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2D numerical and experimental study of the drying kinetics of Compressed earth blocks

Etienne Le Mire, Rudy Bui, Antonin Fabbri, Marie-Sarah Force

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Etienne Le Mire, Rudy Bui, Antonin Fabbri, Marie-Sarah Force. 2D numerical and experimental study of the drying kinetics of Compressed earth blocks. Construction and Building Materials, 2025, 490, 10.1016/j.conbuildmat.2025.141850. hal-05364598

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# 2D numerical and experimental study of the drying kinetics of Compressed earth blocks

Etienne Le Mire<sup>a, b, *</sup>, Rudy Bui<sup>b, d</sup>, Antonin Fabbri<sup>a, d</sup>, Marie-Sarah Force<sup>a, d</sup>

## ARTICLE INFO

### Keywords:

Compressed earth blocks

Drying

Hygrothermal model

Sensitivity analysis

## ABSTRACT

The drying kinetics of earthen materials has recently been experimentally studied, but no modelling of the phenomenon has been proposed yet. This article presents an analysis of 60 days of drying of two Compressed earth blocks, both numerically and experimentally. The experiment consists in the monitoring of the internal relative humidity of the blocks during their drying forced along one direction only. Numerically, the contribution is twofold: first, the hygrothermal model is numerically solved in 2D, using an Alternate Implicit Direction scheme. Then, its robustness is tested through an uncertainty and sensitivity analysis of 1500 simulations of 60 days of drying, with 7 uncertain parameters. When compared to the experimental data, the results show a good predictive power, especially after 20 days of drying. The sensitivity analysis' main conclusion is that boundary conditions, through the use of convective heat and mass transfer coefficients, are what drive the drying kinetics. Other studied material parameters do not significantly influence the kinetics in that specific period of the material's life. Finally, the use of heat and mass transfer analogies (e.g. Lewis) is, again, proven deficient, if not wrong.

## 1. Introduction

Despite being a very ancient construction material, an increasing interest is developing throughout the scientific community towards raw earth [1,2]. This is mainly due to its good hygrothermal, durability and renewable (when unstabilized) properties [3], even though they are yet to be fully understood [4]. In spite of these observations, there are still lots of barriers that prevent the diffusion of earthen materials in construction, among which one can cite technical issues in construction itself (e.g. lack of quality control), economical issues in design (e.g. lack of client demand) or in-use organizational difficulties (e.g. lack of coordination between stakeholders), as pointed out by Pelé-Peltier et al. [5]. In France, the government has launched a national project on earthen construction called PNTerre,<fnref n="1" /> whose goal is to bring together skilled people from all construction fields (masons, architects, engineers, researchers, etc.) in order to promote unstabilized raw earth construction in all its forms and diversity.

Practically, when earth is used for construction, it is moistened to a moisture content that varies according to the construction technique, but generally ranges from around 10% by mass for “dry” techniques such as Compressed earth blocks and rammed earth, to over 20% for wetter techniques such as light earth [6]. As the compressive

strength evolves drastically during drying, through its dependence on the water content [7–9], it is necessary to be able to predict the drying kinetics of earthen structures, so as to predict the evolution of the mechanical properties. Hygrothermal parameters such as hydraulic conductivity [10] or thermal conductivity [11] are also highly dependent on the water content. To that end, drying process simulation of earthen materials is a solution: it requires the derivation of a hygrothermal model, a numerical method of resolution, and experimental measures to validate the model. Hygrothermal models have already been derived for studying heat and moisture transfers in buildings [12–14], mainly for in-service uses and energy consumption calculations. On another hand, drying experiments on cob and light earth [15], on rammed earth [16] or on adobe samples [17] have been done. A recent research has worked on the prediction of the mechanical properties of stabilized and unstabilized compressed earth blocks with regards to three moisture contents [18], but to the knowledge of the authors, no study has, to date, focused on the prediction of the drying kinetics of raw earth by comparing experimental results with numerical predictions.

<footnote><sup>a</sup> ENTPE, École Centrale de Lyon, CNRS, LTDS, UMR 5513, 3 rue Maurice, Audin, Vaulx-en-Velin, 69120, France</footnote>

<footnote><sup>b</sup> Université Claude Bernard Lyon 1, CETHIL, UMR5008, CNRS, INSA, Lyon, Villeurbanne, F-69100, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: etienne.lemire@entpe.fr (E. Le Mire).</footnote>

<footnote n="1">¹ https://projet-national-terre.univ-gustave-eiffel.fr/</footnote>
Hence, this paper proposes a comparison between experimental drying results and numerical predictions computed by a 2D implementation of a hygrothermal model. Numerical simulations of the hygrothermal behaviour of a structure (say a wall) requires the knowledge of some material properties (e.g. density, thermal conductivity, etc.), which are usually measured at lab scale. Even though necessary, these measures have some drawbacks: they come with inevitable uncertainty (e.g. measurement accuracy), and they may not be able to be representative of the heterogeneity due to the change of scale [19]. This is why, in order to fully grasp the power of the model and its robustness to inputs unavoidable uncertainties, an uncertainty and sensitivity analyses was performed. Several studies were published using these methods, on gypsum boards [20], on spruce plywood [21], on hemp concrete on a purely thermal model [22], and on a hygrothermal model [23] among others. As performing sensitivity analysis one input parameter at a time can lead to non explored areas of the inputs space [24], another method relying on the analysis of the variance (ANOVA) can be employed, which has effectively been used on a moisture penetration depth model by Goffart et al. [25], for instance. The ANOVA based sensitivity method is recommended as it provides accurate results within acceptable ranges of computational time [26].

The purpose of this article is to present comparisons between drying experiments and numerical simulations of drying. Section 2 first presents the mathematical hygrothermal model and specifies the boundary conditions used, then introduces the uncertainty and sensitivity analysis method. Section 3 presents the case study of this paper: it consists in the measurements of internal relative humidity of Compressed earth blocks during 60 days. The specimen characteristics as well as the experiment parameters are detailed, and the numerical resolution method is briefly presented. Section 4 shows the results of the sensitivity analysis, along with the experimental data, and Section 5 discusses those results and proposes future developments.

## 2. Hygrothermal model and sensitivity analysis

### 2.1. Governing equations

The hygrothermal models consists of the coupling of two equations: a water mass balance and a heat balance. The model used in this study has already been derived and used in its 1D version in several publications [27–29], hence its derivation will not be detailed here. The principal hypotheses on which it is based are recalled hereafter:

* The only water phases considered are the water vapour and the liquid water. The presence of ice is not considered.

* Among the several transport mechanisms that drive the total water transport, only two are taken into account:

- for the water vapour, only the diffusion (Fick's law) is considered

- for the liquid water, only the capillarity due the presence of a pressure gradient is considered.

The model consists of two coupled equations that link the two driving potentials: the capillary pressure $P_c$ and the temperature $T$. Multiple driving potentials can be used to govern the mass transport. The capillary pressure was chosen based on Bui et al. [28] study. The gravity terms are neglected. The mass balance equation then reads

$$ \frac{\partial w}{\partial P_c} \frac{\partial P_c}{\partial t} = -\bar{\nabla} \left\{ \left[ \delta_l + \delta_p \frac{\rho_v}{\rho_l} \right] \bar{\nabla} P_c - \delta_p \left[ -\frac{P_v}{T} \ln \phi + \phi \frac{dP_v^{\text{sat}}}{dT} \right] \bar{\nabla} T \right\}, \quad (1) $$

where $w$ (in kg m⁻³) is the gravimetric water content, and $t$ (in s) is the time. It is assumed that the amount of water in the material is directly related to the capillary pressure, regardless of the temperature. This negligible impact of temperature, within the ranges encountered in building science, has notably been demonstrated for earthen plasters by Fabbri et al. [30]. $\delta_l$ (in kg m⁻¹ s⁻¹ Pa⁻¹) is the liquid water

permeability, $\delta_p$ (in kg m⁻¹ s⁻¹ Pa⁻¹) is the water vapour permeability, $\rho_l$ (in kg m⁻³) and $\rho_v$ (in kg m⁻³) are the liquid water and water vapour densities, respectively. $P_v$ (in Pa) is the vapour pressure, $P_v^{\text{sat}}$ (in Pa) is the saturation vapour pressure, and $\bar{\nabla}$ is the nabla operator. $\phi = P_v / P_v^{\text{sat}}$ (no unit) is the relative humidity. For the sake of clarity, RH (for Relative Humidity) is sometimes used instead of $\phi$ in the following sections of the paper. The heat transfer equation reads

$$ \begin{aligned} (\rho_d c_{p,\text{mat}} + w c_{p,l}) \frac{\partial T}{\partial t} &= \bar{\nabla} \left\{ \left[ k + (c_{p,v} \bar{T} - L_v) \delta_p \left( -\frac{P_v}{T} \ln \phi + \phi \frac{dP_v^{\text{sat}}}{dT} \right) \right] \right. \\ & \qquad \left. \bar{\nabla} T + \left( \delta_l c_{p,l} T - (c_{p,v} \bar{T} - L_v) \delta_p \frac{\rho_v}{\rho_l} \right) \bar{\nabla} P_c \right\}, \end{aligned} \quad (2) $$

where $\rho_d$ (in kg m⁻³) and $c_{p,\text{mat}}$ (in J kg⁻¹ K⁻¹) are the material dry density and mass heat capacity, respectively. $\bar{T}$ is the temperature expressed in degrees Celsius, $c_{p,v} = 1850$ J kg⁻¹ K⁻¹ is the heat capacity of water vapour, $c_{p,l}$ (in J kg⁻¹ K⁻¹) is the water heat capacity, $k$ (in W m⁻¹ K⁻¹) is the material thermal conductivity and $L_v = 2500$ kJ kg⁻¹ is the latent heat of evaporation. It should be noted that Eq. (2) relies on rather strong assumptions, which are not always fully satisfied, namely the conservation of water flux (both liquid and vapour) and the homogeneity of the enthalpy of phase change. However, this formulation was chosen because it facilitates, from a numerical perspective, the incorporation of the latent heat term as well as the heat advection terms associated with vapour and water transport within the medium, without fundamentally altering the overall trends obtained, which is sufficient given the objectives of this paper.

The relationship between capillary pressure $P_c$ and water content $w$ is given by the Van Genuchten model [31], which must be fitted on experimental data. The model has the following mathematical form

$$ w(P_c) = \frac{w_{\text{sat}}}{\left(1 + \left(\alpha h(P_c)\right)^n\right)^m}, \quad (3) $$

where $\alpha$ (in m⁻¹), $m$ and $n$ are fitting parameters and $h(P_c) = \frac{P_c}{\rho_l g}$ (in m) is called the pressure head. $w_{\text{sat}}$ (in kg m⁻³) is called the saturated water content, and is the maximum theoretical amount of liquid water the material can contain. It is in practice not a directly measurable parameter, in particular for earthen material, for which achieving a fully saturated state while maintaining the same pore structure is an unrealistic expectation. Therefore, as it is recommended by Van Genuchten et al. [32], $w_{\text{sat}}$ should rather be considered as a parameter to be fitted – while staying within an acceptable range of values – than a parameter to be measured independently. Note that Eq. (3) assumes the absence of residual liquid water. An expression for the liquid permeability is also proposed by Van Genuchten [31]. It depends on the amount of water, and reads

$$ \delta_l(w) = K_0 \kappa_l(w) \frac{\rho_l}{\eta_l}, \quad (4) $$

with the liquid relative permeability $\kappa_l$ expressed as

$$ \kappa_l(w) = \left( \frac{w}{w_{\text{sat}}} \right)^{\frac{1}{2}} \left[ 1 - \left( 1 - \left( \frac{w}{w_{\text{sat}}} \right)^{\frac{1}{m}} \right)^{m} \right]^{\frac{1}{2}}. \quad (5) $$

$m$ is the same coefficient as in Eq. (3), $K_0$ is the intrinsic permeability, and $\eta_l = 10^{-3}$ Pa s is the dynamic viscosity of liquid water at 20 °C.

### 2.2. Boundary conditions

The boundary conditions for the simulation are chosen to be representative of the experimental setup (see Section 3). On the bottom (resp. top) edge, i.e. at $y = 0$ (resp. $y = H$), a zero flux Neumann boundary condition is applied to both the temperature $T$ and the capillary pressure $P_c$. Mathematically, it reads

$$ \left\{ \begin{array}{l} \frac{\partial T}{\partial y}(x, 0) = 0, \quad \forall x \in [0, L] \\ \frac{\partial T}{\partial y}(x, H) = 0, \quad \forall x \in [0, L], \end{array} \right. \quad (6) $$
$$ \left\{ \begin{array}{l} \frac{\partial P_c}{\partial y}(x, 0) = 0, \quad \forall x \in [0, L] \\ \frac{\partial P_c}{\partial y}(x, H) = 0, \quad \forall x \in [0, L]. \end{array} \right. \qquad (7) $$

On the left (resp. right) edge, i.e. at $x = 0$ (resp. $x = L$), a convective flux is applied to both the temperature $T$ and the capillary pressure $P_c$. The boundary conditions for the capillary pressure are expressed in terms of water vapour pressure $P_v$ as

$$ \left\{ \begin{array}{l} g_{\text{conv}}(0, y) = h_m(P_v(0, y) - P_{v,\text{ext}}), \quad \forall y \in [0, H] \\ g_{\text{conv}}(L, y) = h_m(P_v(L, y) - P_{v,\text{ext}}), \quad \forall y \in [0, H], \end{array} \right. \qquad (8) $$

where $g_{\text{conv}}$ (in kg m$^{-2}$ s$^{-1}$) is the mass flux. $h_m$ (in kg m$^{-2}$ s$^{-1}$ Pa$^{-1}$) is called the convective mass transfer coefficient (CMTC). In the present study, the CMTC $h_m$ has been directly related to $h$ through the so-called Lewis analogy

$$ h_m = 7.7 \ 10^{-9} h, \qquad (9) $$

which has also been used in Bui et al. [28] for instance. The relationship between $P_c$ and $P_v$ is given by the combination of the Gibbs and Kelvin laws as follows

$$ P_c = -\rho_l R_v T \log(\phi), \qquad (10) $$

where $R_v = 461.5$ J kg$^{-1}$ K$^{-1}$ is the ideal gas constant of water vapour. The boundary conditions for the temperature read

$$ \left\{ \begin{array}{l} q_{\text{conv}}(0, y) = h(T(0, y) - T_{\text{ext}}) + (c_{p,v}\bar{T}(0, y) + L_v) g_{\text{conv}}, \quad \forall y \in [0, H] \\ q_{\text{conv}}(L, y) = h(T(L, y) - T_{\text{ext}}) + (c_{p,v}\bar{T}(0, y) + L_v) g_{\text{conv}}, \quad \forall y \in [0, H], \end{array} \right. \qquad (11) $$

where $q_{\text{conv}}$ (in W m$^{-2}$) is the heat flux, $h$ (in W m$^{-2}$ K$^{-1}$) is called the convective heat transfer coefficient (CHTC). The term $(c_{p,v}\bar{T}(0, y) + L_v) g_{\text{conv}}$ is the latent contribution due to water exchanges at the surface of the material [33].

## 2.3. Uncertainty and sensitivity analysis method

Uncertainty and sensitivity analysis (hereafter called SA for the sake of simplification) for mathematical modelling has several objectives, among which the understanding of the behaviour of the model, and the prioritization of the uncertainty sources to reduce the model output uncertainty [34,35]. It consists in four main steps:

1. Definition of the problem (inputs, mathematical model, variables of interest);

2. Quantification of the uncertainty sources (measurement accuracy, missing data, statistics, etc.);

3. Propagation of the uncertainty of the inputs to the outputs through the model;

4. Identification of the most influential inputs on the outputs of interest.

Let us consider a model $f$, that computes, for a set of inputs $X = (x_1, x_2, ..., x_p)$, the output $Y = f(X)$. Practically, the SA method consists in sampling $N$ sets of inputs $X$, and to decompose $Y$ as follows [36,37]

$$ Y = f(x_1, \dots, x_p) = f_0 + \sum_{i=1}^{N} f_i(x_i) + \sum_{i=1}^{N} \sum_{j=i+1}^{N} f_{ij}(x_i, x_j) + \dots + f_{1, \dots, p}(x_1, \dots, x_p), \qquad (12) $$

where, if the inputs are independent and each term is chosen with zero average and is square integrable, then $f_0$ is the mean value and the decomposition is unique. $f_i$ is only a function of $x_i$, $f_{ij}$ is only a function of $(x_i, x_j)$, etc. This means that the independence of the inputs has to

be verified before going through any interpretation of the results of the SA. The decomposition in Eq. (12) allows to also decompose the total variance $V$ as a sum, as follows

$$ V = \sum_{i=1}^{N} V_i + \sum_{i=1}^{N} \sum_{j=i+1}^{N} V_{ij} + \dots + V_{1, \dots, p}. \qquad (13) $$

Dividing each side of Eq. (13) by $V$ (assuming $V \neq 0$, i.e. not all the outputs are identical), leads to the following equation

$$ 1 = \sum_{i=1}^{N} \frac{V_i}{V} + \sum_{i=1}^{N} \sum_{j=i+1}^{N} \frac{V_{ij}}{V} + \dots + \frac{V_{1, \dots, p}}{V}, \qquad (14) $$

where $S_i = V_i/V$ is the first Sobol index related to $x_i$, whose value explains the influence of the input $x_i$ alone of the output $Y : S_i = 0$ implies that the input $x_i$ has no influence of the value of the output, whereas $S_i = 1$ implies that the output value entirely depends on the value of $x_i$.

Two main choices remain in the hands of the user: the sampling method and the computation method of the sensitivity indices $S_i$. The chosen sampling method is called Latin hypercube: it efficiently samples the whole space of variation of each input, without leaving areas unexplored. It is particularly useful when the computation time of each simulation makes the SA time consuming [38]. As for the computation of the Sobol indices, the function RBD-FAST [39] from the Python library SALib [40,41] has been chosen, as it has been developed for building performances, and validated in the case of a rammed earth wall [29].

# 3. Case study: drying of compressed earth blocks

## 3.1. Experimental setup

The experiment consists in the drying of raw earth samples, whose internal relative humidity has been monitored at 8 locations of interest. Details are given in the following sections.

### 3.1.1. Compressed earth blocks characteristics

The samples are two compressed earth blocks CEB made from a soil extracted in Saint-Antoine-l'Abbaye (a village close to Grenoble, France). The CEBs have been manufactured without any stabilizers, using a gravimetric water content $w_{ini} = 327.6$ kg m$^{-3}$, and then wrapped in hermetic plastic bags for 3 weeks so as to let the water content homogenizes. Most of the properties of this soil have been experimentally determined by Chabriac [42], Soudani [43], Fabbri et al. [44], and are discussed in Section 3.3. The geometrical properties of the CEBs are shown in Fig. 1:

* For 1D drying, four sides of the CEBs have been covered with aluminium sheets (Fig. 1(a)), which prevent heat and moisture exchanges with the environment. The two faces left uncovered are those parallel to the (y, z) plane;

* Eight holes have been drilled on the top surface of each CEB, at specific locations in order to cover most of the surface without interfering with each other (Fig. 1(b)). The holes are also covered with aluminium, which is removed only during relative humidity measurements (which takes approximately 5 min);

* The 5 mm diameter holes have been drilled at a depth of 7cm from the top surface of the CEBs;

* Table 1 indicates the exact drilling coordinates: due to the drilling method, the points are not perfectly symmetrical, but owing to the very small difference and to aid the analysis, they considered symmetrical from a numerical point of view. Hence, due to the symmetry of both the geometry and of the boundary conditions, points A, B, G and H are considered numerically equivalent, points C and F also, and points D and E as well (Fig. 1(b)). In the remaining of the paper, the three groups of points will be referred to as the letters D, F and H, respectively.
<figure><img src="image_4.png" /></figure>

(a) Picture of a CEB with the holes drilled on the top surface. The drying occurs on the two surfaces parallel to the (y, z) plane (hatched surface).

<figure><img src="image_5.png" /></figure>

(b) Holes positions on the CEB. Exact coordinates are given in Table 1.

<caption>Fig. 1. Characteristics of the CEBs and position of the relative humidity probes.</caption>

<table><caption>**Table 1**

Positions of the holes drilled on the top surface of the CEB. The origin is taken at the bottom left corner, as shown in Fig. 1(b).</caption><thead><tr><th>Name</th><th>x [cm]</th><th>y [cm]</th></tr></thead><tbody><tr><td>A</td><td>4.7</td><td>4.4</td></tr><tr><td>B</td><td>9.6</td><td>4.5</td></tr><tr><td>C</td><td>7.3</td><td>9.9</td></tr><tr><td>D</td><td>2.4</td><td>14.5</td></tr><tr><td>E</td><td>12.6</td><td>14.5</td></tr><tr><td>F</td><td>7.45</td><td>19</td></tr><tr><td>G</td><td>5.1</td><td>24.5</td></tr><tr><td>H</td><td>9.9</td><td>24.6</td></tr></tbody></table>

### 3.1.2. Drying experiment parameters

The CEBs have then been placed into a room with controlled temperature and relative humidity, and unwrapped to begin the drying experiment. The initial conditions of the CEBs are $w_{ini} = 327.6$ kg m$^{-3}$ and $T_{ini} = 20^{\circ}$C. The experiment has been carried out for 60 days, and the CEBs internal relative humidity have been manually measured every 2 to 5 days, using Rotronics special humidity probes HC2-C04 (accuracy of $\pm 1.5\%$ in the range $RH \in [0, 100]\%$). In total, 275 relative humidity measurements have been carried out.

The CEBs have undergone a one direction drying process, i.e. only two faces were left uncovered and free to exchange water and heat with the environment (see Fig. 1(a)).

### 3.2. Numerical simulation parameters

The whole code, e.g. hygrothermal model and uncertainty and sensitivity analysis methods, was written in Python 3. The coupled Eqs. (1) and (2) are numerically solved with a finite difference scheme called Alternating Direction Implicit (ADI) [45,46], using a dummy variable $U = \{P_c, T\}$. The ADI method consists in dividing the time step between $t$ and $t + dt$ into two substeps:

1. During the first time step between $t$ and $t + \frac{dt}{2}$, the x-direction is solved implicitly, and the y-direction is solved explicitly. The input variable is the solution at the previous time step $U(t)$, and the output is an intermediate solution usually called $U^*(t + \frac{dt}{2})$;

2. During the second time step between $t + \frac{dt}{2}$ and $t + dt$, the x-direction is solved explicitly, and the y-direction is solved implicitly. The input is the intermediate solution $U^*(t + \frac{dt}{2})$ and the output is the solution $U(t + dt)$.

Fig. 2 shows the stencil used, illustrated with the variable $T$. The computation of one time step consists in solving the following matrix

<figure><img src="image_6.png" /><figcaption>Fig. 2. 2D stencil used for the ADI scheme, illustrated with the temperature $T$ — during the first (resp. second) half time step, the x-direction is solved implicitly (resp. explicitly) and the y-direction explicitly (resp. implicitly).</figcaption></figure>

systems one after another, for all rows and all columns,

$$R_x U^* = B_x, \qquad (15)$$

$$R_y U = B_y, \qquad (16)$$

where $R_x$ and $R_y$ contain the coefficients of the implicit part of the equation, and $B_x$ and $B_y$ those of the explicit part for each time step.

A mesh sensitivity analysis was conducted: simulations with elements of size ranging from 4x4 mm$^2$ to 2x2 cm$^2$ were performed. The output compared was the vapour pressure $P_v$ at the centre of the CEB, which is a direct function of RH and $T$ as detailed in Section 2.1. It was found that using 1x1 cm$^2$ elements induced a 5% mean relative error compared to the finest mesh size, while reducing the computation time by a factor 3. As the purpose of the article is to compare 1500 simulations with each other, it was concluded that this choice of mesh size was optimal with regards to computation time, while inducing an acceptable prediction error. The duration of each simulation is 60 days, and the time step used has been arbitrarily set to $dt = 1400$ s, which is a total of 4937 time steps.

### 3.3. Inputs choices

In this section, the ranges of variation of the sensitivity analysis inputs are detailed: methods of measurements, references containing
<figure><img src="image_7.png" /><figcaption>Fig. 3. Experimental suction data/isotherm, with two fits of the van Genuchten model (Eq. (3)) corresponding to the lower and upper boundaries for $w_{sat}$. The inserted graph is a zoom in the region $P_c \in [5.10^8, 10^9]$ Pa.</figcaption></figure>

the data when necessary, and detailed reasons for the range chosen. Parameter variability was estimated based on previous experimental work, in particular that of Chabriac [42], Soudani [43]. A summary is proposed in Table 2:

* To establish the relationship between water content and capillary pressure, the van Genuchten model (Eq. (3)) was fitted to adsorption/desorption test data performed on the studied material. This test has consisted to subject the sample to controlled humidity cycles, following the ISO 12571 standard, using the DVS Intrinsic device from SMS group (Surface Measurement Systems), which measures the material's mass change in response to varying relative humidity levels. Additional suction measurements were performed using the filter paper method, conducted at 20 °C with Whatman No. 42 filter paper, following the ASTM D5298-16 standard. Then, Eq. (10) establishes the relationship between the sorption/desorption isotherm data and the material's suction data. During this fitting process, for the reasons previously mentioned, the parameter $w_{sat}$ appeared to be weakly constrained, with very different values leading to similar discrepancies between the experimental data and the van Genuchten model's curve. To address this issue, the variation range of $w_{sat}$ was restricted to plausible values. To do so, as proposed by Fabbri et al. [44], the relationship linking the theoretical $w_{sat}$ to the dry bulk density and the densities of water and solid grains was used:

$$w_{sat} = \rho_l \left( \frac{1}{\rho_d} - \frac{1}{\rho_s} \right) \qquad (17)$$

where $\rho_l = 1000$ kg m⁻³ and $\rho_d = 1680$ kg m⁻³ (measured by Soudani [43]). It is assumed that there is no uncertainty on these values. In accordance with the correlation between density and intrinsic permeability for unstabilized raw earth highlighted by Fabbri et al. [44], the intrinsic permeability has been set to $K_0 = 10^{-17}$ m². The skeleton density $\rho_s$ has been measured by Fabbri et al. [44] using a pycnometer Ultrapyc 1200e. The device used has a maximum uncertain of ±4%, and the measured skeleton density is $\rho_s = 2700 \pm 4$ kg m⁻³, which leads to

$$w_{sat} \in [351, 402] \text{ kg.m}^{-3} \qquad (18)$$

For each value of $w_{sat}$ that is randomly picked in the range of Eq. (18), a van Genuchten model is derived using Eq. (3), based on sorption/desorption and suction experimental data, as explained above. Results are shown in Fig. 3. Quite obviously, the influence of $w_{sat}$ is predominant at low capillary pressures, but

still persists at high capillary pressures, as shown in the zoomed area.

* The calorific heat capacity $c_{p,mat}$ has been measured by Chabriac [42], using Differential Scanning Calorimetry. The two values measured at 20 °C are taken as the lower and upper bounds of the range used for the sensitivity analysis: $c_{p,mat} \in [580, 670]$ J kg⁻¹ K⁻¹.

* $k_{dry}$ has been measured using the transient hot wire method. The results were $k_{dry} \in [0.64, 0.90]$ W m⁻¹ K⁻¹, with k being linearly dependent on the volumetric water content $\Theta$, as shown by Chabriac [42]:

$$k = k_{dry} + 5.06\Theta. \qquad (19)$$

The relationship between the volumetric and gravimetric water contents is

$$\Theta = \frac{w}{\rho_l}. \qquad (20)$$

* The water vapour resistance factor $\mu$ was computed from wet cup tests, using the corrected expression proposed by McGregor et al. [47]. This procedure leads to $\mu \in [6, 7]$. Then, the water vapour permeability can be computed as follows [12]

$$\delta_p(\mu, T) = \frac{2.10^7 T^{0.81}}{\mu P_{\text{atm}}}, \qquad (21)$$

where $P_{atm} = 101325$ Pa is the atmospheric pressure.

* The convective heat transfer coefficient (CHTC) $h$ can for example be derived from the air velocity in the room [48] or the wall temperature [49,50]. Fig. 4 shows the variation of the CHTC for different models found in the literature, within the ranges of air velocities and surface/air temperature difference considered during the drying experiments. As having $h = 0$ W m⁻² K⁻¹ leads to zero flux boundary conditions, it has been arbitrarily decided to consider a minimum value for $h$ of $h_{min} = 0.5$ W m⁻² K⁻¹. This leads to $h \in [0.5, 5]$ W m⁻² K⁻¹. The convective mass transfer coefficient (CMTC) $h_m$ can be assessed for each simulation, using the Lewis analogy (Eq. (9)). The impact and the relevance of using this relationship will be discussed later in the paper.

* Fig. 5 shows measurements of the room temperature and relative humidity during the 60 days of drying, with three different sensors. The sensors are all temperature and relative humidity sensors Tinytag view 2 TV-4500, whose accuracy is ±0.4°C for the temperature, and ±3% for the relative humidity. The ranges for both the relative humidity and temperature measured are therefore:
<figure><img src="image_8.png" /></figure>

(a) Convective heat transfer coefficient evolution as a function of the temperature difference between the air and the considered surface, for different models found in the literature [Awbi, 1998, Camci et al., 2021].

<figure><img src="image_9.png" /></figure>

(b) Convective heat transfer coefficient evolution as a function of air velocity for a laminar flow and a turbulent flow [Peeters et al., 2011].

<caption>**Fig. 4.** Values of the convective heat transfer coefficient for different models. The hatched areas correspond to the measured air velocity and temperature difference ranges considered during the drying experiments.</caption>

<table><caption>**Table 2**
Inputs variation ranges.</caption><thead><tr><td>Parameter</td><td>Range</td><td>Mean value</td></tr></thead><tbody><tr><td>Heat capacity c<sub>p,mat</sub> [J kg<sup>-1</sup> K<sup>-1</sup>]</td><td>[580, 670]</td><td>610</td></tr><tr><td>Dry thermal conductivity k<sub>dry</sub> [W m<sup>-1</sup> K<sup>-1</sup>]</td><td>[0.64, 0.90]</td><td>0.77</td></tr><tr><td>Water vapour resistance factor μ [-]</td><td>[6, 7]</td><td>6.5</td></tr><tr><td>Saturated water content ω<sub>sat</sub> [kg m<sup>-3</sup>]</td><td>[351, 402]</td><td>376.5</td></tr><tr><td>Exterior relative humidity RH<sub>ext</sub> [%]</td><td>[50, 60]</td><td>55</td></tr><tr><td>Exterior temperature T<sub>ext</sub> [°C]</td><td>[21, 25]</td><td>23</td></tr><tr><td>Convective heat transfer coefficient h [W m<sup>-2</sup> K<sup>-1</sup>]</td><td>[0.5, 5]</td><td>2.75</td></tr></tbody></table>

$$- RH_{ext} \in [50, 60] \%;$$

$$- T_{ext} \in [21, 25] \,^{\circ}\text{C}.$$

Let us underline that the external temperature and relative humidity that intervene in Eqs. (8) and (11) are assumed constant throughout every drying simulation.

## 4. Results

Thanks to the use of a multiprocessing mapping distributed on 15 CPU, the total CPU time for the 1500 simulations of 60 days of drying is around 15h.

First, the normalized variance for every simulation computed at point D is displayed in Fig. 6: the normalized variance stabilizes at ±5% around 1 after 1000 simulations, which validates that the sample size considered is large enough to compute Sobol indices. The same behaviour has been observed for both points F and H.

Fig. 7 shows the evolution of the CEBs internal relative humidity during 60 days. As described in Section 3.1.1, for every measurement time, points C and D each embed four measurements (two equivalent points for the two CEBs), and point H embeds eight measurements (four equivalent points for the two CEBs). Note that not all the points clearly appear on Fig. 7, due to overlapping with each other. The experimental data are shown on top of the predictions of the 1500 simulations run. The mean is displayed as a black solid line, and the hatched area represents all the predictions that falls within the mean plus or minus the standard deviation (which represents ≈ 68.27 % of the predictions as the results follow a normal distribution).

Fig. 8 shows the evolution of the sum of the Sobol indices relative to each input, as a function of the drying time. The three plots highlight a sum close to 1 from day 10 for points F and H meaning that most of the outputs variations are explained by the first order sensitivity indices. Point D exhibits a slightly different behaviour, as the sum gets close to 1, then decreases to around 0.85. Nevertheless, these results validate
<figure><img src="image_10.png" /><figcaption>Fig. 5. Time evolution of the room temperature and relative humidity throughout the drying campaign, using three different sensors.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 6. Normalized variance at point D.</figcaption></figure>

the hypothesis of independence of the inputs that allows us to perform
the SA, as described in Section 2.3.

Fig. 9 shows the evolution of the first Sobol indices $S_i$ relative to each input, as a function of the drying time. Again, points F and H exhibit a similar behaviour: in the first 5 days, the influence of $w_{sat}$ is predominant, then $h$ takes over ($S_i \approx 0.95$) until day 40, where $RH_{ext}$ starts to have an increased influence ($S_i \approx 0.2$) that is stable until day 60. Regarding point D, the main difference lies in the last 20 days, where there is a slight decrease in the Sobol index of $h$.

5. Discussion

To begin with, a discussion about the experimental setup should be
made. In this study, internal RH has been measured because the probes
were available at the lab. However, it may be more relevant to directly
measure the water content *w* instead of the RH for several reasons. The
first is that RH is a function of both the temperature and the capillary
pressure, meanwhile *w* is only a function of *P*<sub>c</sub>. This means that working
with RH induces the risk of a biased interpretation of the results in some
cases. Then, the sensitivity of measurement close to saturation is very
small, as shown in Fig. 3: in the first 20 days of drying, *w* decreases
(in mean) from *w*<sub>ini</sub> = 327.6 kg m<sup>-3</sup> to *w*<sub>20d</sub> ≈ 150 kg m<sup>-3</sup>. In terms
of capillary pressure, this means that *P*<sub>c</sub> goes from *P*<sub>ini</sub> = 8.10<sup>5</sup> Pa to
*P*<sub>c20d</sub> ≈ 6.10<sup>6</sup> Pa, and using Eq. (10), the relative humidity drops from
RH<sub>ini</sub> = 99.4 % to RH<sub>20d</sub> ≈ 95.7 % which is a variation close to the
±1.5 % accuracy of the sensors, as detailed in Section 3.1.2. Then, the
stabilization time of the sensors is longer as RH drops, which can lead

to increased measurements errors as the drying process goes on. At last,
room relative humidity and temperature greatly vary. This has to be
rendered in the model through convective boundary conditions. One
possible workaround could be the use of climatic chambers that control
both the temperature and the relative humidity: such apparatus would
be modelled through Dirichlet boundary conditions which are easier to
deal with, and prevent the use of convective analogies.

The predictions of the internal relative humidity (Fig. 7) are overall
promising: even though during the first 15–20 days, the drying rate
is overestimated by the model, the predicted mean output for the
remaining 40 days fits well the measurements. The predictions show
a faster drying rate on the edges (Fig. 7(a)), which is expected. Point H
is close to a zero flux boundary, which makes it dry at nearly the same
rate as point F (Figs. 7(b) and 7(c)).

The SA has shed light on several interesting points. First, apart
from $w_{sat}$ in the first 5 days, no other material parameter studied
here has an influence on the drying kinetics ($S_i \approx 0$). In fact, the
drying kinetics is mainly driven by the boundary conditions, the chosen
material parameters having close to no influence: it is shown by the
predominance of the CHTC $h$ in the variation of the predicted RH (Fig.
9). This was to be expected: indeed, this parameter is very complex to
determine accurately, and the scientific community has been debating
about it for a long time now, proposing new methods and formulas
to compute $h$ (see [49–51] for example). For instance, [52] showed
that a 10 % error on $h$ could lead to a 5 % error on the predicted
energy consumption of a building. Moreover, empirical correlations to
<figure><img src="image_12.png" /><figcaption>Fig. 7. RH computation and measurements at the three points of interest.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 8. Sum of the first Sobol indices $\sum S_i$ of the inputs for the three points of interest.</figcaption></figure>

determine *h* are derived from specific geometries, rendering their use
geometry dependent and not fully applicable to any case [48].

Another questionable point is the use of the so-called Lewis analogy
(Eq. (9)) as used in this article. In fact, during the early stages of
<figure><img src="image_14.png" /><figcaption>Fig. 9. First Sobol indices $S_i$ of the inputs for the three points of interest.</figcaption></figure>

drying, the majority of the water transport is done through liquid movement, which reaches the edges of the sample to evaporate. In that case, it seems essential to correctly assess the mass transfer with the environment through the value of $h_m$. The Lewis analogy has been widely used (see [28,33] for instance.), however, other formulas are available such as the Chilton–Colburn analogy. This analogy has for example been used by Mauffré et al. [53] for a direct comparison between convective mass and heat transfer coefficients conducted on earthen plasters exposed to laminar and turbulent flows under controlled humidity conditions in a ventilation tunnel. It reads, for laminar flows,

$$ \frac{1}{h_m} = \frac{\rho_a c_{p,a} R_v T}{h} \left( \frac{S_c}{P_r} \right)^{\frac{1}{3}}, \qquad (22) $$

where $\rho_a = 1.293 \text{ kg m}^{-3}$ is the density of dry air at 20 °C, $c_{p,a} = 10^3 \text{ J kg}^{-1} \text{ K}^{-1}$ at 20 °C and T is the surface temperature at the interface between the sample and the environment. $S_c$ and $P_c$ are the Schmidt

and Prandtl numbers, respectively. Eq. (22) gives

$$ h_m(T) \approx 6.1 \cdot 10^{-9} h, \quad \forall T \in [20, 25]^{\circ}\text{C}. \qquad (23) $$

The coefficient in Eq. (23) is very close to the coefficient of the Lewis analogy (Eq. (9)). As concluded by Mauffré et al. [53], these correlation laws lack accuracy and their relevance is questionable. In particular, they showed that mass transfer appeared to be more sensitive to the surface roughness of the samples than heat transfer. Moreover, the use of such analogies completely hides the impact of the mass transfer with the environment. Indeed, it is possible that the output variation attributed to h is actually due to the variation of $h_m$, but the use of usual analogies prevents us from drawing any conclusion on this crucial point. The applicability of the heat and mass transfer analogy is also dependent on the choice of the reference values $T_{ext}$ and $RH_{ext}$ [54]: these values have been set identical for the whole domain, which could artificially imply a zero-flux condition when there is a flux in practice. Other conditions should also be fulfilled, as for example the
need for the studied object temperature to be close to the equilibrium temperature [55]. Potential solutions are direct measurements of $h_m$, even though having a constant value does not seem physically relevant, or coupled fluid-solid simulations (CFD-HAM) as done by Defraeye et al. [56], which would greatly increase the computation time.

The last aspect of the model boundary conditions to be discussed is the variability of $T_{ext}$ and $RH_{ext}$ (Eqs. (8) and (11)): they should be made time and space dependent when possible. This is especially important when dealing with on-site drying, where the weather can drastically vary, hence greatly modifying the heat and mass exchanges between the material and the environment within small periods of time.

To improve the model's predictive power and realism, it would be relevant to add a mechanical layer. Indeed, drying shrinkage can highly affect the porosity of the material, hence modifying its water uptake capacities [19]. Recently, [57] measured up to 15 % of linear shrinkage on unstabilized earth mortars. Such models have already been derived and numerically implemented using finite volumes methods, for studying the drying of wood [58,59] or to develop a poro-elasto-plasticity model for soils [60], for instance.

## 6. Conclusion

This paper presents the extension of a previously derived 1D hygrothermal model to 2D, in order to study the drying kinetics of Compressed earth blocks. Coupled to an uncertainty and sensitivity analysis, the model predictions have been compared to experimental measures of an 60 days drying campaign. The experimental setup consisted in 2 Compressed earth blocks exposed to a controlled lab environment (controlled temperature and relative humidity), whose four out of six faces have been isolated to force the drying along one axis only. Many interesting comments can be drawn from this study:

* The mean model predictions fit well the experimental data, except for the first 15 days of drying. This is due to the high relative humidity (close to saturation) where the model accuracy may be questionable as the mass transfer is mainly driven by the liquid permeability. Moreover, the sensors accuracy may be questionable in this region;

* The drying kinetics spatial distribution are physically meaningful: the point closer to the free edge dries faster than the points in the bulk;

* When dealing with convective boundary conditions implying convective transfer coefficients, what prevails is the knowledge of those coefficients because they are the main parameters influencing the predicted drying kinetics, while the material parameters chosen in this study have very little to no influence. Practically, if such boundary conditions type is to be used, it is crucial to have the most accurate environmental data as possible so as to reduce the prediction errors;

* The sensitivity analysis was performed on the 1500 simulations: the hypotheses of independence of the inputs has been validated, and the evolution of first Sobol indices has shown a predominance of the convective heat transfer coefficient value on the variation of the output. It was to be expected, as it has been proved by previous studies that it is a critical parameter in hygrothermal modelling;

* The Lewis analogy, often used in hygrothermal models, is highly questionable. From its mathematical form to its validity, its use prevents one to decouple the influence of both the heat and mass transfer with the environment;

* Improvement of the model's predictive power could be achieved by using time and space dependent boundary conditions, then by taking into account shrinkage through the addition of mechanical balances equations.

* 2D drying experiments (i.e. four faces are open to the environment) are soon to be realized, in order to build up the drying experiments database.

These promising results highlight the robustness of the hygrothermal model, and prove it ready for 2D studies of drying.

## CRediT authorship contribution statement

**Etienne Le Mire:** Writing – original draft, Visualization, Software, Methodology, Investigation, Conceptualization. **Rudy Bui:** Writing – review & editing, Software, Methodology, Conceptualization. **Antonin Fabbri:** Writing – review & editing, Validation, Resources, Investigation. **Marie-Sarah Force:** Writing – review & editing, Validation, Data curation.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Antonin Fabbri reports financial support was provided by The French Agency for Ecological Transition. Antonin Fabbri reports financial support was provided by French National Research Agency. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This work has been funded by two french projects: Dynaterre, financed through the call for projects ANR 2023, ANR-23-CE04-0015 funded by the french National Research Agency (ANR in french), and Qualit'ere, financed through the call for projects "Towards responsible buildings 2022" ("Vers des Bâtiments Responsables 2022" in french), funded by the french national ecological transition agency ADEME. The authors would also like to thank Agnès Legris, who realized the Compressed earth blocks as well as the relative humidity measurements during her Master internship.

## Data availability

Data will be made available on request.

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