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Characterization of Concrete by Calibrating Thermo-Hydraulic Multiphase Flow Models


Evaporation tests in concrete columns have been analysed by numerical models to characterize the thermo-hydraulic properties and the processes in concrete. Two evaporation tests were performed: a column heated by a lamp and a column kept in room conditions. The conceptual model considers unsaturated liquid flow and transport of vapour and energy. We also calculated models that take into account the dissolved salts to study its effect on vapour pressure and evaporation. A retention curve has been obtained from relative humidity and gravimetric water content measured after dismantling the tests. The models have been calibrated by adjusting the model’s results to the measured data of water loss, relative humidity and temperature inside the concrete. The parameters obtained with the calibration are the permeability, thermal conductivity, boundary conditions and a tortuosity factor for vapour diffusion. Results show that the vapour diffusion is the dominant water transport process above an evaporation front, and liquid advection is dominant below it.

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\(A\) :

Constant for relative permeability

\(a_\mathrm{w}\) :

Activity of water

\(D\) :

Parameter for diffusion

\(D_\mathrm{m}\) :

Molecular diffusion coefficient (m\(^{2}\) s\(^{-1}\))

\(D^{\prime }_{\alpha }\) :

Mechanical dispersion coefficient (m\(^{2}\) s\(^{-1}\))

\(d_\mathrm{l}\) :

Longitudinal dispersivity (m)

\(E_{\alpha }\) :

Internal energy per unit of mass for each phase (J kg\(^{-1}\))

\(E^\mathrm{i}_{\alpha }\) :

Specific internal energy (J kg\(^{-1}\))

\(f\) :

External supply of component

\(g\) :

Gravity (m s\(^{-2}\))


Relative humidity

\(i\) :

(superscript) Component index, w water, a dry air and h salt

\(i_{\alpha }\) :

Diffusive or dispersive flux (J s\(^{-1}\) m\(^{-2}\))

\(i_\mathrm{c}\) :

Heat conductive flux (J s\(^{-1}\) m\(^{-2}\))

\(j^{i}_{\alpha }\) :

Mass flux of component in each phase (J s\(^{-1}\) m\(^{-2}\))

\(j^\mathrm{w}_\mathrm{g}\) :

Vapour flux (kg s\(^{-1}\) m\(^{-2}\))

\(j_\mathrm{e}\) :

Energy flux (J s\(^{-1}\) m\(^{-2}\))

\(j^{0}_\mathrm{e}\) :

Radiation (J s\(^{-1}\) m\(^{-2}\))

\(k_\mathrm{i}\) :

Intrinsic permeability (m\(^{2}\))

\(k_\mathrm{r}\) :

Relative permeability

\(M^{i}\) :

Molecular weight of component \(i\) (kg mol\(^{-1}\))

\(m\) :

Shape parameter for retention curve

\(n\) :

Parameter for relative permeability

\(P_{\alpha }\) :

Pressure of phase \(\alpha \) (MPa)

\(P^{i}_{\alpha }\) :

Partial pressure of component \(i\) in phase \(\alpha \) (MPa)

\(P^{w}_\mathrm{g,sat}\) :

Saturated vapour pressure (MPa)

\(P_{0}\) :

Entry pressure (MPa)

\(P_{l0}\) :

Reference pressure (MPa)

\(q_{\alpha }\) :

Flow rate (m s\(^{-1}\))

\(R\) :

Ideal gas constant (J mol\(^{-1}\) K\(^{-1}\))

\(S_{\alpha }\) :

Saturation of phase \(\alpha \)

\(S_\mathrm{e}\) :

Effective water saturation

\(T\) :

Temperature (\(^{\circ }\)C)

\(T\) :

Gravimetric humidity

\(\alpha \) :

(subscript) Phase index, l liquid, s solid and g gas

\(\beta \) :

Compressibility (MPa\(^{-1}\))

\(\beta _\mathrm{g}\) :

Boundary vapour exchange coefficient (m s\(^{-1}\))

\(\gamma \) :

Solute variation coefficient

\(\gamma _\mathrm{g}\) :

Boundary gas exchange coefficient (kg s\(^{-1}\) m\(^{-2}\) MPa\(^{-1}\))

\(\gamma _\mathrm{e}\) :

Boundary heat exchange coefficient (J s\(^{-1}\) m\(^{-2}\) \(^{\circ }\)C\(^{-1}\))

\(\epsilon \) :

Volumetric thermal expansion coefficient for liquid (\(^{\circ }\)C\(^{-1}\))

\(\lambda \) :

Thermal conductivity (W m K\(^{-1}\))

\(\mu _{\alpha }\) :

Viscosity of phase \(\alpha \) (MPa s)

\(\theta _{\alpha }\) :

Volumetric phase content of \(\alpha \)

\(\rho _{\alpha }\) :

Density of phase \(\alpha \) (kg m\(^{-3}\))

\(\rho _{l0}\) :

Reference density of the liquid (kg m\(^{-3}\))

\(\sigma _{l0}\) :

Surface tension at temperature \(T\) (N m\(^{-1}\))

\(\tau \) :


\(\phi \) :


\(\omega ^{i}_{\alpha }\) :

Mass fraction of solute \(i\) in phase \(\alpha \) (kg kg\(^{-1}\))


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This work was funded by ENRESA (Spanish Nuclear Waste Management Company) and a Research Grant from the Technical University of Catalonia (UPC).

Author information

Correspondence to M. Carme Chaparro.

Appendix: Constitutive Laws

Appendix: Constitutive Laws

Definition Constraints

Mass fractions

$$\begin{aligned}&\omega ^\mathrm{w}_\mathrm{l}+\omega ^\mathrm{a}_\mathrm{l}+\omega ^\mathrm{h}_\mathrm{l}=1 \end{aligned}$$
$$\begin{aligned}&\omega ^\mathrm{w}_\mathrm{g}+\omega ^\mathrm{a}_\mathrm{g}=1 \end{aligned}$$

Volumetric content

$$\begin{aligned} \theta _\mathrm{l}=\dfrac{V_\mathrm{l}}{V_\mathrm{t}}=\dfrac{1}{\rho _\mathrm{l}}w\rho _\mathrm{s}( 1-\phi ) \end{aligned}$$


$$\begin{aligned}&S_\mathrm{l}+S_\mathrm{g}=1 \end{aligned}$$
$$\begin{aligned}&S_\mathrm{l}=\dfrac{V_\mathrm{l}}{V_\mathrm{p}}=\dfrac{\theta _\mathrm{l}}{\phi } \end{aligned}$$

Partial pressures

$$\begin{aligned} P^\mathrm{a}_\mathrm{g}+P^\mathrm{w}_\mathrm{g}=P_\mathrm{g} \end{aligned}$$

Equilibrium Constraints

Vapour–liquid water (psychrometric law)

$$\begin{aligned} P^\mathrm{w}_\mathrm{g}= & {} a_\mathrm{w}P^\mathrm{w}_\mathrm{g,sat}\exp \left( \frac{-(P_\mathrm{g}-P_\mathrm{l})M^\mathrm{w}}{R\rho _\mathrm{l}(273.15+T)}\right) \end{aligned}$$
$$\begin{aligned} P^\mathrm{w}_\mathrm{g,sat}= & {} 136075\exp \left( \frac{-5239.7}{273.15+T}\right) \end{aligned}$$
$$\begin{aligned} a_\mathrm{w}= & {} 1-\left( \left( \omega ^\mathrm{h}_\mathrm{l} \frac{1000}{M^\mathrm{h}-3}\right) 1.9775\times 10^{-5}T+0.035\right) \omega ^\mathrm{h}_\mathrm{l} \frac{1000}{M^\mathrm{h}-3} \end{aligned}$$

where \(M^\mathrm{h}\) = 49.37 g mol\(^{-1}\)

$$\begin{aligned} \hbox {HR}= & {} \frac{P^\mathrm{w}_\mathrm{g}}{P^\mathrm{w}_\mathrm{g,sat}} \end{aligned}$$
$$\begin{aligned} \omega ^{w}_\mathrm{g}= & {} \frac{P^\mathrm{w}_\mathrm{g}M^\mathrm{w}}{R\rho _\mathrm{g}(273.15+T)} \end{aligned}$$

Dissolved-gaseous air (Henry’s law)

$$\begin{aligned} \omega ^\mathrm{a}_\mathrm{l}=\frac{P^\mathrm{a}_\mathrm{g} M^\mathrm{a}}{\textit{HM}^\mathrm{w}} \end{aligned}$$

Phase and Interphase Properties

Retention curve (van Genuchten law)

$$\begin{aligned} S_\mathrm{e}=\left( 1+ \left( \frac{P_\mathrm{g}-P_\mathrm{l}}{P_{0}\frac{\sigma _{T}}{\sigma _{0}}}\right) ^{\frac{1}{1-m}}\right) ^{-m} \end{aligned}$$

where \(P_{0}=7.7\) Mpa, \(m=0.34\)\(\sigma _{T}\) is the surface tension at temperature T and \(\sigma _{0}=0.072\) N m\(^{-1}\) at 20 \(^{\circ }\)C.

Relative permeability

$$\begin{aligned} k_\mathrm{rl}=\textit{AS}^\mathrm{n}_\mathrm{el} \end{aligned}$$

where \(A=0.01\) and \(n=7\)

$$\begin{aligned} k_\mathrm{rg}=\textit{AS}^\mathrm{n}_\mathrm{eg} \end{aligned}$$

where \(A=1\) and \(n=3\).

Properties of liquid

$$\begin{aligned} \rho _\mathrm{l} = \rho _{l0} \exp \left( \beta \left( P_\mathrm{l}-P_\mathrm{l0}\right) + \epsilon T+ \gamma \omega ^\mathrm{h}_\mathrm{l} \right) \end{aligned}$$

where \(\rho _{l0}=1002.6\) kg m\(^{-1}\)\(P_{l0}=0.1\,\hbox {MPa}\), \(\beta =4.5\times 10^{-4}\) MPa\(^{-1}\)\(\gamma =0.6923\) and \(\epsilon =3.4\times 10^{-4}\,^{\circ }\)C\(^{-1}\)

$$\begin{aligned} \mu _\mathrm{l}= & {} 2.1\times 10^{-12}\left( \frac{1808.5}{273.15+T}\right) \end{aligned}$$
$$\begin{aligned} \sigma _{T}= & {} \left( 1-0.625 \left( \frac{273.15+T}{647.3} \right) \right) \left( 0.2358 \left( \frac{273.15+T}{647.3} \right) ^{1.256} \right) + 0.04055 \omega ^\mathrm{h}_\mathrm{l} \qquad \end{aligned}$$
$$\begin{aligned} E_\mathrm{l}= & {} E^\mathrm{w}_\mathrm{l} \omega ^\mathrm{w}_\mathrm{l}+ E^\mathrm{a}_\mathrm{l} \omega ^\mathrm{a}_\mathrm{l} \end{aligned}$$

where \(E^\mathrm{w}_\mathrm{l}=4184\) T J kg\(^{-1}\) and \(E^\mathrm{a}_\mathrm{l}=1000T\) J kg\(^{-1}\).

Properties of gas

$$\begin{aligned} \rho _\mathrm{g}=\frac{P^\mathrm{w}_\mathrm{g}M^\mathrm{w}}{R(273.15+T)}+\frac{P^\mathrm{a}_{g}M^\mathrm{w}}{R(273.15+T)} \end{aligned}$$

where \(M^\mathrm{a}_\mathrm{g}=0.02895\) kg mol\(^{-1}\) and H \(=\) 10,000 MPa

$$\begin{aligned} \mu _\mathrm{g}= & {} \frac{1.48 \times 10^{-12} \sqrt{273.15+T}}{1+ \dfrac{119.4}{T+273.15} } \dfrac{1}{1+ \dfrac{0.14-1.2 \times 10^{15}k_\mathrm{i}}{P_\mathrm{g}}} \end{aligned}$$
$$\begin{aligned} E_\mathrm{g}= & {} E^\mathrm{w}_\mathrm{g} \omega ^\mathrm{w}_\mathrm{g}+ E^\mathrm{a}_\mathrm{g} \omega ^\mathrm{a}_\mathrm{g} \end{aligned}$$

where \(E^\mathrm{w}_\mathrm{g}=2.5 \times 10^{6}+1900T\) J kg\(^{-1}\) and \(E^\mathrm{a}_\mathrm{g} =1000T\) J kg\(^{-1}\).

Properties of solid

The density of the solid phase (\(\rho _\mathrm{s}\)) is 2360 kg m\(^{-1}\) and \(E_\mathrm{s} =780T\) J kg\(^{-1}\).


Darcy’s flux

$$\begin{aligned} q_{\alpha }=-\frac{k_\mathrm{i} k_\mathrm{r\alpha }}{\mu _{\alpha }}(\bigtriangledown P_{\alpha } - \rho _{\alpha }\mathrm g) \end{aligned}$$

where \(k_\mathrm{i}=4.2 \times 10^{-18}\) m\(^{2}\).

Diffusive flux (Fick’s law)

$$\begin{aligned} i^\mathrm{i}_{\alpha , \mathrm{dif}}=-(\tau \phi \rho _\alpha S_{\alpha } D^\mathrm{i}_\mathrm{m}) \bigtriangledown \omega ^\mathrm{i}_{\alpha } \end{aligned}$$

where \(\tau _\mathrm{non{\text {-}}heated}=0.08\) and \(\tau _\mathrm{heated}=0.3\)

$$\begin{aligned} D^\mathrm{vapor}_\mathrm{m}=D \left( \frac{(273.15+T)^{n}}{P_\mathrm{g}}\right) \end{aligned}$$

where \(D=5.9 \times 10^{-6}\) m s\(^{-1}\) K\(^{-n}\) Pa and \(n=2.3\)

$$\begin{aligned} D^\mathrm{solute}_\mathrm{m}=D \exp \left( \frac{-24530}{R(273.15+T)}\right) \end{aligned}$$

where \(D=1.1 \times 10^{-4}\) m\(^{2}\) s\(^{-1}\).

Dispersive flux (Fick’s law)

$$\begin{aligned} i^\mathrm{i}_{\alpha , \mathrm{dis}}= & {} -\left( \rho _{\alpha } D^{\prime }_{\alpha }\right) \bigtriangledown \omega ^\mathrm{i}_{\alpha } \end{aligned}$$
$$\begin{aligned} D^{\prime }_{\alpha }= & {} d_\mathrm{l}q_{\alpha } \end{aligned}$$

where \(d_\mathrm{l}=0.015\) m.

Mass flux

$$\begin{aligned} j^\mathrm{i}_{\alpha }=q_{\alpha }\rho _{\alpha }\omega ^\mathrm{i}_{\alpha } + i^\mathrm{i}_{\alpha ,\mathrm{dis}} + i^\mathrm{i}_{\alpha ,\mathrm{dif}} \end{aligned}$$

Conductive flux of heat (Fourier’s law)

$$\begin{aligned} i_\mathrm{c}= & {} - \lambda \bigtriangledown T \end{aligned}$$
$$\begin{aligned} \lambda= & {} \lambda _\mathrm{sat}^{S_\mathrm{l}} \lambda _\mathrm{dry}^{1-S_\mathrm{l}} \end{aligned}$$

where \(\lambda _\mathrm{sat}=1.14\) W m K\(^{-1}\) and \(\lambda _\mathrm{dry}=0.66\) W m K\(^{-1}\).

Advective flux of heat

$$\begin{aligned} j_\mathrm{E\alpha }=q_{\alpha }E_{\alpha } \end{aligned}$$

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Chaparro, M.C., Saaltink, M.W. & Villar, M.V. Characterization of Concrete by Calibrating Thermo-Hydraulic Multiphase Flow Models. Transp Porous Med 109, 147–167 (2015).

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  • Drying
  • Retention curve
  • Permeability
  • Concrete
  • Modelling