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Petrophysics
• November 27, 2019
• Oilfield Knowledge Training Centre

#### NMR T1 & T2 relaxation

This article discusses NMR T1 & T2 relaxation. To get basic understanding of NMR, we refer readers to go through article NMR Introduction.

T2 relaxation is related to the detection of proton spins in the transverse plane aligned with the B1field.

T1 relaxation relates the time taken for proton spins to translate from a random alignment to an alignment with the B0 field.

The difference between T1 and T2 can be considered as the measurement of relaxation of the proton spins to B0 in two different measurement directions.  T1 looks in the direction of B0 and normal to B1. T2 looks in a direction normal to B0and parallel to B1 (i.e. T1 refers to the yz plane and T2 refers to the xy plane).

### T1 Relaxation

In a magnetic field, after time t the proton spins are aligned parallel to the magnetic field. At this time, the net magnetization vector lies along the direction of the applied magnetic field B0 and is referred to as the equilibrium magnetization, M0.

In this configuration, the z-axis component of magnetization MZ equals M0. MZ is referred to as the longitudinal magnetization. There is no transverse magnetization in the x-y plane (MX ,MY ). It is possible to change the equilibrium magnetization by exposing the nuclear spin system to energy of a frequency equal to the energy difference between the spin states.  If enough energy is put into the system, it is possible to saturate the spin system and make MZ = 0.

The time constant which describes how MZ returns to its equilibrium state is called the “spin lattice relaxation time (T1)“. The equation governing this behaviour is a function of the time, t, after the initial displacement as given in equation : Mz=Mo (1-e(-t/T1))

T1 is therefore defined as the time required to change the MZ component of magnetization by a factor of e.If the net magnetization is perturbed such that the proton spins are aligned anti-parallel to the z-axis (-z), it will gradually return to its equilibrium position along the +z axis at a rate governed by T1. The equation governing this behaviour as a function of the time, t, after its displacement is defined as Mz=Mo (1-2e(-t/T1))

The spin lattice relaxation time (T1) is the time taken to reduce the difference between the longitudinal magnetization (MZ ) and its equilibrium value by a factor of e.

### T2 Relaxation

T2 relaxation is measured in the transverse plane parallel to the B1 field used to tip the proton spins from the B0 field. For spin tipping to take place, the radio frequency (rf) pulse used to tip the spins into the transverse plane must be tuned to the Larmor frequency of the proton spins. The angle through which the protons are tipped depends on the length of time that the B1 field is turned on. The B1 is usually designed to tip the spins through 900 into the x-y plane.

After the 900 B1 pulse has been applied, the protons’ spins are aligned in the direction of B1 (at which time the detected signal is at a maximum). After a short time the proton spins begin to dephase. Dephasing causes the proton spins to spread out in the T2 plane and, as the spins are no longer aligned in the same direction, the signal begins to decay. After a certain amount of time, the proton spins are completely dephased and there is no signal. The time taken for dephasing is referred to as T2* or Free Induction Decay (FID).

Two factors contribute to dephasing in the T2 plane:

1. Molecular interactions (the pure T2 molecular effect).
2. Variations in the B0 field (the inhomogeneous T2 effect).

In terms of porous media (i.e. fluid bearing rocks), the T2 molecular effect can be attributed to molecular interactions in the fluid, and molecular interactions between the fluid and the pore wall. As discussed earlier, these interactions impart information about the fluid type and the pore size. Unfortunately for petrophysics, the dephasing process is mainly dominated by the inhomogeneous T2 effect.

To isolate the more useful molecular interactions, the inhomogeneous effects are negated using a special sequence of rf pulses – known as a Carr-Purcell-Meiboom-Gill (CPMG) sequence.

The dephasing caused by the inhomogeneity of the B0 field is reversible.  Reversal can be achieved by applying a second B1 pulse in the opposite direction to the pulse responsible for initial spin tipping. This reverse pulse causes the proton spins to precess in the opposite direction and leads to a realignment of the proton spins.

After refocusing by the reverse pulse, the proton spins begin to dephase again. However, the proton spins can be refocused yet again by applying a third pulse in the opposite direction to the second.  A train of 1800 pulses is used to continually refocus the proton spins, and this train of pulses is referred to as the CPMG pulse sequence.

The CPMG pulse sequence consists of a 900 pulse that tips the spins into the transverse plane followed by a series of 1800 pulses that refocus the proton spins.

The result of the CPMG pulse sequence is a series of spin echoes, referred to as an echo-train. Magnetization decay between the 1800 pulses is due to FID.  The decay of the complete echo-train describes irreversible dephasing due to the molecular decay associated with molecular interactions in the fluid and interaction between the fluid and the pore walls. Irreversible dephasing is referred to as the T2 decay.