### Nuclear Magnetic Resonance  
#### 1.3.1 Concept and Application  
Nuclear Magnetic Resonance (NMR) is the response of atoms' nuclei, having an odd number of proton and neutron, in the presence of applied magnetic fields at a specific resonance frequency [12]. As a result of nuclei angular momentum, in this case, hydrogen proton, due to its large magnetic moment and abundance in both water and oil, the proton experiences a magnetic moment. The proton then behaves like a spinning bar magnet. It is polarised in the presence of an externally applied magnetic field, resulting in an induced frequency (Larmor frequency) that is a quantum property and can be measured [22]. The schematic nuclei response to the applied magnetic fields is illustrated in Figure 1. Therefore, NMR manipulates the wetting fluids' hydrogen protons to measure its relaxation times (NMR signal decay) due to its interaction with the pore surface [23; 24]. Due to this interaction, the wetting fluid experiences a longer relaxation time (smaller diffusion coefficient) due to the surface relaxation acceleration from the surface-liquid interaction (surface relaxivity). As a result, the non-wetting fluid relaxes at a single exponential time (bulk relaxation time). This mechanism has been reported and recognised by several authors [8; 22; 25; 26; 27] that for a rock not being thoroughly water wet, the surface relaxation will accelerate the bulk relaxation time of oil. This acceleration will be dependent on the volume of oil interacting with the pore surface.  
The general relaxation time of the fluids in the pores of a sample with a uniform magnetic field can be defined as [28]:  
\[\frac{1}{\mathrm{T}_{2}}=\frac{1}{\mathrm{T}_{2B}}+\rho\frac{\mathrm{A}}{\sqrt{ \mathrm{S}_{\mathrm{i}}}}\frac{\mathrm{f}_{\mathrm{i}}}{\mathrm{S}_{\mathrm{i}}} \tag{2}\]  
where \(\mathrm{T}_{2}\) is the observed relaxation time, \(\mathrm{T}_{2B}\) is the relaxation rate of bulk fluid, \(\rho\) is the surface relaxivity, \(\mathrm{A}/\mathrm{V}\) is the pore ratio of the surface area to volume and is used to represent the pore size, \(\mathrm{f}_{\mathrm{i}}\) is the fraction of pore wetted by fluid and \(\mathrm{S}_{\mathrm{i}}\) is the saturation of pore by fluid. This equation was then used to measure the Nuclear Wetting Index, NWL as [11]:  
\[\mathrm{NWI}=\frac{\mathrm{Surface\ wetted\ by\ water\ -\ Surface\ wetted\ by\ oil}}{\mathrm{Total\ Surface}} \tag{3}\]  
To correlate to the Amott-Harvey index, results vary from \(-1\) for strongly oil-wet rocks to +1 from strongly water-wet rocks, with 0 representing neutrally wet rocks. This, however, is not fundamentally equal to the Amott-Harvey and the USBM indices. The relationship between the wetting fluid decay time and its interaction with the pore spaces can also be expressed as [3]:  
\[\frac{1}{\mathrm{T}_{\mathrm{i}}}\approx\rho\frac{\mathrm{A}}{\sqrt{\mathrm{V}}} \tag{4}\]  
where the surface relaxivity \(\rho\) is expressed as [3]:  
\[\rho=\frac{\delta}{\mathrm{T}_{\mathrm{IS}}} \tag{5}\]  
\(\delta\) represents the surface area thickness, and \(\mathrm{T}_{\mathrm{IS}}\) represents the surface relaxation time. Measurement of wettability using measurements of \(\mathrm{T}_{2}\) is complicated in that it also relies on the measurements of fluid viscosity, rock relaxivity, and pore size distribution. The  
Figure 1: Polarisation of hydrogen nuclei illustrating nuclei excitation/relaxation in the presence of applied external magnetic energy.  
most pragmatic way to circumvent this is to quantitatively apply numerical modelling to measure wettability and saturation [5]. The use of the NMR tool, a non-invasive and non-destructive tool, as part of the wireline logging suite makes it possible to locate pay zones in a thinly laminated sandstone reservoir that would have otherwise been missed by the resistivity, density, and gamma-ray logs. This helps for volumes to be quantified and identified, improving completion strategy and maximising the reservoir production.