Benchmark solution
=====================

A reference solution relating to volume deformation :math:`\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 :math:`p_c`:

.. math::
\ varepsilon_v =\ varepsilon_ {v,1} =\ underbrace {-\ tilde\ kappa_s\ ln\ left (\ frac {p_c+k_s\ tilde\ kappa_s} {p_c (0) +k_s\ tilde\ kappa_s}\ left)} _ {\ varepsilon_v_s}\ right)} _ {\ varepsilon_v_s ^e},\ quad\ text {with}\ quad\ tilde x =\ frac {x} {1+e_0}
    :label: volumical_deformation_saturation

where :math:`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 :math:`p''=-\sigma_m''=-(\sigma_m+p_g)` (:math:`p_g` being the gas pressure):

.. math::
\ varepsilon_v =\ varepsilon_ {v,2} =\ varepsilon_ {v,1}
     \ underbrace {-\ tilde\ kappa\ kappa\ ln\ left (1+\ frac {p "} {K\ tilde\ kappa}\ right)} _ {\ Delta\ varepsilon_v^e}\ underbrace {- (- (\ tilde\ lambda_0-\ tilde\ kappa}\ right)} _ {\ delta\ varepsilon_v^e}\ underbrace {- - (- (\ tilde\ lambda_0-\ tilde\ kappa}\ right)} _ {\ delta\ varepsilon_v^e}\ underbrace {- - (- (\ tilde\ lambda_0-\ tilde\ kappa}\ right)} _ {\ delta\ varepsilon_v^e} ^0_ {con}\ rangle} {p_ {con} ^0} ^0}\ right)} _ {\ Delta\ varepsilon_v^p},\ quad\ text {with}\ quad\ langle x\ rangle =\ frac {x+|x|} {2}
    :label: volumical_deformation_compression

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

.. math::
\ varepsilon_v =\ varepsilon_ {v3,} =\ varepsilon_ {v,2}\ underbrace {-\ tilde\ kappa_s\ ln\ left (1+\ frac {p_c} {1+\ frac {p_c} {k_s} {k_s\ tilde\ kappa_s}\ right)} _ {\ Delta\ varepsilon_v^e}\ left (1+\ frac {p_c} {k_c} {k_s} {k_s\ tilde\ kappa_s}\ right)} _ {\ Delta\ varepsilon_v^e}\ underbrace {- (\ tilde\ lambda_s-\ tilde\ kappa_s)\ tilde\ kappa_s)\ ln\ left (1+\ frac {\ langle p_c-s_0 (\ varepsilon_ {v,2} s_0 (\ varepsilon_ {v,2} ^p)}\ right)} _ {\ Delta\ varepsilon_ {v,2} ^p)} _ {\ Delta\ varepsilon_ {v,2} ^p)} _ {\ Delta\ varepsilonon_ {v,2} ^p)} _ {\ Delta\ varepsilonon_ {v,2} ^p)} _ {\ Delta\ varepsilonon_ {v,2} ^p)} _ {\ Delta\ vare_v^p},\ quad\ text {with}\ quad s_0 (\ varepsilon_ {v,2} ^p) = s_0^0\ exp\ left (-\ frac {\ varepsilon_ {v,2} ^p} {\ tilde\ lambda_s-\ tilde\ kappa_s}\ left (-\ frac {\ varepsilon_ {v,2} ^p} {v,2} ^p} {\ tilde\ lambda_s-\ tilde\ kappa_s}\ right)
    :label: volumic_deformation_desaturation