2. Benchmark solution

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

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

\[\]

varepsilon_v =varepsilon_ {v,1} =underbrace {-tildekappa_slnleft (frac {p_c+k_stildekappa_s} {p_c (0) +k_stildekappa_s}left)} _ {varepsilon_v_s}right)} _ {varepsilon_v_s ^e},quadtext {with}quadtilde x =frac {x} {1+e_0}

label:

volumical_deformation_saturation

where \(p_c(0)=100\) kPa is the initial capillary pressure.

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_0-tildekappa}right)} _ {deltavarepsilon_v^e}underbrace {- - (- (tildelambda_0-tildekappa}right)} _ {deltavarepsilon_v^e}underbrace {- - (- (tildelambda_0-tildekappa}right)} _ {deltavarepsilon_v^e} ^0_ {con}rangle} {p_ {con} ^0} ^0}right)} _ {Deltavarepsilon_v^p},quadtext {with}quadlangle xrangle =frac {x+|x|} {2}

label:

volumical_deformation_compression

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

\[\]

varepsilon_v =varepsilon_ {v3,} =varepsilon_ {v,2}underbrace {-tildekappa_slnleft (1+frac {p_c} {1+frac {p_c} {k_s} {k_stildekappa_s}right)} _ {Deltavarepsilon_v^e}left (1+frac {p_c} {k_c} {k_s} {k_stildekappa_s}right)} _ {Deltavarepsilon_v^e}underbrace {- (tildelambda_s-tildekappa_s)tildekappa_s)lnleft (1+frac {langle p_c-s_0 (varepsilon_ {v,2} s_0 (varepsilon_ {v,2} ^p)}right)} _ {Deltavarepsilon_ {v,2} ^p)} _ {Deltavarepsilon_ {v,2} ^p)} _ {Deltavarepsilonon_ {v,2} ^p)} _ {Deltavarepsilonon_ {v,2} ^p)} _ {Deltavarepsilonon_ {v,2} ^p)} _ {Deltavare_v^p},quadtext {with}quad s_0 (varepsilon_ {v,2} ^p) = s_0^0expleft (-frac {varepsilon_ {v,2} ^p} {tildelambda_s-tildekappa_s}left (-frac {varepsilon_ {v,2} ^p} {v,2} ^p} {tildelambda_s-tildekappa_s}right)

label:

volumic_deformation_desaturation