2. Reference solution (A, B and C models)#

2.1. Validation of the water exchange term from the heat exchange term#

A certain number of hypotheses are put forward in order to make the analogy with the thermal case:

  • the temperature is constant (no thermal calculation);

  • the gas pressure is constant;

  • Vapor pressure is negligible;

  • We are neglecting the broadcast of Fick;

  • The material is non-deformable (no mechanical calculations);

  • Gravity is zero;

  • Water is incompressible \({\rho }_{w}=\mathit{cste}\).

Recall that Darcy’s law (see R7.01.11) for the gas phase is written as:

(2.1)#\[ {M} _ {\ mathit {gz}} =- {\ rho}} =- {\ rho} _ {\ mathit {gz}} {\ mathit {gz}}} ^ {H}\nabla {p} _ {\ mathit {gz}}\]

with:

  • \({M}_{\mathit{gz}}\) the gas flow;

  • \({\rho }_{\mathit{gz}}\) the density of the gas;

  • \({\mathrm{\lambda }}_{\mathit{gz}}^{H}\) the hydraulic conductivity of the gas;

  • \({p}_{\mathit{gz}}\) gas pressure.

Gold:

(2.2)#\[ {p} _ {\ mathit {gz}} =\ mathit {cste}\ Rightarrow\nabla {p} _ {\ mathit {gz}} =0\]

So:

\[\]

: label: eq-4

{M} _ {mathit {gz}} = {M}} = {M} _ {mathit {as}} + {mathit {gz}} = 0

Since we’re neglecting the Fick broadcast, then:

\[\]

: label: eq-5

{M} _ {mathit {ad}} = {M} _ {mathit {VP}} =0

By () and ():

(2.3)#\[ {M} _ {\ mathit {as}} =0\]

Thus we can simplify the problem, which boils down to the equation for the conservation of the mass of liquid water:

(2.4)#\[ \ dot {{m} _ {w}}} +\ text {Div} ({M} _ {w}) =0\]

Or \({m}_{w}={\rho }_{w}\mathrm{\varphi }S(1+\mathit{Tr}(\epsilon ))-{\rho }_{w}^{0}{\mathrm{\varphi }}^{0}{S}^{0}\), with:

  • \({\rho }_{w}\) the density of water, \({\rho }_{w}^{0}\) the density in the initial state;

  • \(\mathrm{\varphi }=\frac{{V}_{\mathit{vide}}}{V}\) the porosity of the material, \({V}_{\mathit{vide}}\) corresponding to the unfilled pores, \(V\) to the porosity of the material in the initial state;

  • \(S=\frac{{V}_{\mathit{lq}}}{{V}_{\mathit{vide}}}\) water saturation, \({S}^{0}\) initial saturation;

  • \(\epsilon\) the deformation of the material;

The material is considered to be non-deformable: \(\epsilon =0\) and \(\mathrm{\varphi }=\mathit{cste}={\mathrm{\varphi }}^{0}\) and that water is incompressible, \({\rho }_{w}={\rho }_{w}^{0}\). So:

(2.5)#\[ {m} _ {w} (t) = {\ rho} _ {w} _ {w} ^ {0}\ mathrm {\ varphi} (S (t) - {S} ^ {0})\]

This involves:

(2.6)#\[ \ dot {{m} _ {w}} (t) = {\ rho} _ {w} _ {w}\ mathrm {\ varphi} {\ partial} _ {t} S ({p} _ {c} (t)) = {\ rho} (t)) = {\ rho} _ {t)))) = {\ rho} _ {t}))) = {\ rho} _ {t}))) = {\ rho} _ {w} _ {w}\ mathrm {\ varphi} {S} ^ {\ text {'}}} ({p} _ {c} (t))) = {\ rho} _ {t}))) = {\ rho} _ {t})) = {\ rho} _ {t})\ dot {{p} _ {c}} (t)\]

Since the effects of gravity are overlooked, Darcy’s law for the liquid phase is written:

(2.7)#\[ {M} _ {\ mathit {aq}} = {M} _ {w}} =- {\ rho} _ {w} {\ lambda} _ {\ mathit {q}}} ^ {H} (S) (S)\nabla {p} _ {w}\]

Gold \({p}_{c}={p}_{\mathit{gz}}-{p}_{\mathit{lq}}={p}_{\mathit{atm}}-{p}_{w}\). So \(\nabla {p}_{c}=-\nabla {p}_{w}\) and therefore \({M}_{w}={\mathrm{\rho }}_{w}{\mathrm{\lambda }}_{w}^{H}(S)\nabla {p}_{c}\).

By replacing \({M}_{w}\) and \(\dot{{m}_{w}}\) with their last expressions in () and assuming \({\lambda }_{w}^{H}(S)=\frac{K\times {K}_{\mathit{rl}}(S)}{\mu }\) to be a constant:

(2.8)#\[ {\ mathrm {\ rho}} _ {w}\ mathrm {\ varphi} {S} {S} ^ {\ text {'}} ({p} _ {c})\ times\ dot {{p} _ {c} _ {c}} _ {c}}} (t) + {\ mathrm {\ lambda}}} _ {w} ^ {H} (S)\ text {Div} _ {c} _ {c})\ times\ dot {p} _ {p} _ {c} _ {c} _ {c} _ {c}} (t) + {\ mathrm {\ lambda}}} _ {w} ^ {H} (S)\ text {Div} _ {c}} (s)\ text {Div} _ {c}\ rho}} _ {w}\ mathrm {.} \nabla {p} _ {c}) =0\]

Water is assumed to be incompressible so:

(2.9)#\[ {\ mathrm {\ rho}} _ {w}\ mathrm {\ varphi} {S}} ^ {\ text {'}} ({p} _ {c})\ times\ dot {{p} _ {c} _ {c}} (t) + {\ mathrm {\ rho}} (t) + {\ mathrm {\ rho}}} _ {w} {\ lambda}} _ {w} ^ {H}} _ {w} {\ lambda}} _ {w} ^ {H} (S)\ text {Div} (\nabla {p} _ {c}) =0\]

\({\rho }_{w}\mathrm{\varphi }{S}^{\text{'}}({p}_{c})\times \dot{{p}_{c}}(t)+{\rho }_{w}{\lambda }_{w}^{H}(S)\text{Div}(\text{}\nabla {p}_{c})=0\)

With the Laplacian operator:

(2.10)#\[ {\ mathrm {\ rho}} _ {w}\ mathrm {\ varphi} {S}} ^ {\ text {'}} ({p} _ {c})\ times\ dot {{p} _ {c} _ {c}} (t) + {\ mathrm {\ rho}} (t) + {\ mathrm {\ rho}}} _ {w} {\ lambda}} _ {w} ^ {H}} _ {w} {\ lambda}} _ {w} ^ {H} (S)\ mathrm {\ Delta} {p} {p} _ {c} =0\]

Finally:

(2.11)#\[ {\ partial} _ {t} {p} _ {c} +\ frac {{\ mathrm {\ lambda}} _ {w} ^ {H} (S)} {\ mathrm {\ varphi} {S} {S} {S}} {S}} ^ {\ text {c}} ^ {c}} ^ {c}} ^ {c})}\ mathrm {\ Delta} {p} {p} {p}} _ {c} =0\]

To close the system, we add:

  • the initial condition: \({p}_{c}(x\mathrm{,0})\mathrm{:}=10000\mathit{Pa}\);

  • the mixed condition on [BC] (or [BCFG] depending on the geometry): \({\mathrm{\lambda }}^{H}\nabla ({p}_{c}\mathrm{.}n)={h}^{H}({p}_{\mathit{ext}}-{p}_{c})\).

This formulation is similar to a heat equation without a source term, of the form:

(2.12)#\[ \ mathrm {\ rho} {C} _ {p} {p} {\ partial} _ {t} T+\ text {Div} (q) =0\]

By taking Fourier’s law that relates heat flow \(q\) to the temperature gradient, namely:

(2.13)#\[ q=-\ mathrm {\ lambda}\nabla T\]

Assuming that \(\mathrm{\lambda }\) is a constant, characteristic of the material:

(2.14)#\[ {\ partial} _ {t} T-\ frac {\ mathrm {\ lambda}} {\ mathrm {\ rho} {C} _ {p}}\ mathrm {\ Delta} T=0\]

Where \(\mathrm{\rho }\) is the density of concrete, \({C}_{p}\) is the specific heat of the material, and \(\mathrm{\lambda }\) is the thermal conductivity.

The thermal problem is thus already treated in THER_NON_LINE. We can then compare the results of the thermal problem with those of problem THM, by taking closure conditions of the same type as in the case THM, i.e. by asking:

\(\begin{array}{c}T(x\mathrm{,0})=\mathit{cste}\\ \mathrm{\lambda }(\text{}\nabla T\mathrm{.}n)={h}_{T}({T}_{\mathit{ext}}-T)\mathit{au}\mathit{bord}\end{array}\)

In addition, ensuring that:

  • \(T(x\mathrm{,0})=\mathit{cste}\mathrm{:}={p}_{c}(x\mathrm{,0})=10000\);

  • \(-\rho {C}_{p}≔\mathrm{\varphi }{S}^{\text{'}}({p}_{c})\)

  • \(\lambda \mathrm{:}={\lambda }^{H}\)

  • \({h}_{T}\mathrm{:}={h}^{H}\)

  • \({T}_{\mathit{ext}}\mathrm{:}={p}_{\mathit{ext}}\)

Subsequently, we then rely on thermal models AXIS, PLAN_DIAG, and 3D, and 3D, which are compared with models AXIS_THH2MS, D_ PLAN_THH2MS and 3D_ THH2MS, respectively.