Local boundary conditions for NMR-relaxation in digitized porous media

Abstract

We narrow the gap between simulations of nuclear magnetic resonance dynamics on digital domains (such as CT-images) and measurements in D-dimensional porous media. We point out with two basic domains, the ball and the cube in D dimensions, that due to a digital uncertainty in representing the real pore surfaces of dimension D − 1, there is a systematic error in simulated dynamics. We then reduce this error by introducing local Robin boundary conditions.

Numerical simulations of NMR relaxation in chalk using local Robin boundary conditions

“…This study extends our previous work [15], that used local boundary conditions (LBC) to locally adapt the probability for surface relaxation, p S , when a NMR excitation encounters the pore surface in the porous medium. For simple domains we have shown that this method closely reproduces the solutions to the corresponding PDE model for which the surface relaxation is controlled by the surface relaxation parameter ρ via a Robin boundary condition [15]. It is clear that development towards more accurate numerical modeling benefit from predictions on well characterised samples, and the purpose of this article is to first carry out benchmarking simulations and then produce results for natural chalk samples.…”

“…With the true (lowest) eigenvalues, λ j , for the ball and cube with physical parameters such as ρS/V = 3 and ρ = R 0 = D 0 = 1 reported in Table 1, we can compare the approximation in (15) 3) and (4)) is increasing by a factor 100 from (a) to (b), and from (b) to (c).…”

“…We see that LLBC improves the result for the ball, while a systematic deviation is still present. To further illustrate the dramatic effect of the digitized surface of the ball (sphere) we have also plotted numerical results obtained from a radial random walk, which is the same data as the "D = 3"-curve in Figure 1 (a) of [15], that converge to the analytic solution given a small enough discretization ∆r of the radius of the ball. 17), with 27 terms included in the sum such that 1− j b j 10 −6 , is shown with (green) dots.…”

“…The second of the two simple three-dimensional geometries we consider is the cube. While the results of the D-dimensional case were derived for dimensionless variables in [15] for different initial conditions with help of Sturm-Liouville theory, we here give explicitly the total magnetisation for the cube with uniform initial conditions…”

“…We will use a simple linear interpolation of the geometry, which is then merged into the boundary conditions for surface relaxation. The general concept was introduced in [15] and was there described in detail for two spatial dimensions under the name linear local boundary conditions (LLBC).…”