Diffusion and Gas

A Third Relaxation

The fluid in a pore relaxes in two ways, at the pore wall and in the bulk (Section 10.1). A logging tool adds a third. Its magnetic field is not uniform but has a gradient $G$, and a molecule that wanders along it between the pulses of an echo train meets a different field, dephases, and so decays faster. The extra decay rate is

where $D$ is the diffusion coefficient of the fluid, $T_E$ the echo spacing, and $\gamma$ = 26,752.2 rad/(s G) the gyromagnetic ratio of the proton (CODATA), with $G$ in gauss per centimetre and $T_E$ in seconds. US Patent 5,498,960 writes the law with half the echo spacing, $t$, as $\gamma^{2}G^{2}D\,t^{2}/3$, which is the same thing. The mechanisms act together, so their rates add, $1/T_2 = 1/T_{2,\mathrm{surface}} + 1/T_{2,\mathrm{bulk}} + 1/T_{2D}$. The term goes as the square of the gradient and of the echo spacing and in proportion to $D$, so the lever is the fluid.

Why Gas Is the Fluid It Finds

US Patent 6,366,087 gives the diffusion coefficients at 200 °F as $7.7\times10^{-5}$ cm²/s for brine and $100\times10^{-5}$ cm²/s for gas, and the figure takes those two, the conditions of a reservoir, which makes gas about 13 times water. Akkurt and co-authors (1995) give water at room temperature as $2.3\times10^{-5}$ cm²/s, against which the same gas would be 43 times: warmer water diffuses faster. They are representative values, not constants: the diffusion coefficient of a gas also falls as its density rises.

Gas also has two properties that make a tool count it short. Its hydrogen index, the hydrogen per unit volume relative to fresh water, is low. For the Rock Physics anchor gas, 0.202 g/cc (50 °C, 25 MPa, gravity 0.65, from Batzle and Wang), taken as the alkane of its molar mass, it is 0.42 (methane at that density would be 0.45); US Patent 5,498,960 gives 0.2 to 0.6 for reservoir gas. And its T1 is long, 3 to 6 s in the same patent, so in a wait of a few seconds it has not recovered its magnetisation. The figure takes 4.4 s, the gas T1 of Table 2 of US Patent 6,366,087 (Section 10.4 takes the same), and, because a gas is a fast-motion fluid, the same for its bulk T2.

The What-If on Ogbon-1

Ogbon-1 has no NMR log and no gas: its hydrocarbon is oil. The figure takes its upper oil sand, whose half feet average 23.0 % porosity and 18.9 % water saturation, and asks what an NMR tool would report if the 81 % of the pores that hold oil held gas instead. The brine, the anchor's, with a hydrogen index of 0.96, is the other 19 %. Its T2 comes from the pore radius you set, 20 μm at the start, and the book's quartz relaxivity of 10 μm/s (Section 10.1), which gives 667 ms; the radius is the figure's own choice. The tool starts at the example of US Patent 5,498,960, 17 gauss per centimetre and an echo spacing of 1.2 ms, with a wait of 8.0 s, the long wait of Akkurt and co-authors' example; Akkurt and co-authors report 17 to 22 gauss per centimetre for that tool over 70 to 300 °F. The tool counts the signal left at the first echo.

The Apparent T2

At an echo spacing of 1.2 ms the diffusion of the gas alone gives a T2 of 40.3 ms, and with its 4.4 s bulk T2 added the gas has an apparent T2 of 39.9 ms, the 40 ms that Table 2 prints for gas. The brine, in the same gradient, keeps 293 ms of the 667 ms its pores give it. Make the pores twice as large and the brine goes to 376 ms and the gas stays at 39.9 ms: the gas has no wall to relax on, so diffusion, not pore size, sets its T2, which is why a gas is found at short T2 whatever the pores.

What the Tool Under-Reads

The tool reads 10.6 % of the 23.0 %, 12.4 porosity units low. Almost none of that is diffusion. The hydrogen index of 0.42 takes 10.7 porosity units, the incomplete recovery in the 8.0 s wait (84 %) takes 1.3, the decay of the gas before the first echo takes 0.2, and the brine 0.2. Shorten the wait to 2.0 s and the gas recovers 37 %, the recovery term rises to 5.0 and the tool reads 6.4 %. A gas zone read as if it were liquid under-reads porosity before diffusion has a say.

Two Echo Spacings

What diffusion does is move the gas, and it moves more as the echo spacing grows. Lengthen the spacing from 1.2 ms to 3.0 ms and the gas T2 falls from 39.9 ms to 6.44 ms, 6.2 times, while the brine goes from 293 ms to 74.4 ms, 3.9 times: warm water diffuses too, but its pore wall carries part of its decay. The tool now reads 8.2 %, and comparing it with the reading at 0.2 ms finds 2.6 porosity units of gas signal that the longer spacing took away. A measurement at one echo spacing sees where the signals are; a pair of spacings shows which of them diffusion moves most, here the gas. Doubling the spacing quadruples the diffusion rate: the gas T2 falls from 39.9 ms to 10.0 ms at 2.4 ms.

Where the Figure Stops

The law above is for free diffusion. In one echo the gas moves $\sqrt{2\,D\,T_E}$, 15 μm at 1.2 ms and 24 μm at 3.0 ms, and where that exceeds the pore radius the walls hold it back, the real diffusion is slower and the real T2 longer. The figure draws those T2 as dashed and calls them a shortest value; with 20 μm pores that is from 2.0 ms. It also fixes the gas coefficient and the hydrogen index and gives the brine no bulk relaxation, so it shows the direction and the order of the effects and not a logged gas zone.

References

• Akkurt, R., Vinegar, H. J., Tutunjian, P. N. and Guillory, A. J. (1995). NMR logging of natural gas reservoirs. SPWLA 36th Annual Logging Symposium, Paris, 26-29 June 1995.
• Vinegar, H. J., Akkurt, R. and Tutunjian, P. N. (1996). NMR logging of natural gas in reservoirs. US Patent 5,498,960, Shell Oil Company. uspto.report/patent/grant/5498960.
• Coates, G. R. and Hou, L. B. NMR logging apparatus and methods for fluid typing. US Patent 6,366,087 B1, filed 28 October 1999. patents.google.com/patent/US6366087.
• Batzle, M. and Wang, Z. (1992). Seismic properties of pore fluids. Geophysics, 57(11), 1396-1408.