Swelling Gel

The simple swelling gel model is based on the idealization of a gel as a volume of individual spherical particles of equal radius, $r_a$. The swelling evolution (discussed in detail in Constitutive behavior in a porous medium) is assumed to be given by

where the value of any grouping of terms in angled brackets $⟨⟩$ is set equal to zero if its mathematical result is not positive, and

$r_a^f$

is the fully swollen radius;

${\tau }_1$

is the relaxation time of the gel particles;

s

is the saturation of the surrounding medium;

$r_a^{\mathrm{dry}}$

is the radius of the gel particles when they are completely dry;

$r_a^t={\left(\frac{n^{0}J}{4\sqrt{2}{k}_a}\right)}^{\frac{1}{3}}$

is the maximum radius that the gel particles can achieve before they must touch;

$r_a^s={\left(\frac{3}{4\pi }\frac{n^{0}J}{k_a}\right)}^{\frac{1}{3}}$

is the effective gel radius when the volume is entirely occupied with gel;

$n^0$

is the initial porosity of the material;

J

is the volume change in the material; and

$k_a$

is the number of gel particles per unit volume.

The second term in the definition of gel growth incorporates the assumption that the gel will swell only when the saturation of the surrounding medium, s, exceeds the effective saturation of the gel. The third term in the growth equation reduces the swelling rate when the surface of gel particles exposed to free fluid is limited by the combination of packing density and gel particle radius.

The swelling gel model is defined by specifying the variables $r_a^{\mathrm{dry}}$, $r_a^f$, $k_a$, and ${\tau }_1$.

GEL