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.

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• Convection in porous media
• Laminar flows
• Porous media
• Stokes equations