The Dual-Water Model

Two Waters in the Pore

Section 8.1 showed that clay carries current beside the brine, and Section 8.4 wrote that current through the cation exchange capacity of the clay. The dual-water model tells the same story through the water. The Schlumberger glossary describes it as a model of shaly formations that considers there to be two waters in the pore space: the far water, the normal formation water, and the near water, the clay-bound water held in the electrical double layer at the clay surface (SLB Energy Glossary, "dual water"). The far water conducts as the formation water does, with conductivity $C_{wf} = 1/R_w$. The near water conducts through the clay's counter-ions and has a conductivity of its own, $C_{wb} = 1/R_{wb}$, that does not depend on the salinity of the far water; it is held to the clay and does not flow. Clavier, Coates and Dumanoir built the model on this picture (SPE Journal, 1984; first presented as SPE 6859 in 1977).

Let $\phi_t$ be the total porosity, which holds both waters (Section 4.6), $S_{wt}$ the total water saturation and $S_{wb}$ the share of the pore space that the bound water takes. For $n = 2$ the conductivity of the rock is

The second form is the meaning of the model. The free water, $S_{wt} - S_{wb}$, and the bound water, $S_{wb}$, conduct in parallel, each at its own conductivity, inside one rock whose geometry is Archie's. With no bound water, or with $C_{wb} = C_{wf}$, it is Archie's equation. The clay term sits where Waxman-Smits puts $B\,Q_v$ (Section 8.4): $S_{wb}(C_{wb} - C_{wf})$ is the extra conductance the clay brings beyond what the same water, free, would carry. The equation is a quadratic in $S_{wt}$, and the kernel (Petro.dualWaterSw) solves it for its positive root.

The Figure's Rock, and Why It Is Not Ogbon-1

Ogbon-1's resistivity was built as clean Archie in every bed, its shaly oil sand included (Section 7.6), so the well has no clay conduction for this model to find. The figure therefore builds a what-if rock that has it. The rock, this section's own, has the porosity of Ogbon-1's shaly oil sand, 17.0 % (the mean of its 144 half feet), and the book's $R_w$ of 0.040 ohm.m. Its clay binds water as Section 4.6 takes it out, $S_{wb} = V_{sh}\,\phi_{sh}/\phi_t$, with $\phi_{sh}$ the density porosity of Ogbon-1's 429 shale half feet, 9.8 %; at the shaly oil sand's 21.9 % clay that is 12.6 % of the pore space, and the effective porosity $\phi_t(1 - S_{wb})$ is 14.9 %, the 14.9 % of Section 4.6. (The glossary writes the bound water from the clay's counter-ion concentration, $S_{wb} = \alpha\,v_q\,Q_v$ with $v_q$ about 0.28 cm3/meq at 25 °C; Ogbon-1 has no core measurement of $Q_v$, so the figure takes the route of Section 4.6.)

The bound water's resistivity comes from the well's own shale. In a shale the bound water fills the pores, so $S_{wt} = S_{wb} = 1$ and the law gives $R_{sh} = a\,R_{wb}/\phi_{sh}^{m}$, that is $R_{wb} = R_{sh}\,\phi_{sh}^{m}/a$. With the well's 2.20 ohm.m shale and its 9.8 % that is 0.0213 ohm.m, a conductivity of 46.9 S/m against the free water's 25 S/m: the bound water conducts 1.88 times as well. That is this book's way of choosing $R_{wb}$ from a shale, not a published value for a clay. The rock's resistivity is then made forward by the law above from a free-water saturation you choose, and read back. A model that reads a rock made by its own law recovers it exactly, which tests the arithmetic and not the physics; a real well tests the model against core.

Reading It Back: Total, Bound, Free

Take the default: 21.9 % clay and 19.6 % water in the effective pore space. The rock holds 12.6 % bound water and 19.6 % of the rest, so its total water is 29.8 %, and its deep log reads 11.3 ohm.m, where a clean rock with the same water would read 15.6 ohm.m. Clean Archie, which knows one water, reads that lower log as 34.9 % water, 1.17 times the total the rock holds. The dual-water model, with the bound water at 0.0213 ohm.m, returns 29.8 % total, 12.6 % of it bound, and the rest is the effective saturation:

The second equality says that the hydrocarbon in the rock, 12.0 % here, is the same whether it is counted in the total pore space or in the effective one. Section 4.6 treats the clay-bound water as immobile, so the free water is the water that can move.

More Current Than Water

Plate (c) shows why the clay matters out of proportion to its volume. In the default rock the bound water is 42.5 % of the water in the pores and carries 58.1 % of the current, because it conducts 1.88 times as well as the free water; at the book's 40 % shale cutoff the figures are 60.5 % and 74.2 %. Clean Archie sees only the lowered log, and puts the extra current down to extra water: that is the over-read of Section 8.1, and with no clay at all (35.9 ohm.m) it vanishes.

Pay That Clean Archie Throws Away

Set the clay to 30 % and the free water to 40 %. The clay binds 17.3 % of the pores, the total water is 50.4 %, and the log reads 4.18 ohm.m. Clean Archie gives 57.5 %, over the book's 50 % cutoff, and the half foot is not pay. The dual-water model gives 40.0 % free water, under the cutoff, with 8.4 % of the rock in oil. Applying the book's cutoff to the free water, on the effective porosity, is this figure's own choice, and the total water of 50.4 % is itself over the cutoff: the model changes which water is counted, and the choice of that is the interpreter's.

Run on Ogbon-1: an Over-Correction

On the shaly oil sand as logged, its 144 half feet read as Section 7.6 reads them (a gamma-ray clay averaging 21.8 % and a density porosity averaging 17.0 %), clean Archie returns the 19.6 % water the well was built with. The dual-water model, told there is 12.6 % bound water on average at the shale's 0.0213 ohm.m, finds clay conduction that is not in the log and reads 14.8 % total and 2.7 % free water on average, and 14.5 % of the rock in oil against the 13.7 % built. At 29 of the half feet the log holds less water than the clay binds, and their free water is set to 0. That is the well's doing, not the model's: Ogbon-1 is a clean-Archie rock with clay in its gamma ray, and every shaly-sand model reads less water there than the truth. Section 8.6 runs the models down the whole well.

What the Model Needs

Two inputs beyond Archie's, and the answer is as good as they are. The first is the bound water, here from the shale volume and the shale's porosity. The second is $R_{wb}$, and plate (b) shows what a wrong one does at the default rock, where the right value gives 19.6 % free water. Put in as the free water's own 0.040 ohm.m, the bound water carries no extra current, the total water read is clean Archie's 34.9 %, and only the subtraction of the bound water is left, 25.5 % free. At 0.010 ohm.m the total is 20.7 % and the free water 9.3 %; at 0.060 ohm.m, 37.1 % and 27.9 %. A bound-water resistivity that makes the log's water less than the clay binds gives a free water of zero, which the figure says rather than print a negative saturation. Shale volumes past the book's 40 % cutoff are shaded in plates (a) and (c): the book calls that rock shale, not pay.

References

• Clavier, C., Coates, G. and Dumanoir, J. (1984). Theoretical and experimental bases for the dual-water model for interpretation of shaly sands. SPE Journal, 24(2), 153-168. SPE-6859-PA. doi:10.2118/6859-PA.
• Schlumberger. Energy Glossary, "dual water". glossary.slb.com.