Multiphase flow in porous media is governed by a system of degenerate parabolic PDEs. We discuss approximation of the scalar degenerate parabolic PDE u_t + div[F(u) - D(u) grad b(u)] = 0 using weighted essentially non-oscillatory (WENO) finite volume reconstructions. The solution may exhibit steep fronts, which are difficult to approximate even in one space dimension, since at the front the normal derivative is essentially a delta function distribution. Generalizing earlier work on the subject [1,2], we present two approaches to approximating the diffusive flux. The first uses a sampling line to approximate the normal diffusive flux on each mesh cell facet. It is a more general and simpler procedure than that described in [2], allowing unstructured meshes. The second approach utilizes a diffusive numerical flux that combines information reconstructed on each side of a facet. These approaches can be used in any spatial dimension on general unstructured computational meshes using, e.g., the newly introduced multilevel WENO reconstruction [3]. Any finite volume discretization will require a mesh or submesh of spacing d, so the normal derivative at a front will be approximated by a value that is O(1/d), independently of the equation being approximated. We present analytical and numerical studies of the problem and show that, although the computed flux may be inaccurate, over a few time steps there is little effect on the computed solution for our methods. [1] Y. Liu, C.-W. Shu, M. Zhang, High order finite difference WENO schemes for nonlinear degenerate parabolicequations, SIAM Journal on Scientific Computing, Vol. 33 (2),pp. 939-965, 2011. [2] T. Arbogast, C.-S. Huang, X. Zhao, Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes, J. Comput. Phys., Vol. 399, pp. 108921, 2019. [3] T. Arbogast, Chieh-Sen Huang, and Chenyu Tian, A finite volume multilevel WENO scheme for multidimensional scalar conservation laws, Comput. Methods Appl. Mech. Engrg., Vol. 421, pp. 11681, 2024.