The effect of the quadratic gradient term on variable-rate well-tests

Conventional welltest models for liquid flow are not consistent with material balance. The quadratic gradient term in the non-linear diffusion equation is usually neglected in light of the slightly compressible fluid assumption. This approach is questionable for live oil and low permeability reservoirs. It is already known that linearization by neglecting terms may lead to errors for large values of time. The material balance error is compounded by the traditional skin model which implies the assumption of incompressible flow into the wellbore. A method that is consistent with material balance is proposed. All terms in the non-linear diffusion equation are retained. In addition a skin model that accommodates compressible flow across the skin zone is used. The conceptual skin models for incompressible and compressible flow are different, but they are characterized by the same numerical value. A short discussion of this difference is provided. The resulting model, which is based on source and Greenʹs functions, is well suited for restricted entry problems. Results for a partially penetrated well in homogeneous reservoirs is presented. The effect of damage skin and wellbore storage is included. This study shows that the linearization error depends on: (1) the magnitude of wellbore damage, (2) the pseudo-skin resulting from partial penetration and (3) a non-linear flow parameter, α. The semi-log slope, which has traditionally been used to compute the reservoir permeability, may depend on the skin factor.