2. Benchmark solution

Remember that the water mass input is written as: \({m}_{w}=\varphi \mathrm{.}{\rho }_{w}\mathrm{.}(1+{\varepsilon }_{v})\), which can be derived in the following form: \({\mathrm{dm}}_{w}=d\varphi {\rho }_{w}(1+{\varepsilon }_{v})+d{\rho }_{w}\varphi (1+{\varepsilon }_{v})+{\rho }_{w}\varphi d{\varepsilon }_{v}\) with \(\varphi\) the Eulerian porosity.

If we assume small trips, we will therefore have:

(2.8)\[ {\ mathrm {dm}} _ {w} =d\ varphi {\ rho} _ {w} +d {\ rho} _ {w}\ varphi + {\ rho} _ {w} _ {w}\ varphi d {\ varepsilon} _ {v}\]

The variation in porosity is written according to the relationship:

(2.8)\[ d\ varphi\ text {=} (b-\ varphi) (d {\ varepsilon} _ {V}\ text {-} 3 {\ alpha} _ {0}\ text {dT}\ text {dT}\ text {+}\ text {+}}\ frac {{\ text {dp}}} _ {s}})\]

with \({\alpha }_{0}\) the linear dilation of the skeleton (assimilable to the porous medium). It is recalled that the Biot coefficient \(b\) and the compressibility module of solid grains \({K}_{s}\) and are linked to the « drained » compressibility module of the porous medium \({K}_{0}\), such that:

\(b\text{=}1\text{-}\frac{{K}_{0}}{{K}_{s}}\)

Moreover, the variation in the density of water is written as:

(2.8)\[ \ frac {d {\ rho} _ {w}} {{\ rho}} {{\ rho}} {{w}}\ text {=}\ frac {{\ text {dp}} _ {w}} {{k}}} {{K} _ {w}} {{K} _ {w}} {K} _ {w}} {K} _ {w}} {K} _ {w} _ {w}} {K} _ {w} _ {w} _ {w} _ {k} _ {w} _ {w} _ {w} _ {w} _ {w} _ {w}}\]

with the water compressibility module \({K}_{w}\) and its expansion module \({\alpha }_{w}\).

Finally, if the law of behavior is elastic, we recall that the deformation is linked to the effective stress such that:

(2.8)\[ d {\ epsilon} _ {V} =\ frac {d\ sigma\ text {'}} {{K} _ {0}} +3 {\ alpha} _ {0} _ {0}\ text {dT}\]

Moreover, the formulation under total constraint tells us that:

\(d\sigma \text{'}\mathrm{=}d\sigma +b{\text{dp}}_{w}\), considering here that the environment is in constant confinement, we therefore have:

\(d\sigma \text{'}\mathrm{=}b{\text{dp}}_{w}\), which in the end gives us

(2.8)\[ d {\ varepsilon} _ {V}\ mathrm {=}\ mathrm {=}\ frac {b {\ text {dp}} _ {0}} _ {0}} +3 {\ alpha}}} +3 {\ alpha}} _ {0}\ text {dT}}\]

We can now inject (2), (3), (4) and (5) into equation (1) and we get that:

(2.8)\[ \ frac {{\ mathit {dm}} _ {w}} {w}} {{w}} {{\ rho}} _ {w}}\ mathrm {=} (\ frac {{b} ^ {2}}} {{K}}} {{K} _ {2}}} {{K} _ {2}}} {{K} _ {2}}} {{K} _ {0}}} +\ frac {0}}} +\ frac {0}}} +\ frac {\ varphi}} +\ frac {\ varphi}} {K} _ {w}}) {\ text {dp}}} _ {w}} +\ varphi (3 {\ alpha} _ {0}\ text {-} 3 {\ alpha} _ {w})\ text {dT})\ text {dT}\]

Considering that the medium is not drained, we therefore have:

(2.8)\[ (\ frac {{b} ^ {2}}} {{K} _ {0}}} +\ frac {(b\ mathrm {-}\ varphi)} {{K} _ {s}}} +\ frac {\ varphi}} {\ varphi}} {{K}} {{K} _ {w}}) {\ text {dp}}} _ {w}\ mathrm {=}} +\ frac {\ varphi}} {\ varphi} {{K}}} +\ frac {\ varphi}} {\ varphi} {{K}}}} _ {w}\ text {-} 3 {\ alpha} _ {0})\ text {dT}\]

This can be written in the form:

(2.8)\[ {\ text {dp}} _ {w} =\ Lambda\ text {dT}\]

With \(\Lambda\) the thermal pressurization coefficient such as:

\(\Lambda =\frac{\phi \left(3{\alpha }_{w}\text{-}3{\alpha }_{0}\right)}{\left(\frac{{b}^{2}}{{K}_{0}}+\frac{\left(b-\phi \right)}{{K}_{s}}+\frac{\phi }{{K}_{w}}\right)}\)

Note that this coefficient reveals the thermal differential \(({\alpha }_{w}-{\alpha }_{0})\)

2.1. Constant thermal expansion of water

In case \({\alpha }_{w}=\mathit{cte}\) the digital application is immediate.

Digital application:

With the data defined above, we obtain:

\(\Lambda =\mathrm{2,25}{10}^{5}\mathit{Pa.}{K}^{-1}\)

This gives for a temperature variation \(\Delta T=40°C\), a pressure variation of \(\Delta p=\mathrm{9,01}\mathit{Mpa}\) (i.e. a pressure of 13.01 MPa).

2.2. Thermal expansion of water as a function of temperature

In this case, we will write

\(\Lambda (T)=\frac{\phi \left(3{\alpha }_{w}(T)\text{-}3{\alpha }_{0}\right)}{\left(\frac{{b}^{2}}{{K}_{0}}+\frac{\left(b-\phi \right)}{{K}_{s}}+\frac{\phi }{{K}_{w}}\right)}\) and

\({\text{dp}}_{w}=\Lambda (T)\text{dT}\)

To be able to integrate this formula, we consider that the thermal expansion of water is considered to be a linear function of temperature such as \({\alpha }_{w}=\mathit{a.}T+b\). \(\Lambda (T)\) is then also a linear function that is integrated between the initial temperature T0 and the final temperature T1, which gives:

\(\Delta p=\frac{A}{2}\mathrm{.}({\mathit{T2}}^{2}-{\mathit{T1}}^{2})+B(\mathit{T2}-\mathit{T1})\) with

\(A=\frac{3\phi a}{\left(\frac{{b}^{2}}{{K}_{0}}+\frac{\left(b-\phi \right)}{{K}_{s}}+\frac{\phi }{{K}_{w}}\right)}\)

and

\(B=\frac{\phi \left(\mathrm{3b}\text{-}3{\alpha }_{0}\right)}{\left(\frac{{b}^{2}}{{K}_{0}}+\frac{\left(b-\phi \right)}{{K}_{s}}+\frac{\phi }{{K}_{w}}\right)}\)

Digital application:

We take it here

\(a=\mathrm{2,63}{10}^{-6}\) and \(b=-7{10}^{-4}\) which correspond to \({\alpha }_{w}(293)=\mathrm{6,67}\mathrm{.}{10}^{-5}{K}^{-1}\) and \({\alpha }_{w}(333)=\mathrm{1,72}\mathrm{.}{10}^{-4}{K}^{-1}\) and which are realistic values for water.

All calculations done, for a temperature variation between 20° C. and 60° C., we obtain a pressure variation of \(\Delta p=\mathrm{10,9}\mathit{MPa}\) and therefore a final pressure of \(\mathrm{14,9}\mathit{MPa}\).