Dipole-flow disturbed by a circular inclusion of conductivity different from the background: From deterministic to a self-consistent analytical solution

Steady dipole-flow through a porous medium, and disturbed by a circular inclusion ${\mathrm{\Omega }}_0$ of conductivity different from the background, is solved analytically. The solution is achieved by means of the circle theorem, which is reformulated to account for the entry/leave of mass and energy through the boundary $\partial {\mathrm{\Omega }}_0$. It is shown that the governing potential is that which one would consider in absence of the disturbance supplemented with an ad hoc (fictitious) dipole laying inside ${\mathrm{\Omega }}_0$. Besides the theoretical interest, the analytical solution is used to compute the effective conductivity ${\mathrm{K}}^{\mathrm{eff}}$, by means of the self-consistent approximation. Overall, ${\mathrm{K}}^{\mathrm{eff}}$ is found to depend upon the flow configuration, and therefore it cannot be sought as a medium's property (nonlocality). In particular, ${\mathrm{K}}^{\mathrm{eff}}$ depends upon the joint probability density function $f$ of the conductivity and the distribution/size of the inclusions. Results, analyzed for a fairly general model of $f$, demonstrate that the coefficient of correlation $\rho$ between the involved random fields is the key parameter characterizing the structure of ${\mathrm{K}}^{\mathrm{eff}}$. Indeed, the latter results larger or smaller than that of the background, depending on whether $\rho$ is negative or positive, respectively. For $\rho =0$, the effective conductivity is a local property and, in this case, one can apply the superposition principle with the homogeneous conductivity replaced by the geometric mean.

Figure 1

Geometry of a dipole with the source (red) and the sink (blue) at (0,0) and $(\ell ,0)$, respectively. The flow is disturbed by a circular inclusion ${\mathrm{\Omega }}_0$ of radius ${\mathcal{R}}_0$ and center $z_0$. The positions $z_0^{\prime }$ and $z_0^{″}$ identify the fictitious dipole lying inside ${\mathrm{\Omega }}_0$, whereas $z$ is the current position

Figure 2

Geometrical sketch pertaining to the fulfillment of the mass and energy conservation along the boundary $\partial {\mathrm{\Omega }}_0$ of the inclusion.

Figure 3

Contour plot of the scaled hydraulic head $h{K}_{\infty }/\sout{Q}$ for two largely different values of the contrast ratio $\kappa =K_0/K_{\infty }$. The center of the inclusion (whose radius is ${\mathcal{R}}_0=\ell /4$) is placed between the sink and the source.

Figure 4

Normalized EC as function of the variance ${\sigma }_Y^2$, and several values of the correlation coefficient $\rho$. The EC pertaining to a mean uniform flow (green, thick, dashed line) is also shown. Other parameter: $\omega \equiv Z_{<}/{\sigma }_Z=0.1$. The inset shows the comparison between numerical simulations (symbols) and analytical self-consistent model (lines) of the scaled EC.