial term on the RHS of (22) negligible compared to the terms on its LHS. Actually, the values reported by Madgwick [16] for experiments on building stones (see Table I therein) correspond to air velocities below 0.05 cm/s even in the fastest experiment lasting less than 100 s. Consideration of standard ranges for the remaining air parameters (characteristic values of air viscosity and density are, as well known, = air ≅ 1.81 × 10 −5 [Pa s], ̂f 0 =̂a ir ≅ 1.225 [kg m −3 ], and as a reference value of atmospheric pressure it can be taken p 0 ≅ 10 5 [Pa]) and considering that an upper bound to concrete and stone permeability within ordinary building applications can be safely selected in K Pi = 10 −15 [m 2 ], the coefficient on the RHS of (22) is at least 9 and 15 orders of magnitude lower than its first and second coefficients on the LHS, respectively.
Accordingly, by neglecting the RHS of (22), the resulting equation yields which substituted into (23) yields one single thirdorder differential equation in the only unknown p: To approach integration of Eq. (25) the magnitude of the coefficient K Pîf 0 R ∕ k T M in (25) is evaluated to check its negligibility. Such coefficient has the dimension of the reciprocal of pressure and is accordingly indicated as 1∕p * . Its magnitude is computed based on the data already recalled above for K Pi , , ̂f 0 , and assuming for the other variables k T = 0.025 W/(mK), M = 0.029 kg/mol: Since 7.76 × 10 −7 is small compared to unity, the first term on the RHS of (25) is neglected and the equation to be solved becomes: A first integration of (27) provides Equation (28), falls within the general Bernoulli form upon setting P(x) = 0 , Q(x) = C 1 + C 2 x and n = −1.
The solution to (28) is promptly computed by introducing the auxiliary function z = p −n+1 = p 2 such that z � = 2pp � whereby the equation pp � = C 1 + C 2 x can be written as z � = 2C 1 + 2C 2 x , thus yielding z = z 0 + 2C 1 x + C 2 x 2 with z 0 integration constant. Accordingly, and from (24) Boundary conditions allow to express z 0 , C 1 and C 2 as function of the inlet and outlet quantities p 1 , p 2 , v f 1 by solution of the system: which is Accordingly, the general pressure field solution to (28) is from (30) while the velocity field from (31) is

From (35) one computes the following relation between velocities
where v inc. is the velocity corresponding to Darcy's Eq. (2) for an incompressible fluid. Equation (36) can be also cast into the more expressive form: from which it can be seen that, if separate measurements provide the same inlet and outlet flows v f 1 = v f 2 , then the continuum model predicts that such velocity must be Darcy's velocity for the incompressible case: v f 1 = v f 2 = v inc. .

Isothermal process
The solution under the hypothesis of isothermal process in the gas during permeation is hereby briefly (37) reviewed. Such solution is considered by many authors a sufficiently reliable description of how pressure distributes along the filter, and thus of practical relevance for laboratory measurement of gas flow in stone materials. In Muskat's treatise [20] it is simply shown that, under the hypothesis that a gas expands isothermally behaving as an ideal gas, the pressure field is such that dp 2 ∕dx is constant. Other authors, like Klinkenberg [14], assume a hypothesis of isothermal process albeit without an explicitly statement, but adopting instead subsidiary hypotheses. Among them, in particular, Madgwick [16], while analyzing both theoretically and experimentally air flow through a tile-like stone specimens, introduces the hypothesis that the product pv f is constant along the filter length, notably without mentioning isothermal conditions. As it will be shown below, such constancy condition can be shown to be equivalent to a hypothesis of isothermal process. Nevertheless, it should be remarked that in his air permeation experiments on Withbed Stone an amount of more than half liter of air flows through a 2.54 cm thick specimen in approximately 1 min and half with actuating pressure corresponding to water heads of less than 150 cm. Such details suggest that a hypothesis of isothermal expansion, while reasonably applicable for most test set-ups with stone materials of sufficiently low permeability, should be more carefully examined when the filter is thin and intrinsic permeability is not sufficiently low or the pressure head is higher.
When the hypothesis of purely isothermal process, T = T 0 , is introduced in system (17)- (21), it can be readily recognized that the chain of inferences holds: pĴ f = const , p∕̂f = const , pv f = const , pp � = const . Indeed, setting T = T 0 in (21) gives: so that from (18) and (38) one infers Indicating by p 1 = p(x = 0) and p 2 = p(x = L) the pressures at the inlet and outlet, respectively, and by ̂f 1 and ̂f 2 the corresponding densities, and taking the density at the inlet as the reference density ̂f 0 ∶=̂f 1 , from (39) one obtains Also, from (17) one infers v f = v f 0̂f 0 ∕̂f and so that, owing to (39) and (40) and setting v f 0 ⋅ v f 1 , the further constancy condition holds: Account of (24) and (42) yields: Equation (43), pp � = −C , is again of Bernoulli form since it can be represented as in (29) with P(x) = 0 , Q(x) = −C and n = −1 . The solution is similar to the previous one for the general case and can be obtained again by introducing the auxiliary function z = p −n+1 = p 2 such that z � = 2pp � whereby the equation pp � = −C can be written as z � = −2C , thus yielding z = z 0 − 2Cx with z 0 another integration constant. Accordingly, where the constant z 0 is recognized to be z 0 = p 2 1 from the boundary condition p(x = 0) = p 1 .
The velocity field, which is computable by (24) or (42), is: Formulas of practical convenience in permeability experiments are those providing v f 1 or v f 2 as explicit expressions of p 1 and p 2 . The one providing v f 1 is promptly obtained from the outlet boundary condition stemming from (44) which yields in turn The corresponding expression for v f 2 under isothermal conditions is obtained from the relation implied by (42) The most general representation (37) encompasses (48) which is actually recovered when the constraint The reliability of (47) in providing a coefficient K Pi independent from the pressures p 1 and p 2 has been experimentally assessed by Madgwick [16] by measuring v f 1 from the time required by a fixed air volume to flow through a tile-like stone specimen. It should be however emphasized that Eq. (44) is not a solution to (23). Actually, substitution of pp � = − ∕K Pi p 1 v f 1 into (23) would imply that the quantity: should be equal to zero. While this is not, in general, analytically true, however such zero condition is approximately met in air-permeability testing applications over stone specimens since, as we have shown in (26) above, 1∕p * is a quantity smaller than one millionth. It should be also considered that a more accurate modelling of the thermomechanical problem might have also addressed heat exchanges between the solid and the fluid (beyond the scope of this study) which are certainly important given the high specific surface in stones. (49)

Adiabatic process
While the hypothesis of an adiabatic process in the gas is certainly unrealistic for a thick filter of low intrinsic permeability, an isentropic process might be approached in the gas when the filter is very thin and the material has higher intrinsic permeability. By thinning the stone filter, a significantly different condition can be approached, characteristic of other thin-filter applications, like the passage of gas through a textile fabric or a wire screen (see, e.g. [24]), in which dissipative terms are clearly much lower than those corresponding to a several centimeters-thick stone filter.
In an adiabatic process the variation of internal energy ΔE is only determined by the work w spent by the system as the heat absorbed is zero, viz., dE = −dw . Since for 1 mol of gas dE = C v dT and dw = pdV = RTdV∕V = RTdĴ f ∕Ĵ f (the last equality stems from the identity dV∕V = dV what yields upon integration, as well known for adiabatic processes, to: The second equation in (50) provides, by integration of (12) the expression of the potential energy function which is admitted by the gas when this undergoes an adiabatic process: Let us incidentally observe that in the adiabatic limit the temperature can be expressed as the following univocal function T Ĵ f of the volumetric strain: (50) Owing to the second of (4) and to (17), and setting = C p ∕C v = R + C v ∕C v , with C p being the constant-pressure capacity, one further has: Equation (54) returns for the velocity: so that its space derivative is: Substitution of (55) and (56) in (22) returns a differential equation which, written in the dimensionless variable y = p∕p 0 , reads: The magnitude of the coefficients corresponding to a typical permeability experiment with air can be estimated again on the basis of characteristic data and bounds relevant to typical air permeation tests, already recalled in the examination of the magnitudes of coefficients entering Eq. (25). By considering again the characteristic values and bounds employed for the magnitudes of the coefficients in (22), the order of magnitude of coefficients in (57) Integration of (58) gives (58) Also, from (61) and (55) one computes the velocity field in the adiabatic case:

Role of temperature on concrete permeability
It is natural to inquiry the relation between the general solution to the thermomechanical problem, provided by (34)-(37), and the special solutions above recalled for the isothermal and adiabatic cases. The generality of (37) has been already pointed out showing that it encompasses both the solution for the incompressible fluid and the solution for the flow of a gas undergoing an isothermal process. Comparison between (61) and (34) shows instead that the adiabatic pressure field does not fall within representation (34), so that it is neither a solution to the general system (17)- (21), nor an approximate solution to Eq. (27) in presence of experimental data admitting the underlying condition 1∕p * ≅ 0.
Finally, from (35) or (71) one computes the sought relation expressing the outlet velocity as function of boundary pressures and temperatures: which recovers, as expected, the special isothermal relation when T 1 = T 2 .

Application to air-permeation tests on stones
This subsection illustrates the relevance of solution (72) in elucidating the role of temperature for the interpretation of results from air-permeation tests on stone specimens, exemplified using data from the fundamental experiments carried out by Madgwick [16] according to the basic testing methodology fine-tuned by the same Author. In his seminal study, Madgwick enforced, over a 2.54 cm thick tile-like specimen of Whitbed Stone placed in a pressure cell, a stationary unidirectional open-flow permeation of air along the thickness. Using as actuator an elementary hydraulic system made of flexible tubes and water reservoirs, and applying in 5 different tests water heads ranging between 145 and 5 cm, he could accurately measure a value of resistivity ∕K Pi corresponding, for an air  Table 1 reports the values, in SI system, of the water head Δh and of the inlet and outlet pressures and seepage velocities of air as computed from Madgwick's data from barometric, manometric and chronometric measurements. As it can be seen, values p 1 almost correspond to the atmospheric pressure, up to some fluctuations relatable to barometric variations. Figure 1 illustrates the pressure profile, as computed from the isothermal Eq. (44), showing the slight deviation from linearity. It is worth being remarked that, although the deviation from a linear pressure profile might appear, visually speaking, not so important, Madgwick's data [16] show that account of Eq. (44) is necessary. Actually, if a most elementary Darcy model with linear pressure profile is employed without accounting for the isothermal one corresponding to the square root profile of Eq. (44), a much higher maximum deviation, of 7.5%, would be wrongly attributed in the repeatibility of the measurement of K Pi with different water heads. Figure 2 compares the profiles for all 5 experiments, vertically shifted so as to adjust p 1 to 0.1 MPa, set as reference inlet value to permit easier comparison.
Finally, Fig. 3 illustrates the dependency of v f 2 upon temperature as computed by straightforward application of Eq. (72) assuming T 1 = 20 °C and T 2 in the range from − 20 to 180 °C (dashed graph). For completeness and comparison, the temperaturedependence of air viscosity is also taken into account considering the linear interpolation T 2 = 10 −6 ⋅ 17.15 + 4.59 ⋅ T 2 ∕100 .
The dashdot line shows the graph that does not account for the temperature gradient-dependency of Eq. (72) and considers only the change of air viscosity with temperature. The solid line accounts for both temperature-dependency of Eq. (72) and the dependency of air viscosity upon temperature. As Fig. 3 shows, the effect of Eq. (72) deserves consideration since a non negligible deviation can be recognized as temperature is changed.

Discussion and conclusions
A consistent variational poroelastic continuum model combined with the ideal gas theory describing the permeation of compressible and incompressible fluids through inert stone filters has been presented.
Integration of differential equations yields general solutions which provide explicit formulas relating the outflow rate to inlet and outlet pressures, encompassing both the special cases of a gas undergoing an isothermal process and the case of an incompressible fluid. It is shown that the general solution does not include the pressure profile stemming from the hypothesis of adiabatic process in the gas.
Quite remarkably, the study shows that, along with the dependence on the pressure gradient, the outflow rate is also dependent on the temperature gradient. Since such dependency stems from the very basic characteristics of the considered continuum thermomechanical model and not from heuristic hypotheses other than an ordinary account of the laws of Darcy and of ideal gases, it can be considered to be a physically objective feature of a solid filter permeated by a gas. To the best of the authors' knowledge such temperature-gradient-dependency has not been reported neither in dedicated classic treatises [5,20,27] nor in very recent studies on gas permeation in concrete materials [13,21]. The applicative relevance of the temperature-gradient-dependency pointed out in this study should deserve attention from the community working in the standardization of permeability testing in stone materials.
Funding Open access funding provided by Università degli Studi della Campania Luigi Vanvitelli within the CRUI-CARE Agreement.

Conflict of interest
The authors declare that they have no conflict of interest.
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