A generalized methodology is presented that obtains analytical solutions for imbibition waterfloods in naturally fractured oil reservoirs undergoing multi-step matrix-to-fracture transfer processes. The phenomenological representation of the oil transfer from matrix to fracture is based on a three-exponential matrix-to-fracture transfer function, the necessity for which is seen by examination of experimental data. The resulting in-tego-differential equation is converted to a fourth-order partial differential equation, linearized by invoking the unit end-point mobility ratio approximation, and then solved analytically by asymptotic means. It is shown that the asymptotic-approximation approach significantly reduces the complexity of the solution process and yields adequate solutions for longtime evaluation of waterfloods in naturally fractured reservoirs. The solution is not only computationally advantageous, but it also provides a physically meaningful interpretation of the propagation speed and diffusive spreading of the progressing wave front, which could not be readily obtained from the usual type of solution methods, such as the direct application and inversion of the Laplace transformation.

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