2. Benchmark solution

A reference solution relating to volume deformation \(\varepsilon_v\) in the two loading phases is presented without difficulty.

In the first phase (desaturation), it is presented as a non-linear function of capillary pressure \(p_c\):

\[\]

varepsilon_v =varepsilon_ {v,1} =underbrace {-tildekappa_slnleft (1+frac {p_c} {k_Stildekappa_s}right)}} _ {varepsilon_v^e}underbrace {- (tildelambda_s}right)} _ {varepsilon_v^e}underbrace {- (tildelambda_s}right)} _ {varepsilon_v^e}underbrace {- (tildelambda_s} s-tildekappa_s)lnleft (1+frac {langle p_c-s^0_0rangle} {s^0_0+k_stildekappa_s}right)}right)}} _ {varepsilon_v^p}} _ {varepsilon_v^p},quadtext {with}quadlangle xrangle =frac {x+|x|} {2},quadtilde x =frac {x} {1+e_0}

label:

volumic_deformation_desaturation

In the second phase (isotropic compression), it is a non-linear function of net pressure \(p''=-\sigma_m''=-(\sigma_m+p_g)\) (\(p_g\) being the gas pressure):

\[\]

varepsilon_v =varepsilon_ {v,2} =varepsilon_ {v,1}

underbrace {-tildekappakappalnleft (1+frac {p « } {Ktildekappa}right)} _ {Deltavarepsilon_v^e}underbrace {- - (tildelambda (p_c) -tildekappa}right)} _ {deltavarepsilon_v^e}underbrace {- - (-tildelambda (p_c) -tildekappa}right)} _ {deltavarepsilon_v^e}underbrace {- - (-tildelambda (p_c) -tildekappa)lnleft (1+frac {lang » -tilde p_ {con} (p_c,varepsilon^p_ {v,1})rangle} {tilde p_ {con} (p_c,varepsilon^p_ {v_ {v_ {v_ {v,1})}}right)} _ {Deltavarepsilon_v^p}

label:

volumical_deformation_compression

where:

\[\]

tildelambda (p_c) =tildelambda_0left ((1-r)exp (-beta p_c) +rright),quad

tilde {p} _ {con} (p_c,varepsilon_ {varepsilon_ {v,1} ^p) = p_rleft (cfrac {p^0_ {con}expleft (-cfrac {varepsilon_ {varepsilon_ {v,1} ^p) = p_rleft (cfrac {p^0_ {con}expleft (-cfrac {varepsilon_ {varepsilon_ {v,1} ^p} {tilde {lambda} _0-tilde {kappa}}right)} {p_r}right) ^cfrac {tilde {lambda} _0-tilde {kappa}} {tilde {lambda} (p_c) -tilde {kappa}} :label: detail_terms