One-dimensional, third power type non-darcy flows trough porous media

Author: M. Sławomirski

In the paper one-dimensional confi ned non-Darcy’s fl ow through porous media has been considered.
Basing on rational considerations and empirical investigations presented in the literature the author has
assumed the third power type relationship between the seepage velocity u and pressure gradient grad P. The
considered relationship represented by Eq. (9) involves two parameters K and β typical of a given porous
material. The fi rst of them is a homologue of permeability coeffi cient whereas the second one expresses the
deviation from the Darcy’s law. The author analysed the rectilinear fl ow, the fl ows through layered porous
medium, the radial fl ow in the vicinity of a single well, and the spherical fl ow. It has been demonstrated
that the third power type equation (9) may effi ciently be applied for the description of non-Darcy’s fl ows
through porous media, and equations describing various one-dimensional non-Darcy’s fl ows may uniquely
be solved applying direct methods. The solutions involve often the Cardano formulae for the roots of
cubic equations. For all cases considered in this paper the discriminants of cubic equations are positive.
Eeach of equations possesses then three roots: two complex roots, and one real root. Complex roots do not
possess a physical sense whereas the real root represents the genuine solution of the problem.The author
proved that for the simple rectilinear non-Darcy’s fl ow the seepage velocity depends on pressure drop ΔP
in the non-linear mode represented by Eq. (15). Two detailed cases of the rectilinear fl ow through layered
medium were considered: the fl ow the direction of which is parallel to the strata plane, and the fl ow the
direction of which is normal to the strata plane For the fi rst case the fl ow rate is the sum of fl ow rates through
successive strata (cf. Eqs. (17), (18)). For the second case the author obtained the non-linear relationship
(Eqs. (30), (31)) between the seepage velocity u, overall pressure drop per distance, and harmonic means
of Ki parameters for successive strata. The radial fl ow has been considered in a vicinity of a single well.
It has been demonstrated that in a well drainage zone the dependence of pressure P on the distance from the well axis r is represented by the sum of the logarithmic and the negative second power functions (Eq. (49)). Moreover, the dependence the pressure drop in the drainage zone Pe – Pw on the well flow rate Q is on-linear, and it is represented by the third power type relationship (Eq. (51)). The formula for a single well production has been obtained in the form (55). Finally, the spherical type fl ow has been considered.The author has demonstrated that for the spherical flow the dependence of pressure P on the distance fromthe centre of the sphere r is represented by the sum of the negative power terms.

Keywords: flow through porous media, non-Darcy’s flow, low Reynolds numbers hydrodynamics