Flow Units: RQI and FZI

Chapter 9, Permeability Estimation

Learning objectives

  • Compute the reservoir quality index, the normalized porosity and the flow zone indicator from a permeability and a porosity
  • Say why a flow unit is a line of unit slope on RQI against the normalized porosity, and read its FZI where it meets a normalized porosity of 1
  • Cut the sorted FZI of a well at its widest gaps into flow units, and say how many units the data support
  • Say what Ogbon-1's FZI is, given that its permeability is Timur's, and what a wrong irreducible water saturation does to it
  • Say what flow units are for: within a unit, a permeability that depends on porosity alone

Making Rock Type Rigorous

Sorting a permeability cloud by rock type helps only if the sorting follows a rule. Amaefule, Altunbay, Tiab, Kersey and Keelan (1993) gave one. From a permeability kk in millidarcies and a porosity ϕ\phi as a fraction they define a reservoir quality index in micrometres, a normalized porosity, and their ratio, the flow zone indicator:

RQI=0.0314kϕ,ϕz=ϕ1−ϕ,FZI=RQIϕz.\text{RQI} = 0.0314\sqrt{\dfrac{k}{\phi}}, \qquad \phi_z = \dfrac{\phi}{1-\phi}, \qquad \text{FZI} = \dfrac{\text{RQI}}{\phi_z}.

The constant 0.0314 is the square root of the millidarcy expressed in square micrometres, 0.987×10−30.987 \times 10^{-3}, so RQI comes out as a length. Amaefule et al. rearranged the Kozeny-Carman equation to give FZI in terms of a shape factor, the tortuosity and the surface area per unit grain volume (Mirzaei-Paiaman et al., 2018, reproduce the rearrangement). FZI is therefore a property of the pore geometry that porosity alone does not carry: it falls as the pores become more tortuous or the grains expose more surface. Rocks that share an FZI make a hydraulic flow unit, which is the idea of the method.

At 5890 ft, in the upper oil sand, the density porosity is 23.2 % and Timur'srelation, with an irreducible water saturation of 18 %, gives 425 mD: RQI is1.35 μm, the normalized porosity φz is 0.302, and FZI, their ratio, is 4.46 μm. Thehalf foot falls in flow unit 3 of 4, which holds 144 half feet from the upper oilsand and has an FZI of 4.45 μm, the height at which its line of unit slope meets φz= 1. Ogbon-1's permeability was made with this relation and this saturation, so theFZI is a model's, and so are the units. Cut at the 3 widest gaps of the sorted FZI(29.8 %, 9.5 % and 1.6 %), the 772 net-reservoir half feet make 4 units, and one FZIper unit estimates a permeability within a factor of 1.15; the widest gap not cut is0.36 %. (a) RQI against normalized porosity0.050.10.20.510.050.10.20.512510Normalized porosityRQI (μm)FZI 4.45RQI 1.35(b) FZI along the well1235105800590060006100620063006400FZI (μm)Depth (ft)Key: in (a) the heavy line is the line of unit slope of the flow unit of the half foot read, drawn to anormalized porosity of 1, where its height is the FZI; the thin lines are the other units; the blue dotsare the half feet of the unit read and the grey dots the other net-reservoir half feet, one dot every 2of them, with a shale left off; the ringed dot is the half foot at the depth read. In (b) the dashedvertical lines are the FZI of each unit, the blue one that of the unit read, and the dotted horizontalline is the depth read. The permeability is Timur's relation, so the FZI is a model's.Porosity 23.2 % (the upper oil sand). Permeability, Timur 425 mD (Timur, 18 %, as built). RQI 1.35 μm(normalized porosity 0.302). Worst k error 1.15 times (one FZI per unit, 4 units). FZI 4.46 μm (unit 3of 4, its FZI 4.45).

Flow Units Are Lines of Unit Slope

Rearrange the definition and RQI=FZI ϕz\text{RQI} = \text{FZI}\,\phi_z. Take logarithms: log⁡RQI=log⁡ϕz+log⁡FZI\log\text{RQI} = \log\phi_z + \log\text{FZI}. On logarithmic axes of RQI against the normalized porosity, a rock of one FZI is therefore a straight line of slope 1, and where it meets ϕz=1\phi_z = 1 its height is the FZI. Rocks of different FZI are parallel lines, and a flow unit is the set of points on one of them.

Ogbon-1 gives 772 net-reservoir half feet to try this on, 386 ft of its 600 ft logged, kept by the book's cutoffs (a shale volume of at most 40 % and a porosity of at least 8 %, set out in Section 13.1). They are the only half feet with a permeability, so the only ones with an RQI. The porosity is the density porosity, and the permeability is Timur's relation of Section 9.2, k=0.136 ϕ4.4/Swi2k = 0.136\,\phi^{4.4}/S_{wi}^{2} with the porosity and the irreducible water saturation in per cent, taken at the 18 % Ogbon-1 was built with. At 5890 ft, in the upper oil sand, the porosity is 23.2 %, Timur's relation gives 425 mD, RQI is 1.35 μm, the normalized porosity is 0.302 and FZI is 4.46 μm. In the tight limestone at 6080 ft the porosity is 8.5 %, the permeability 5.1 mD, RQI 0.24 μm, the normalized porosity 0.093 and FZI 2.63 μm: the limestone's point is lower on the plot, and so is its line.

How Many Units?

A flow unit needs a boundary, and the boundary has to come from the data. The figure finds boundaries with a rule of its own, not Amaefule et al.'s procedure: sort the net-reservoir half feet by FZI and cut the sorted log FZI at its widest gaps. The widest gaps in Ogbon-1 are 29.8 %, 9.5 % and 1.6 % (the FZI rises by that much from one half foot to the next in the sorted list), and then 0.36 %, 0.36 % and 0.33 %. Three gaps stand out, and cutting them gives four units, one to a bed of the net reservoir: the tight limestone (156 half feet, FZI 2.72 μm), the shaly oil sand (144, 3.88 μm), the upper oil sand (144, 4.45 μm) and the main sand (328, 4.66 μm). Giving each half foot its unit's FZI in place of its own misestimates its permeability by at most a factor of 1.15, while one FZI for the whole well is out by up to 2.49, in the tight limestone at 6114.5 ft.

The number of units has to be read from the gaps, and the data have to have them. The third gap, 1.6 %, separates the upper oil sand from the main sand, and a gap that small would be lost in any scatter of the FZI larger than 1.6 %. Ask for five or six units and the further cuts fall inside the tight limestone, in gaps of 0.36 % each: they cut a smooth run of FZI, 2.54 to 2.90 μm, into units of 53, 2 and 101 half feet, and the worst error falls only from 1.15 to 1.09.

A Model's FZI

Ogbon-1 has no core. Its permeability is Timur's relation applied to its porosity with one saturation, so its RQI and FZI are a model's, not a measurement's. Timur's permeability goes as the porosity to the power 4.4 and as the inverse square of the saturation, which makes RQI proportional to ϕ1.7/Swi\phi^{1.7}/S_{wi} and puts the FZI at

FZI∝ϕ0.7 (1−ϕ)Swi.\text{FZI} \propto \dfrac{\phi^{0.7}\,(1-\phi)}{S_{wi}}.

FZI rises with porosity alone: the main sand, at 26.0 % porosity, has 1.71 times the FZI of the tight limestone at 9.0 %, and the closed form gives 1.71. The four units are porosity classes. Grain size, sorting and tortuosity, the things FZI is meant to carry, are nowhere in Ogbon-1's permeability, and the units say nothing about them: that is what a core is for. The saturation is a second thing the model brings. Double it to 36 % and every permeability falls to a quarter, the 425 mD at 5890 ft becoming 106 mD, and every FZI to a half, 4.46 μm becoming 2.23 μm. Every half foot moves by the same factor, so the gaps, the four units and the 1.15 do not change: a classification cannot see a wrong saturation, and only the height of the lines can, which needs a core.

What Flow Units Are For

Within a unit FZI is one number, so the definitions give the permeability as a function of the porosity alone:

k=ϕ(FZI ϕz0.0314)2.k = \phi\left(\dfrac{\text{FZI}\,\phi_z}{0.0314}\right)^{2}.

That is the payoff the method promises: a transform that needs no more than the porosity log, with one FZI for each unit. At a fixed porosity kk goes as the square of the FZI, which is why an FZI 20 % off puts the permeability 44 % off. The title of the paper says what it is for: to use core and log data to identify hydraulic flow units and to "Predict Permeability in Uncored Intervals/Wells". Nothing in this figure tests that, because Ogbon-1 has no core: on Ogbon-1 the porosity log already gives Timur's permeability, since that is how the permeability was made. Section 9.5 turns to NMR, which comes closest to measuring the pore sizes that FZI only infers.

References

  • Amaefule, J. O., Altunbay, M., Tiab, D., Kersey, D. G., and Keelan, D. K. (1993). Enhanced reservoir description: using core and log data to identify hydraulic (flow) units and predict permeability in uncored intervals/wells. SPE 26436. doi:10.2118/26436-MS.
  • Mirzaei-Paiaman, A., Ostadhassan, M., Rezaee, R., Saboorian-Jooybari, H., and Chen, Z. (2018). A new approach in petrophysical rock typing. Journal of Petroleum Science and Engineering, 166, 445-464. doi:10.1016/j.petrol.2018.03.075.
  • Timur, A. (1968). An investigation of permeability, porosity, and residual water saturation relationships. SPWLA 9th Annual Logging Symposium, paper J.
  • PetroWiki. Estimating permeability based on surface area and water saturation. petrowiki.spe.org.

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