## 2 Theoretical Background of NMRI Analysis  
The result from NMRI analysis includes the \(T_{2}\) spectrums from low field nuclear magnetic resonance (LFNMR) measurements and \(T_{2}\)-weighted image from magnetic resonance imaging (MRI) measurements [27]. The NMR scan was consistent with the physical phenomenon that a spin nuclei system (e.g., protons) in a strong magnetic field absorbs energy from externally matched radio frequency (RF) field energy, which is identical to the spin nuclear system [28; 29]. The NMRI technique is easily affected by hydrogen fluids in porous media as a non-invasive method to characterize the pore distribution of residual soils. The signal amplitude can indicate the present protons in water or can be associated with pore characteristic while the relaxation time (\(T_{2}\)) is a measure of the rate at which the precession of hydrogen nuclei in the formation pore water gradually decay in the presence of an inhomogeneous magnetic field, providing information of pore structures [30]. The value of \(T_{2}\) is complex since it is regulated by three relaxation mechanisms (the bulk, surface and diffusion relaxations). Given this, the total transverse relaxation time in the porous media can be expressed by:  
\[\frac{1}{T_{2}}=\frac{1}{T_{2B}}+\frac{1}{T_{2S}}+\frac{1}{T_{2D}}, \tag{1}\]  
where \(T_{2}\) denotes the total relaxation time of the water in pores that was ascertained by the Carr-Purcell-Meiboom-Gill (CPMG) sequence; \(T_{2B}\) is the bulk fluid relaxation time; \(T_{2S}\) is the surface relaxation time and \(T_{2D}\) indicates the diffusion relaxation time and accounts for the transverse relaxation in a uniform magnetic field. Equation (1) suggests that the total \(T_{2}\) relaxation time is determined by pore sizes, fluid properties and testing parameters, as well as the magnetic field properties. Nevertheless, the diffusion relaxation can be ignored in a low and uniform magnetic field. Besides, the bulk relaxation can also be ignored because it is much more time-consuming than surface relaxation for non-viscous fluid within nonmagnetic materials. Thus, the \(T_{2}\) relaxation time ascertained in this test is primarily regulated by the surface relaxation, so Equation (1) can be simplified as [31]:  
\[\frac{1}{T_{2}}=\frac{1}{T_{2S}}=\rho_{2}(\frac{S}{V})_{\textit{pore}}, \tag{2}\]  
where \(\rho_{2}\) refers to the surface relaxation coefficient, associated with the specific combination of pore fluid and mineral grains for a particular soil. It is also proved that \(\rho_{2}\) values are proportional to the pore sizes in a linear manner. \((\frac{S}{V})_{\textit{pore}}\) denotes the ratio of the surface area \(S\) to the pore volume \(V\). and the value of \((\frac{S}{V})_{\textit{pore}}\) is dependent of the pore shape. According to a study by [PERSON] et al. [32], pores in soils can be considered some cylindrical pores, so Equation (2) can be written as:  
\[\frac{1}{T_{2}}=\rho_{2}\frac{4}{D}, \tag{3}\]  
where, \(D\) (\(\mu\)m) is the diameter of a cylindrical pore. Equation (3) suggests that the value of \(\rho_{2}\) acts as a critical parameter to characterize the pore sizes. Though \(\rho_{2}\) varies with the mineralogical composition of soil, it is a constant for a specific soil and has not association with temperatures and pressures in external environments. Equation (2) indicates that there is a linear relation between the relaxation time \(T_{2}\) and the pore diameter D. Thus, it is recognized that to calculate the pore diameters in a porous mediums based on its \(\rho_{2}\) value. In several researches, \(\rho_{2}\) values were calculated by comparing the \(T_{2}\) spectrum curves with POSD results from mercury injection porosimetry technique as presented by [PERSON] et al. [33] and nitrogen gas adsorption technique as presented by [PERSON] et al. [34]. Besides, the \(\rho_{2}\) values can be calculated by empirical formulas of permeability.