The Poro-Perm Crossplot

Chapter 9, Permeability Estimation

Learning objectives

  • Plot permeability on a logarithmic axis against porosity on a linear axis and read the straight line as an exponential law
  • Say why Ogbon-1's crossplot is one curve and not a cloud, and what a what-if of varying irreducible water adds
  • Measure the scatter about the line as a factor and read a permeability with its band
  • Sort by rock type, and say how far that helps

The Workhorse Plot

The most-used permeability transform is just a graph. Plot core permeability on a LOGARITHMIC axis against core porosity on a LINEAR axis, and in many consolidated sandstones and carbonates the plugs fall roughly along a straight line (Nelson, 1994); the slope, the intercept and the scatter of that line differ from one formation to the next. A straight line on that plot is an exponential law:

log⁡10k=a+b ϕ⟺k=10a⋅10 b ϕ,\log_{10} k = a + b\,\phi \quad\Longleftrightarrow\quad k = 10^{a}\cdot 10^{\,b\,\phi},

with ϕ\phi in per cent, so each porosity unit multiplies the permeability by 10b10^{b}. A handful of core plugs fixes aa and bb, and the line then turns the log porosity, which every well has, into a permeability, which only the cored intervals have.

At 23.0 % porosity a straight line through the 76 plugs reads 357 mD, and 96 % ofthe plugs lie within a factor of 4.5 of it (two standard deviations), between 80 mDand 1594 mD. The plugs differ in irreducible water, which Ogbon-1's permeabilitynever does, and that is all the scatter there is. With a line for each rock type thecoarse rock reads 580 mD here and the fine rock 227 mD, and the factor falls to 3.2.(a) Permeability against porosity610152025300.1110100100010 000Porosity (%)k (mD)357 mD(b) How far the lines miss÷10÷3.21×3.2×10one lineper typeError, a factorKey: in (a) the heavy black line is Ogbon-1's own relation, Timur's equation with one irreducible watersaturation, and the grey dots are the well's permeability log; the orange dots are the what-if plugs ofthe coarse rock type and the blue dots those of the fine rock type; the dashed line is the straight linethrough all the plugs, with its band of two standard deviations shaded, and the orange and blue linesare the lines through each type; the dotted line is the porosity read. In (b) the first row is the errorof the one line at each plug and the second row the error of the line for its own type, each with itsband of two standard deviations.Scatter, one line for all 4.5 × (two standard deviations, 0.33 decades each; 96 % of the plugs inside).Scatter, a line per rock type 3.2 × (two standard deviations, 0.25 decades each; 96 % of the plugsinside). Coarse over fine here 2.56 × (the lines at 23.0 %; Timur's equation gives 2.56). Gain per 5porosity units 4.1 × (one line for all: 0.12 decades a porosity unit). Permeability the line reads 357mD (96 % of the plugs between 80 mD and 1594 mD; per type 580 mD and 227 mD).

Ogbon-1 Shows a Curve, Not a Cloud

Ogbon-1 has no core, and its permeability is not a measurement. It was computed, in net reservoir only, from each half foot's porosity and one irreducible water saturation, SwiS_{wi} = 18 %, by Timur's equation (Section 9.2). So a poro-perm crossplot of the well is a single curve by construction. The figure takes a plug every 5 ft of the net reservoir, 76 of them, with porosities from 8.1 to 27.0 % and permeabilities from 4.1 to 833 mD, and every one sits on that curve. The straight line fitted to them gains a factor of 3.9 in permeability for every 5 porosity units, but even here it is not exact: it is within a factor of 1.65 of the plugs at two standard deviations (1.55 at the worst plug), because Timur's law is a power of the porosity and not an exponential. Fitted on log-log axes the same plugs give the exponent 4.40 and no misfit. At 23.0 %, the upper oil sand's mean porosity, the line reads 357 mD where the curve gives 412 mD, and the bed's own 144 half feet have a geometric mean of 410 mD: a line read at a bed's mean porosity says what one transform predicts for a rock of that porosity, not what the bed averages. A tight curve on a poro-perm plot is therefore no evidence that porosity predicts permeability; for Ogbon-1 it is evidence of how the well was made.

The Scatter Real Rock Shows: a What-If

Real core scatters, because porosity does not fix the throat size (Section 9.1). The figure shows how with a labelled what-if, not a measurement: it keeps the 76 plugs and their porosities, and lets each plug have an irreducible water saturation of its own. Timur's permeability goes as Swi−2S_{wi}^{-2}, so a plug with twice the irreducible water has a quarter of the permeability at the same porosity, and that alone makes a cloud. With the settings the figure opens on, a spread of 1.30 within a rock type and a contrast of 1.60 between a coarse and a fine type, the plugs hold 9 % to 38 % irreducible water and permeabilities from 1.4 to 2867 mD. At 23.0 % porosity the straight line through all of them reads 357 mD, as the line through Ogbon-1's own plugs does, and 96 % of the plugs lie within a factor of 4.5 of it (two standard deviations), between 80 and 1594 mD. How much scatter a formation really has is measured on its core: Nelson (1994) finds that it differs from one formation to the next, and the spread and the contrast set here are the figure's own, not measured rock.

Sort by Rock Type First

Rock types fall on different, roughly parallel lines, so the plugs are best sorted before they are fitted. In the what-if the fine rock holds 1.60 times the coarse rock's irreducible water, so at 23.0 % porosity the two lines read 580 mD for the coarse rock and 227 mD for the fine, 2.56 times apart, and a line for each type brings the factor from 4.5 to 3.2. Sorting helps only as far as the offset between the types exceeds the scatter inside each: with a spread of 1.50 and a contrast of 1.70 it takes the factor from 7.4 to 5.4, and with one rock type it does nothing, 4.2 against 4.1 at a spread of 1.40. Section 9.4 builds a rule for the sorting that does not rest on a guess.

Read a Permeability With Its Band

A line on a logarithmic axis has a band of constant factor, not of constant width. At 9.0 % porosity, where the tight limestone lies (its own half feet average 6.6 mD), the line reads 6.9 mD, between 1.5 and 31 mD; at 27.0 %, in the main sand, it reads 1101 mD, between 246 and 4921 mD. A permeability quoted from a poro-perm line without its band says more than a log can know.

References

  • Nelson, P. H. (1994). Permeability-porosity relationships in sedimentary rocks. The Log Analyst, 35(3), 38-62.
  • Timur, A. (1968). An investigation of permeability, porosity, and residual water saturation relationships for sandstone reservoirs. SPWLA 9th Annual Logging Symposium, paper J.

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