Polynomial solutions of Bloch NMR flow equations for classical and quantum mechanical analysis of fluid flow in porous media

Abstract

In many NMR experiments it is noticed that liquids confined in porous materials exhibit properties that are different from those of the bulk fluid. Determining the relationship between macroscopic properties and the microscopic structure of porous materials and their components has been difficult. Despite decades of study, researchers’ understanding is generally limited to empirical correlations based on laboratory measurements. Only recently have researchers been able to calculate a few of the macroscopic physical properties of rocks from first principles. In this contribution, we have presented polynomial solutions of the Bloch NMR flow equations for classical and quantum mechanical analysis of porous media applicable in oil and gas industry. The NMR polynomials are derived based on the condition that 1 2 2 1 2 1 ( ) T T B x << . We transform the transverse magnetization My, into porous medium with a transformation constant a , which must be positive. The direction of the positive constant a, can only take on one of = (2n+1) possible values. The lowest allowed value of a, with n = 0 is not zero. This implies that, even in a very complicated porous medium, the lowest allowed value of a always has a T2 relaxation parameter. This is valid both in classical and quantum mechanical analysis of porous media. In addition to relating the porosity with depth, the mathematical procedure yield information on the dynamics of various parts of the porous media. The transverse relaxation rates R2n are the eigenvalues of the porous medium with n as the quantum number. It determines the specific porous state which the fluid particles occupy within the pore. The state, and therefore T2 relaxation rate is quantized, determined by the quantum number n which is any integer greater than or equal to zero.