SPE 89429 Theoretical Development of the Brooks-Corey Capillary Pressure Model from Fractal Modeling of Porous Media

The capillary pressure model proposed empirically by Brooks and Corey has been used widely for several decades. However it is not clear why the Brooks-Corey capillary pressure model works so well. In this study, it has been found that the empirical Brooks-Corey capillary pressure model can be derived theoretically from fractal modeling of porous media. Also found was the correlation between the pore size distribution index in the Brooks-Corey capillary pressure model and the fractal dimension. The pore size distribution index increases with the decrease in fractal dimension of the porous media. Capillary pressure curves of different types of rock samples were measured using a mercury intrusion technique. The values of pore size distribution index and fractal dimension were calculated. The relationship between the two parameters obtained from the experimental data was consistent with the relationship derived theoretically. This implies that the fractal dimension of porous media may be inferred directly using the Brooks-Corey capillary pressure model instead of the fractal model. The theoretical development in this study demonstrates that the Brooks-Corey capillary pressure model, once considered as empirical, has a solid theoretical base. This may be why the Brooks-Corey capillary pressure model works satisfactorily in many cases. Introduction Capillary pressure plays an important role in many recovery processes. It is essential to represent capillary pressure curves properly. The frequently-used model to express a capillary pressure curve mathematically is the Brooks-Corey capillary pressure model. Brooks and Corey conducted analysis for the capillary pressure curves of a large number of consolidated core samples. The capillary pressure curves were measured using a desorption approach. Brooks and Corey found that the relationship between the capillary pressure and the normalized or effective wetting-phase saturation was a straight line on a log-log plot. The mathematical expression of this relationship was known as the Brooks-Corey capillary pressure model later. Residual wetting-phase saturation must be known or assumed to calculate the normalized or effective wetting-phase saturation. In the analysis by Brooks and Corey, the residual wetting-phase saturation was chosen such that the data fit as closely as possible to a straight line when plotted on log-log paper. The Brooks-Corey capillary pressure model works satisfactorily in many cases and has been utilized widely for several decades in petroleum and other industries. However it is not clear why the Brooks-Corey capillary pressure model works so well. Note that the capillary pressure model was proposed empirically by Brooks and Corey. Many researchers have studied the fractal nature of reservoir rocks and other porous media in the past two decades. It has been found that most natural porous media such as reservoir rock are fractals and can be characterized using a fractal model or a fractal curve which represents the relationship between the number of pores and the radius of pores. Such a fractal curve is a straight line on a log-log plot and the slope of the straight line is referred to as the fractal dimension of the porous media. The magnitude of fractal dimension is a representation of the heterogeneity of the porous medium. The greater the fractal dimension, the greater the heterogeneity of the porous media. Note that the pore size distribution index in the Brooks-Corey capillary pressure model is also a representation of the heterogeneity of porous media. The greater the pore size distribution index, the more homogeneous the porous medium. Attention has also been paid to the application of fractal modeling of porous media in reservoir engineering. The applications include the development of relative permeability models, capillary pressure models, and the models to predict oil production rate, etc. The author reviewed the literature in this area in a previous paper. The review shows that the fractal modeling of porous media is a powerful tool to characterize the heterogeneity of porous media and to study fluid flow mechanisms. In this study, we conducted a theoretical development based on the fractal geometry to derive the Brooks-Corey capillary pressure model. Capillary pressure curves of Berea, chalk, and reservoir sandstone were measured using a mercury intrusion technique to infer the fractal dimension. The values of fractal dimension were calculated using the fractal model