EVAPORATION FRONT AND ITS MOTION

. The evaporation demands upon a rock or soil surface can exceed the ability of the profile to bring sufficient amount of liquid water. A dry surface layer arises in the porous medium that enables just water vapor flow to the surface. The interface between the dry and wet parts of the profile is known as the evaporation front. The paper gives the exact definition of the evaporation front and studies its motion. A set of differential equations governing the front motion in space is formulated. Making use of a set of measured and chosen values, a problem is formulated that 5 illustrates the obtained theory. The problem is solved numerically and the results are presented and discussed.

as containing liquid water and without liquid water, respectively. Such a part of the domain's boundary which is open to the atmosphere is considered as the dry contact. 60 Under these conditions, there necessarily exists a set of points inside the studied domain or upon its boundary that makes an interface between the wet medium (porous medium) and the dry medium (porous medium or air). In view of the above introduced terms, these points can be considered as points of the evaporation front.
Generally, the porous medium profile can be divided into three parts: (a) the dry zone, where just two phases, solid and gaseous (air), are present and water exists in the form of vapor as a component of the gaseous phase, (b) the wet zone, where the movable liquid water exists, and (c) the intermediate zone, where the liquid water is present but only in such a contact with the solid phase, that makes it unmovable. Here, such liquid phase water is understood as movable that moves due to the hydraulic head gradient.
The evaporation front does not exist in itself; it is a matter of definition. It seems natural to place the evaporation front to the intermediate zone or to an interface between the intermediate zone and one of the neighboring zones. It can be expected 70 that during the process of evaporation, the depth of the intermediate zone becomes small. The present water evaporates quickly due to its immobility, its small amount and contact with the solid phase. In view of this and of the fact that experimentally the evaporation front can be indicated as a sharp interface between two neighboring zones, we assume: the extend of the zone (c) can be neglected and the evaporation front is defined as the common boundary of the zone without liquid water and the zone with movable liquid water. The concept evidently enables existence of a jump in water content values. 75 We do not consider the temperature distribution and heat flow and balance; it is only supposed that the heat conditions : , ::::: since, :: in :::::: wirtue :: of :: its ::::::::: definition, ::: the :::::::::: evaporation ::::: front ::::: results ::::: from ::: the ::::: water :::::::: transport :::: data. ::::::: Though ::::::::: unknown, ::: the :::: heat :::: flow within the profile are sufficient to provide the amount of the ::::::: provides ::: the : latent heat of vaporization that is necessary for the evaporation resulting from the actual process of water transport.
In virtue of its definition, the ::: The : evaporation front changes its position with time according to the outer conditions. Its shape 80 and motion results from mutual relations (water transfer) between the wet zone and the dry zone. The front moves towards the wet region if the evaporation exceeds the flow of the liquid water towards the interface through the wet zone and vice versa.
Since the evaporation front inside porous media, e.g. in a rock massif, is difficult to detect, mathematical modelling :::::::: modeling becomes an important tool always if the knowledge of its position and motion is required.
In what follows, all the introduced characteristics are macroscale porous-medium characteristics; e.g. a domain is a macroscale 85 domain, a surface is a macroscale surface, etc.. Denote by Ω, Ω ⊂ R 3 , the domain in space, and by (0, T ) the time interval in which we study the transport process and suppose that the movable liquid-phase water occupies an open part G w of the time-space domain G (i.e. the water content θ is positive and sufficient to enable the water flow in G w ), where We further define :::::::::::: (2) in virtue of our assumptions, θ = 0 in G d ::: and ::::: θ > 0 :: in :::: G w , :::::: where : θ ::::::: denotes ::: the ::::: water :::::: content.
To any time t ∈ (0, T ) we define the wet zone Ω w t and the dry zone Ω d t by putting

It holds
100 We define the evaporation front γ t at time t ∈ (0, T ) as The image of the evaporation front in time-space is :::::::::::::::::::::::::::::: We assume that Γ is a smooth hypersurface in R 4 and denote by ν Γ ∈ R 4 the unit vector normal to Γ that points out of the wet part or into the dry part of G. Since we defined the wet and dry boundary B w t and B d t of Ω at time t with respect to the outer conditions, this orientation has sense everywhere on Γ. :::::::::: . Then the hypersurface Γ can be in a certain neighborhood of (t, ξ) expressed by a function τ in the form of equation :::::::::::::: Then ν Γ t < 0 implies the existence of a positive value τ : ϵ such that (t + ϑ, ξ) ∈ G w for ϑ ∈ (0, τ ) ::::::: ϑ ∈ (0, ϵ). Consequently, the evaporation front γ t moves at its point ξ towards the dry zone if ν Γ t is negative at (t, ξ) and vice versa.

135
The position of the evaporation front results from the mutual relations between the water transport in the wet zone and in the dry zone. Denote by : n :::: the ::::::: porosity, ::: by w the volumetric flux density of liquid water in the wet zone, by v d and v w the volumetric flux density of the gaseous phase in the dry zone and in the wet zone, and by b d and b w the water vapor flux density by diffusion in the gaseous phase within the dry zone and the wet zone. Let further c d and c w denote the water vapor concentration in the gaseous phase within the dry zone and the wet zone. We suppose that functions b d , c d , v d are continuous The evaporation front is not connected with a fixed set of mass points and the problem of its motion is not a problem of the particle tracking. The evaporation front moves in such a direction and with such a velocity that are given by the balance of mass of water. Since the tangential motion of the evaporation front at its point does not change the front's position, the evaporation front moves at each point in the direction of its normal.

Problem formulation
The front's motion reflects the proportions between the water flow and transport out of the front and towards the front which 190 are given by the laws of flow and transport in the wet zone and in the dry zone. In order to evaluate flow and transport in porous media, Darcy's law and Fick's law can be utilized where the third coordinate x 3 is oriented vertically upwards, h is the pressure head, k is the hydraulic conductivity, and D d and D w are the coefficients of water diffusion in air within the porous medium.

195
In the dry zone, water is present in the form of water vapor and its motion is governed by the continuity equation with the use of Fick's law: The vapor motion in the wet zone is governed by the same laws 200 and the motion of liquid phase water in the wet zone is governed by Richards' equation In virtue of the introduced theory, the evaporation front motion is governed by Eqs. (20), (21), (22) and (15) or (18). The unknown functions are θ and h, connected by the retention curve, c w , c d and ξ defined in G w , G d and [0, T ). The functions n, k, D, v and ρ are supposed to be known or given by additional equations.

205
In this way, Eqs. (21) and (22)  is then given as the position of the moving boundary.
Another possible formulation of the problem is to solve the ordinary differential equation (15) in the interval [0, T ), where the right-hand side of the equation is given as the solution of the problems (21), (22) and (20) defined in G w and G d .

210
The latter approach was utilized when solving the problem presented in the 5-th section.
In virtue of the introduced theory, the law of the evaporation front motion is given by Eq. (15). In one space dimension, since ν(t, ξ) is either 1 or -1, the equation reads

A solved problem
In the frame of a wider research that concerns evaporation from rock surfaces and the time dependence of evaporation front positions, an experiment was carried out. It was an unpublished auxiliary experiment performed by several of the author's colleagues and since it was not sufficiently documented from the point of view of this study, it cannot be simulated. Nevertheless, part of its results can be utilized here in order to present an example of possible use of the achieved theoretical results. The missing data were simply chosen, not optimized. The following description should be understood as a problem formulation not as a documentation of measurements.
A cylinder shaped sample of the studied rock was put to the position with horizontal axis. The jacket of the cylinder was insulated so that no water (of any phase) could penetrate and the motion of water through the sample was possible only in the horizontal direction along the cylinder's axis.

250
The length of the sample was L = 49 mm, and the length of the time interval was T = 63 days. One open end of the sample, say x = 0, was equipped so that it was possible to measure the pressure head and the rate of water inflow into the sample at this point. The obtained discrete data were approximated by smooth functions h 0 (t) and w 0 (t) (pressure head and volumetric flux density), h 0 , w 0 ∈ C 1 (0, T ), that were utilized as the imposed boundary conditions. Soil moisture retention data and the hydraulic conductivity (at saturation) K s were obtained elsewhere using samples similar 260 material. Making use of these characteristics, the Mualem and van Genuchten parameters θ r , θ s , α, m and n were determined and hence, the hydraulic conductivity and the soil moisture retention curve as functions of either pressure head or water content were defined.
Similarly the value of the diffusion coefficient of water in the gaseous phase within the porous medium, D d , was obtained measured on samples of the utilized material. The characteristic surface layer ε, was added from outside to the domain Ω, its 265 value was taken from the paper (Slavík et al., 2020), where the layer is referred to as "calibrated L" and also as "boundary layer".
The fluorescein visualization method, (Weiss et al., 2018), was utilized to detect the front's position during the experiment.
Fluorescein was applied at the water input side of the sample and a set of couples (t, ξ), time and the front's position, was registered. The data were calibrated (using a simple linear transformation) to agree with the value measured after finishing the 270 experiment.
The following problem was formulated. In the interval [0, T ], the solution t → ξ(t), ξ ∈ C 1 (0, T ) ∩ C[0, T ], of equation (28) is sought that satisfies the initial condition where ρ and n (density and porosity) are known constants.
Similar result has already been expected even for more general cases, see the estimations utilized when replacing Eq. (15) by Eq. (18). Making use of these results, Eq. (28) becomes where the unknown parameters at the right-hand side of (32) are solutions of the following two initial-boundary value problems and, since b w , c w and D w do not appear in what follows, b, c and D stand for b d , c d and D d .
To determine functions θ and w, function h defined in G w is sought that satisfies equation (22), now in the form 300 and the conditions in (0, L), ::::::::::::::::::::::::: and where θ(h), is the retention curve. Functions h 0 and w 0 are known from the experiment and function h i , the initial pressure head distribution, was not measured and has to be determined.

320
where c i is the initial distribution of the water concentration in the gaseous phase. The values c e and c f were chosen with respect to these requirements: the relative humidity at the front was 100%, the outer relative humidity was between 30% and 50% and the temperature between 18 • C and 23 • C.
In order to define the initial state of the sample, functions c i and h i , it was assumed: the process started from the steady state water vapor diffusion determined in (L, L + ε) by the boundary values c f and c e and from the steady state water flow 325 determined in (0, L) by the initial conditions h 0 (0) and w 0 (0). The error involved to functions c i and h i vanishes soon, since the sample is small and the imposed boundary conditions take over the dominant role.

340
The solution to the problem (42), (43) is equivalent to the solution to the problem ::::::::::::::::::::: with the initial condition h i (0) = h 0 (0). Let the maximum solution to this initial-value problem be defined in [0, λ). It can be 345 shown that, in our case, λ > L and the wet zone reaches the point x = L. Hence, our choice of the initial condition h i does not contradict the initial condition (29).
The problem (32), (29) was solved numerically using a predictor-corrector method. The values of the right-hand side were determined using the method of Rothe to solve problems (33) to (35) and (37) to (39).
The evaporation front has been defined as an interface separating two different zones, wet and dry, inside or upon the boundary of a porous medium domain. The exact definition of these zones, presented in this paper, is based on the form of water they 380 contain. Subsequently, the law of the evaporation front motion was formulated in the form of the vector equation (15). Since the law is based on the complete mass balance of water, i.e. liquid water and water vapor, it holds generally and does not need any additional account of energy. The laws of heat transfer and heat balance do not affect the presented equations which define the evaporation front motion. On the contrary, solving problems that are fully determined by water transport data, the equations of the evaporation front motion can give certain insight into the energy requirements of such processes, e.g. the final part of 385 section 5. Smits et al. (2011) andNuske et al. (2014) studied the process of evaporation from soils with the particular attention the phase change, and found that nonequilibrium models yield better agreement with experimental data than equilibrium models.
The nature of the phase change process does not affect the results presented here directly, since the equation of the evaporation front motion requires other kind of data. The process of phase change enters Eq. (15) through its actual effect on the transport 390 of water. On the other hand, the constitution laws like Darcy's law or the retention curve, that may be utilized when solving problems with Eq. (15), are equilibrium laws. In the example presented in section 5, equilibrium laws were utilized. However, the governing equation (15), being general, makes it possible to use nonequilibrium laws as well. Mls (1999) presented a general nonequilibrium approach to two-phase systems that keeps Darcy's law valid. Lehmann et al. (2008) and Or et al. (2013) investigated the process of evaporation from the top of an initially saturated 395 vertical column. They introduced the term characteristic length as the distance between the surface and the receding drying front (interface between the saturated zone and the unsaturated zone) and described different stages of the evaporation process.

415
The characteristic surface layer ε was found experimentally, see (Slavík et al., 2020), and accepted in this paper as a part of the measured data. Note that Song et al. (2018) studied similar problems and introduced also a special diffusion layer outside the porous medium. From the viewpoint of moving front equations, the characteristic surface layer prevents infinite value of function b d at x = 0 and t = 0 which may be obtained when solving a problem with equation (20). This possibility origins in the fact that the equation is a balance equation that contains an equilibrium law -the Fick law; for more on this problem and 420 an alternate approach see (Mls and Herrmann, 2011).
The process of evaporation alone does not determine the direction of the evaporation front motion. Since the denominator of the right-hand side of (15) is positive, the direction of the evaporation front motion is determined by the sign of the scalar product in the numerator. Consequently, both the processes of wetting or drying (increasing or decreasing the wet zone) can take place while evaporating water out of the profile; compare also Eqs. (32) and (48). The presented theory is now prepared to prove its reliability on such problems that are fully documented and to be used when solving a wide range of problems of evaporation from a rock or soil profile. Pressure head (cm) Figure 2. The x = 0 boundary condition -the pressure head at the boundary. The squares show the measured values, the smooth function is their approximation h0(t) that was used in the solved example. Volumetric flux density (cm/d) Figure 3. The x = 0 boundary condition -the volumetric flux density at the boundary. The step function shows the measured values, the smooth function is its approximation used in the solved example. Depth of the evaporation front (cm) (2) (3) (2) c f = 2.41 × 10 −5 , ce = 9.94 × 10 −6 , (3) c f = 2.56 × 10 −5 , ce = 1.06 × 10 −5 , all in g/cm 3 .