Identification of local contact angle distribution inside a porous medium from an inverse optimization procedure

The local contact angle distribution within the pore space of a porous medium of mixed wettability is identified by combining pore network simulations and an optimization method. The latter is used to solve the inverse problem consisting of determining the local contact angle distribution from the capillary pressure and relative permeability curves. The inverse optimization method combines a genetic algorithm and the hill-climbing algorithm. The method is illustrated considering a Boolean distribution of the local contact angle in which the pore bodies and throats in the pore space are either hydrophilic or hydrophobic.

Figure 1

A $12\times 5$ square pore network. Throats and pore bodies in brown are hydrophilic ($\theta ={80}^{\circ }$). Throats and pore bodies in gray are hydrophobic ($\theta ={115}^{\circ }$). A hydrophilic element percolating path, connecting the network inlet (at the bottom) to outlet (at the top), is illustrated.

Figure 2

Fluid distribution for various values of the hydrophilic pore body and throat fraction, $f$, in a 2D $30\times 30$ square network. (a) $f=0$; (b) $f=0.1$; (c) $f=0.2$; (d) $f=0.3$; (e) $f=0.4$; (f) $f=0.5$; (g) $f=0.6$; (h) $f=0.7$; (i) $f=0.8$; (j) $f=0.9$; and (k) $f=1$. The displacing fluid (in blue) is injected from the bottom using the capillary-pressure algorithm. The displaced fluid (in gray) escapes from the top. Patterns correspond to fluids distributions at breakthrough.

Figure 3

Capillary-pressure curves computed for a $30\times 30$ network for several values of the fraction, $f$, of hydrophilic elements in the network. The curve corresponding to $f=0.5$ is represented with a thicker black line. (a) The top curve corresponds to $f=0$ (fully hydrophobic network). (b) The bottom curve corresponds to $f=1$ (fully hydrophilic network).

Figure 4

Displacing fluid relative permeability curves computed for the same $30\times 30$ network as for Fig. 3 for several values of the fraction $f$ of hydrophilic elements in the network. The curve at the far right in the figure on the left corresponds to $f=0$ (fully hydrophobic network). The curve at the far left in the figure on the right corresponds to $f=1$ (fully hydrophilic network). The curve corresponding to $f\approx 0.5$ is represented with a thicker black line.

Figure 5

Comparison between the relative permeability curves (left column) and capillary-pressure curves (right column) of the reference and optimized networks. (a), (b) $f=30\%$; (c), (d) $f=50\%$; and (e), (f) $f=70\%$.

Figure 6

Distribution of hydrophilic (in blue) and hydrophobic (in light gray) elements in the reference network (left column) and optimized network (right column). (a), (b) $f=30\%$; (c), (d) $f=50\%$; and (e), (f) $f=70\%$.