# Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Nuclear magnetic resonance in porous media

The nuclear magnetic resonance (NMR) method has been intensively used to study the structure of porous media and various processes occurring in them. This technique allows to determine characteristics such as the porosity and pore size distribution, the permeability, the water saturation, the wettability, etc. NMR is also a very active area of experimental research.

## Theory of relaxation time distribution in porous media

Microscopically the volume of a single pore in a porous media may be divided into two regions; surface area S and bulk area V.

The surface area is a thin layer with thickness ${\displaystyle \delta }$ of a few molecules close to the pore wall surface. The bulk area is the remaining part of the pore volume and usually dominates the overall pore volume. With respect to NMR excitations of nuclear states for hydrogen-containing molecules in these regions, different relaxation times for the induced excited energy states are expected. The relaxation time is significantly faster for a molecule in the surface area, compared to a molecule in the bulk area. This is an effect of paramagnetic centres in the pore wall surface that causes the relaxation time to be faster, as reported by Brown and Fatt (1956). The inverse of the relaxation time, ${\displaystyle {T_{i}}}$, is expressed by contributions from the bulk area V, the surface area S and the self diffusion d:

${\displaystyle {\frac {1}{T_{is}}}=\left(1-{\frac {\delta S}{V}}\right){\frac {1}{T_{ib}}}+{\frac {\delta S}{V}}{\frac {1}{T_{is}}}+{\frac {\left({\gamma Gt_{E}}\right)^{2}}{12}}}$ with i=1,2

where δ is the thickness of the surface area, S is the surface area, V is the pore volume, ${\displaystyle {T_{ib}}}$ is the relaxation time for bulk, ${\displaystyle {T_{is}}}$ is the relaxation time for the surface, γ is the gyromagnetic ratio, G is the magnetic field gradient (assumed to be constant), ${\displaystyle {t_{E}}}$ is the time between echoes and D is the self diffusion coefficient of the fluid.

The NMR Signal Intensity in the ${\displaystyle {T_{2}}}$ distribution plot reflected by the measured amplitude of the NMR signal is proportional to the total amount of hydrogen nuclei, while the relaxation time depends on the interaction between the nuclear spins and the surroundings. In a characteristic pore containing e.g. water, the bulk water exhibits a single exponential decay. The water close to the pore wall surface exhibits faster ${\displaystyle {T_{2}}}$ relaxation time for this characteristic pore size

## NMR Permeability Correlations

The connection between NMR relaxation measurements and petrophysical parameters such as permeability stems from the strong effect that the rock surface has on promoting magnetic relaxation. For a single pore, the magnetic decay as a function of time is described by a single exponential:

${\displaystyle M(t)=M_{0}\mathrm {e} ^{(-t/T_{2})}}$

where ${\displaystyle {M_{0}}}$ is the initial magnetization and the transverse relaxation time ${\displaystyle {T_{2}}}$ is given by:

${\displaystyle {\frac {1}{T_{2}}}={\frac {1}{T_{2b}}}+\rho {\frac {S}{V}}}$

${\displaystyle S/V}$ is the surface-to-volume ratio of the pore, ${\displaystyle {T_{2b}}}$ is bulk relaxation time of the fluid that fills the pore space, and ${\displaystyle \rho }$ is the surface relaxation strength. For small pores or large ${\displaystyle \rho }$, the bulk relaxation time is small and ${\displaystyle 1/T_{2}=\rho S/V}$

Real rocks, of course, contain an assembly of interconnected pores of different sizes. The pores are connected through small and narrow pore throats (i.e. links) that restrict interpore diffusion. If interpore diffusion is negligible, each pore can be considered to be distinct and the magnetization within individual pores decays independently of the magnetization in neighbouring pores. The decay can thus be described as:

${\displaystyle M(t)=M_{0}\sum _{i=1}^{n}{a_{i}}\mathrm {e} ^{(-t/T_{2})}}$

where ${\displaystyle a_{i}}$ is the volume fraction of pores of size i that decays with relaxation time ${\displaystyle {T_{2i}}}$. The multi-exponential representation corresponds to a division of the pore space into n main groups based on ${\displaystyle S/V}$ values. Usually, a logarithmic mean, ${\displaystyle {T_{2lm}}}$, of the relaxation times is used for permeability correlations:

${\displaystyle T_{2lm}=\mathrm {e} ^{(\sum {a_{i}}log{T_{2i}}/\sum {a_{i}})}}$

${\displaystyle {T_{2lm}}}$ is thus related to an average ${\displaystyle S/V}$ or pore size. Commonly used NMR permeability correlations are of the form:

${\displaystyle k\approx a\Phi ^{b}\left(T_{2lm}\right)^{c}}$

where ${\displaystyle \Phi }$ is the porosity of the rock.The exponents b and c are usually taken as four and two, respectively. Correlations of this form can be rationalized from the Kozeny–Carman equation:

${\displaystyle k\approx {\frac {\Phi }{\tau }}\left({\frac {V}{S}}\right)^{2}}$

by assuming that the tortuosity ${\displaystyle \tau }$ is proportional to ${\displaystyle \Phi ^{1-b}}$. However, it is well known that tortuosity is not only a function of porosity. It also depends on the formation factor ${\displaystyle F=\tau /\Phi }$. The formation factor can be obtained from resistivity logs and is usually readily available. This has given rise to permeability correlations of the form:

${\displaystyle k\approx aF^{b}\left(T_{2lm}\right)^{c}}$

Standard values for the exponents b and c are –1 and 2, respectively. Intuitively, correlations of this form are a better model since it incorporates tortuosity information through F. The value of the surface relaxation strength ${\displaystyle \rho }$ affects strongly the NMR signal decay rate and hence the estimated permeability. Surface relaxivity data are difficult to measure, and most NMR permeability correlations assume a constant ${\displaystyle \rho }$. However, for heterogeneous reservoir rocks with different mineralogy, ${\displaystyle \rho }$ is certainly not constant and surface relaxivity has been reported to increase with higher fractions of microporosity. If surface relaxivity data are available it can be included in the NMR permeability correlation as

${\displaystyle k\approx aF^{b}\left(T_{2lm}\right)^{c}}$

## T2 Relaxation

For fully brine saturated porous media, three different mechanisms contribute to the relaxation: bulk fluid relaxation, surface relaxation, and relaxation due gradients in the magnetic field. In the absence of magnetic field gradients, the equations describing the relaxation are

on S

with the initial condition

where D0 is the self-diffusion coefficient. The governing diffusion equation in the reconstructed media is solved by a random walk algorithm. The algorithm simulates the Brownian motion of a diffusing magnetized particle or random walker. Initially, the walkers are launched at random positions in the pore space. At each time step, t, they advance from their current position, x(t), to a new position, x(t+t), by taking steps of fixed length in a randomly chosen direction. The time step is given by

The new position is given by

x(t t) x(t ) ε sinθ cos  y(t t) y(t)  ε sinθ sin  z(t t) z(t)  ε cosθ (24)

The angles (0 θ π) and (0 2π) are selected randomly. It can be noted that cosmust be distributed uniformly in the range (-1,1). If a walker encounters a pore-solid interface, it is killed with a finite probability . The killing probability is related to the surface relaxation strength by

If the walker survives, it simply bounces off the interface and its position does not change. At each time step, the fraction p(t) of the initial walkers that are still alive is recorded. Since the walkers move with equal probability in all directions, the above algorithm is valid as long as there is no magnetic gradient in the system.

When protons are diffusing, the sequence of spin echo amplitudes is affected by inhomogeneities in the permanent magnetic field. This results in an additional decay of the spin echo amplitudes that depends on the echo spacing 2. In the simple case of a uniform spatial gradient G, the additional decay can be expressed as a multiplicative factor

where is the ratio of the Larmor frequency to the magnetic fie ld intensity. The total magnetization amplitude as a function of time is then given as

## NMR as a Tool to Measure Wettability

The wettability conditions in a porous media containing two or more immiscible fluid phases determine the microscopic fluid distribution in the pore network. Nuclear Magnetic Resonance measurements are sensitive to wettability because of the strong effect that the solid surface has on promoting magnetic relaxation of the saturating fluid. The idea of using NMR as a tool to measure wettability was presented by Brown and Fatt (1956). They based their theory on the hypothesis that, at the solid-liquid interface, molecular motion is slower than in the bulk liquid. In this solid-liquid interface the diffusion coefficient is reduced, which in turn is equivalent to a zone of higher viscosity. In the zone of higher viscosity the magnetically aligned protons can more easily transfer their energy to their surroundings. The magnitude of this effect depends upon the wettability characteristics of the solid with respect to the liquid in contact with the surface. Brown and Fatt started their work by measuring proton spin-lattice relaxation time of water in uncoated sand packs as water-wet media, and Dri-film treated sand packs as oil-wet porous media. Their experiments resulted in a linear relation between relaxation rate and fractional oil wetted surface area.