Discussion on seepage governing equations for low permeability reservoirs with a threshold pressure gradient

Low permeability reservoirs account for an increasing proportion of oil production. Threshold pressure gradient is an important factor that governs the flow in low permeability porous media. The 1-D seepage governing equation (SGE) for low permeability porous media can be derived from the 1-D core flooding experimental rule. In the literature, for isotropic porous media, the SGE with a threshold pressure gradient (TPG) in Cartesian and cylinder coordinate systems are incompatible to each other. In addition, irrational results were found in simulation using SGEs in the Cartesian coordinate system. In this study, 3-D SGEs with a TPG in the Cartesian coordinate system and for radial flow in the cylindrical coordinate system are derived from the vector form of the seepage velocity in 3-D domain which is transformed from the 1-D seepage velocity vector. The 1-D equation degenerated from the 3-D SGE of low permeability media is in accordance with the 1-D SGE. The derived SGE of low permeability porous media in Cartesian coordinate systems is consistent with that in cylindrical coordinate systems. So, the contradiction of SGEs with a TPG in literature is resolved. For anisotropic reservoirs with a TPG, with the assumption that the impeding of a TPG to flow in porous media occurs in the opposite direction of the seepage velocity vector, the general seepage initiation condition for anisotropic porous media with a TPG is derived. The SGEs for anisotropic porous media with a TPG under a specific condition in Cartesian coordinate systems and for radial flow in cylindrical coordination the systems are derived, and then are degenerated to isotropic cases. It is found that a simple form of the SGE anisotropic porous media with a TPG can only be derived when the flow is radial. So, it is suggested that numerical simulations for anisotropic porous media with a TPG should use the equation set composed by the pressure and seepage velocity vector. The analysis also indicates that a TPG of anisotropic reservoirs is a two-order tensor, and cannot be represented by a vector. However, the current form of effective pressure gradient requires further investigation.