Why Permeability Is Hard

Chapter 9, Permeability Estimation

Learning objectives

  • Say why a porosity log cannot tell two rocks of different permeability apart
  • Use the Kozeny-Carman equation to show that permeability goes as the square of the grain size at any porosity
  • Show that a gain in porosity cannot make up for finer grains beyond a modest factor
  • Explain that Ogbon-1's permeability is computed from its porosity, and that every permeability from logs is an estimate

A Volume and a Rate

Section 1.4 showed that permeability is set at the pore throats, the narrow necks between pores, and gave the permeability of the simplest rock there is, a pack of equal spheres of diameter dd and porosity ϕ\phi, in Carman's form of the Kozeny-Carman equation:

k=ϕ3 d2180 (1−ϕ)2.k = \frac{\phi^{3}\, d^{2}}{180\,(1-\phi)^{2}}.

This section asks what that means for a log. Every porosity tool, the density, neutron and sonic logs of Chapters 4 and 5, responds to the pore space as a whole, the bodies and the throats together: it reads how much pore a rock has, not how narrow the way through it is. Figure 9.1 sets two packs of one porosity side by side, a coarse one and a fine one, and asks what a log could tell them apart by.

At 26 % porosity a coarse rock of 200 μm grains passes 7227 mD and a fine rock of20 μm grains, 10 times finer, only 72 mD, 100 times less. A density log reads2.221 g/cc in both. To flow as the coarse rock does, the fine rock would need 68 %porosity, more open than the 47.6 % of the loosest cubic packing of equal spheres:no gain in porosity can make up the loss. Ogbon-1's permeability here is Timur'sestimate, 706 mD, from the porosity and a fixed irreducible water saturation alone,between the two: the well holds one value at each porosity, a log's view, not acore's.(a) Two rocks and the well5102030400.01110010000Porosity (%)k (mD)7227 mD72 mD(b) What porosity would have to do10305070900.11001000001e8Porosity of the fine rock (%)k (mD)68 %Key: the orange line is the coarse rock and the teal line the fine rock; the dotted bar joins them atthe chosen porosity; the dashed black line and dots in (a) are Ogbon-1's permeability (Timur) and itsnet-reservoir half feet, one in ten; the dashed orange line in (b) is the coarse rock's value, and theshaded porosities are those above the simple cubic packing of equal spheres, 47.6 %.Coarse over fine 100 times (the grain ratio, 10, squared). Coarse rock 7227 mD (coarse rock, 200 μm,26 %). Fine rock 72 mD (fine rock, 20 μm, 26 %). Density log, both 2.221 g/cc (both rocks: sandstonematrix 2.65, filtrate 1.0). Porosity the fine rock needs 68 % (over the cubic packing's 47.6 %).

One Porosity, Two Permeabilities

Take 26 % porosity, the main sand's, and spheres of 200 μm and 20 μm, a ratio of 10. The coarse pack passes 7227 mD and the fine one 72 mD, 100 times less. A density log, with Ogbon-1's sandstone matrix of 2.65 g/cc and logging fluid of 1.0 g/cc, reads 2.221 g/cc in both, because it sees the volume of the pores and the two packs have the same. The factor between them is the square of the ratio of the grains, and it does not depend on the porosity: at 36 % the packs pass 25,648 mD and 256 mD, still 100 times apart. A porosity log that is exactly right leaves the permeability uncertain by the square of however much the grains vary.

For scale, take the porosities of Ogbon-1's net reservoir, 8.0 to 27.0 %. A pack of fixed 200 μm grains changes in permeability by a factor of 61 across that whole range; a change of 10 in the grain size alone changes it by 100. Nelson (1994) reviews the permeability-porosity relationships of sandstones and carbonates and relates the differences in their slope, intercept and scatter to grain size, sorting, diagenesis and compaction.

Why Porosity Cannot Buy It Back

Could a higher porosity make up for finer grains? In this model only for a modest factor. For the 20 μm pack to pass what the 200 μm pack passes at 26 %, it would need 68 % porosity, more open than the 47.6 % of the simple cubic packing of equal spheres, 1−π/61 - \pi/6, the loosest of the three cubic lattices. With a ratio of 3 in the grains the fine pack needs 45 %, 19 porosity units more and under that limit; with a ratio of 100 it needs 95 %. A real rock has cement and clay as well as grains, and the figure has neither; what carries over is that the grains and the throats they make, not the pore volume, decide how fast fluid moves.

What Ogbon-1 Shows

A figure that reads Ogbon-1's permeability reads a model. The well was built forward, and its permeability is Timur's estimate (Timur, 1968), k=0.136 ϕ4.4/Swi2k = 0.136\,\phi^{4.4}/S_{wi}^{2} with porosity and irreducible water saturation in per cent, applied to each half foot of its net reservoir with one irreducible water saturation, 18 %; Section 9.2 sets the equation out. So at a porosity of 26 % the equation gives the well 706 mD, and the main sand's net pay, of mean porosity 26.0 %, has a geometric-mean permeability of 704 mD. Plate (a) draws that line between the two packs and puts the well's half feet on it: they lie on the line because the permeability was computed from the porosity. Ogbon-1 therefore shows what an estimate from porosity gives; it cannot show how wrong that estimate can be.

Why Logs Struggle

Permeability cannot be read from a log as porosity can, because the log adds up the pore volume and permeability is set by how narrow the path through it is. It has to come from a transform calibrated to core, from the one extra clue the porosity logs can add, or from a measurement sensitive to pore size. The rest of Chapter 9 is that toolkit: the Timur and Tixier equations (Section 9.2), the poro-perm crossplot (Section 9.3), flow units (Section 9.4), permeability from NMR (Section 9.5), and the estimate along Ogbon-1 (Section 9.6).

References

  • Carman, P. C. (1937). Fluid flow through granular beds. Transactions of the Institution of Chemical Engineers, 15, 150-166.
  • Kozeny, J. (1927). Ueber kapillare Leitung des Wassers im Boden. Sitzungsberichte der Akademie der Wissenschaften in Wien, 136(2a), 271-306.
  • 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. The constants: PetroWiki, Estimating permeability based on surface area and water saturation.

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