Modeling Transverse Imbibition in Double-Porosity Simulators

Abstract

Abstract Published imbibition experiments of an advancing fracture water level surrounding a single matrix block are simulated using various double porosity models. These double porosity formulations are inherently unable to represent transverse imbibition with an advancing water level in the fracture as they are based essentially on one dimensional flow. Gross matches of the experimental and double porosity simulations required shape factors that were unrepresentative of the matrix block size. Increasing the fracture water injection rate required increasing the shape factor in order to achieve gross matches, hence a constant shape factor does not give the best matches with experimental data at different rates. A new double porosity transfer function for transverse imbibition is presented based on a nonlinear diffusion equation model for imbibition and the effective exposure time of elements of the matrix block to fracture water. This formulation uses the physical dimensions of the matrix block directly, i.e. no shape factor, and was effective in matching the experimental data without tuning any parameters. A comparison of a cubical single matrix undergoing fracture waterflooding was made between the various double porosity formulations and a fine grid, single porosity simulation. The conventional double porosity simulations did not this fine grid simulation as well as the proposed double porosity formulation. Introduction Naturally fractured reservoirs present many challenges from a numerical modeling point of view. The bulk of the storage of a naturally fractured reservoir resides in the matrix blocks while the fracture network makes up the dominant reservoir flow paths. The overall performance of any naturally fractured reservoir should then be quite sensitive to rate of fluid exchange between the storage body and the reservoir flow paths. This exchange, the matrix/fracture transfer function, is a critical component of any mathematical model used for the simulation of these reservoirs. In this paper, we will present a model for incorporating the effect of transverse imbibition into the matrix/fracture transfer function. We will test the proposed model with two cases, the published experimental work of Kleppe and Morse and a fine grid simulation of a cubical matrix block undergoing fracture waterflooding. Comparisons of the double porosity imbibition modeling methods presented by Litvak and by Thomas et al. will also be made to these two test cases. Warren and Root presented equations for unsteady-state single phase flow in double porosity reservoirs. Their double porosity domain assumes a continuous. uniform fracture network oriented parallel to the principal axes of permeability. The matrix blocks in this system occupy the same physical space as the fracture network and are assumed to be identical rectangular parallelepipeds with no communication between matrix blocks. Matrix blocks are also assumed to be isotropic and homogeneous. Assuming pseudosteady-state flow between the matrix elements and the fracture network, Warren and Root presented analytical solutions of these double porosity equations for pressure build-up tests. Their matrix/fracture transfer rate is controlled by a geometrical shape factor which is a function of the surface-volume ratio of the matrix blocks. Extension and numerical solution of the Warren and Root double porosity model to multiphase flow in three dimensions was done by Kazemi et al. Attempts to match water imbibition into artificially fractured cores by Kazemi and Merrill showed some success if the matrix block was divided into subdomains. As capillary pressure curves were not measured on the cores, capillary pressure was a parameter that was modified to help fit the recovery curves. cater enhancements of the matrix/fracture transfer function by Gilman and Kazemi included a more realistic gravity potential between the matrix and fracture and the ability to account for fluid transfer to the matrix system due to an imposed pressure gradient in the fracture. The improved gravity transfer was accomplished by gridding the matrix block into subdomains. P. 155^