fronts.D.letxs

fronts.D.letxs(Lw, Ew, Tw, Ls, Es, Ts, Ks=None, k=None, nu=1e-06, g=9.81, alpha=1.0, theta_range=(0.0, 1.0))

Return a diffusivity function that combines the LETx relative permeability correlation and the LETs capillary pressure correlation for spontaneous imbibition. Both correlations are part of the LET family of hydraulic functions.

Given the saturated hydraulic conductivity \(K_S\), irreducible water saturation \(S_{wir}\), capillary pressure \(P_{cir}\) at irreducible saturation, and shape parameters \(L_w\), \(E_w\), \(T_w\) and \(L_s\), \(E_s\), \(T_s\); the LET-based diffusivity function \(D\) is defined as:

\[D(\theta) = -K_S K_{rw}(\theta) d P_c/d \theta\]

where the variable \(\theta\) is moisture content.

The functions \(K_{rw}\) and \(P_c\) are, respectively, relative permeability and capillary pressure, defined as:

\[K_{rw} = \frac{S_{wp}^{L_w}}{S_{wp}^{L_w} + E_w (1 - S_{wp})^{T_w}}\]
\[P_c = P_{cir} \frac{(1 - S_{ws})^{L_s}}{(1 - S_{ws})^{L_s} + E_s S_{ws}^{T_s}}\]

with:

\[S_{wp} = S_{ws} = \frac{\theta - \theta_r}{\theta_s - \theta_r}\]

and:

\[P_{cir} = \frac{\rho g}{\alpha}\]
Parameters:

  • Lw (float) – \(L_w\) parameter for the LETx correlation.

  • Ew (float) – \(E_w\) parameter for the LETx correlation.

  • Tw (float) – \(T_w\) parameter for the LETx correlation.

  • Ls (float) – \(L_s\) parameter for the LETs correlation.

  • Es (float) – \(E_s\) parameter for the LETs correlation.

  • Ts (float) – \(T_s\) parameter for the LETs correlation.

  • Ks (None or float, optional) – \(K_S\), the saturated hydraulic conductivity. Must be positive. If neither Ks nor k are given, the saturated hydraulic conductivity is assumed to be 1.

  • k (None or float, optional) – Intrinsic permeability of the porous medium. Can be given in place of Ks, which results in the saturated hydraulic conductivity being computed using \(K_S = kg/\nu\). Must be positive.

  • nu (float, optional) – \(\nu\), the kinematic viscosity of the wetting fluid. Only used if k is passed instead of Ks. Must be positive. Defaults to 1e-6, approximately the kinematic viscosity of water at 20°C in SI units.

  • g (float, optional) – Magnitude of the gravitational acceleration. Only used if k is passed instead of Ks. Must be positive. Defaults to 9.81, the gravity of Earth in SI units.

  • alpha (float, optional) – \(\alpha\) parameter. The default is 1. Must be positive.

  • theta_range (sequence of two floats, optional) – (\(\theta_r\), \(\theta_s\)), where \(\theta_r\) is the minimum (also known as residual) water content and \(\theta_s\) is the maximum water content. The default is (0, 1). \(\theta_s\) must be greater than \(\theta_r\).

Returns:

D –

Function to evaluate \(D\) and its derivatives:

  • D(theta) evaluates and returns \(D\) at theta

  • D(theta, 1) returns both the value of \(D\) and its first derivative at theta

  • D(theta, 2) returns the value of \(D\), its first derivative, and its second derivative at theta

In all cases, the argument theta may be a single float or a NumPy array.

Return type:

callable

References

[1] LOMELAND, F. Overview of the LET family of versatile correlations for flow functions. In: Proceedings of the International Symposium of the Society of Core Analysts, 2018, p. SCA2018-056.

[2] GERLERO, G. S.; VALDEZ, A.; URTEAGA, R; KLER, P. A. Validity of capillary imbibition models in paper-based microfluidic applications. Transport in Porous Media, 2022, vol. 141, no. 7, pp. 1-20.