Waxman-Smits

Brine and Clay in Parallel

Simandoux and the Indonesia equation (Sections 8.2 and 8.3) fit the clay term to a shale volume and a shale resistivity. Waxman and Smits (1968) went to cores instead, and gave the clay term its own physical measure: the exchange cations the clay holds at its surface. They called their concentration per unit pore volume $Q_v$, the cation-exchange capacity of the rock divided by its pore volume, in meq/cm3 (the same as meq/ml), and gave the cations an equivalent conductance $B$. In a rock full of water the cations conduct beside the brine, so the two conductances add. Waxman and Smits defined $F^{*}$ as the reciprocal of the slope of the rock's conductivity $C_o$ against the water's, $C_w$, and for a clean sand it is the usual formation factor:

With hydrocarbon in the pores, and a saturation exponent $n^{*}$ of 2, the brine term falls as $S_w^2$ and the clay term only as $S_w$:

Solved for $S_w$ it is a quadratic, as Simandoux's was, but every piece now has a meaning. With $Q_v = 0$ it is Archie's equation. For $n^{*}$ other than 2 there is no closed root and the equation is solved numerically.

The Rock the Figure Uses

Ogbon-1 has no clay conduction: its resistivity was built as clean Archie in every bed, the shaly sand included. So the figure does not run the model on the well and call it a correction. It builds a what-if rock with the porosity of Ogbon-1's shaly oil sand, 17.0 %, and a clay you set (this section's own rock: its clay is a $Q_v$, not a shale volume), works out the resistivity that rock would log, and reads it back two ways, with clean Archie and with Waxman-Smits. The truth is known, so the error of each is too. A clay of 0.20 meq/cm3 in that rock is a cation exchange capacity of 1.5 meq/100 g of rock (the capacity of the rock over its pore volume, turned back into a capacity per gram with the well's sandstone grain density). In the well's water, 0.040 ohm.m, the brine conducts 25.0 S/m and the clay 0.920 S/m. The rock holds 19.6 % water and logs 30.3 ohm.m, where the same rock without clay would log 35.9 ohm.m. Clean Archie, with no clay term, takes the extra current for water and reads 21.4 %, 1.8 points too much. Waxman-Smits, given the clay the rock has, reads the 19.6 % back.

Why the Salinity Decides

The clay's conductance, $B\,Q_v$, belongs to the rock; the brine's, $C_w = 1/R_w$, to the water, and it changes far more. At 0.040 ohm.m the brine carries 25.0 S/m against the clay's 0.920 S/m, and the clay takes 3.5 % of the current in a rock full of water. In water of 2.00 ohm.m the brine carries 0.500 S/m, a fiftieth as much, and the clay 0.544 S/m, because $B$ is not a constant: the fit of Waxman and Smits (1968) to their cores at 25 °C, their Equation 30, $B = 4.6\,(1 - 0.6\,e^{-0.77/R_w})$ in (S/m)/(meq/cm3), falls from 4.60 to 2.72 between the two waters. Its curve was fitted to cores in brines of 0.19 to 4.8 ohm.m, and it is flat at the saltiest brine they measured, 0.040 ohm.m, so the figure's water runs from there. The clay now takes 52.1 % of the current, and clean Archie reads 50.2 % water in a rock that holds 19.6 %, 30.6 points too much. Waxman and Thomas (1974) went on to the temperature coefficient of the conductivity; the book gives Ogbon-1 no formation temperature, so every value here is the 25 °C one, and the well's 0.040 ohm.m is read as a 25 °C value too.

Oil Makes the Clay Count

The share is also larger with oil in the pores. The clay's share of the current is $B Q_v/(S_w C_w + B Q_v)$, which rises as the saturation falls, because the brine term goes as $S_w^2$ and the clay term as $S_w$. The same clay that takes 3.5 % of the current in the rock full of water takes 15.8 % at 19.6 % water, so a shaly sand matters most exactly where the oil is.

What the Model Needs, and What Ogbon-1 Shows

Waxman-Smits needs $Q_v$, and Waxman and Smits measured it on cores. A well without core has to estimate it, and the model reads only as well as the estimate: given half the rock's clay, 0.10 meq/cm3 instead of 0.20, it reads 20.5 %, 0.9 points too high, and given none it is clean Archie. Run on Ogbon-1's own log it shows the other side. A log carries no $Q_v$, so the chapter sets it on the well one way, from the shale line: the $Q_v$ whose clay conductance equals the shale term of Simandoux's equation, $B\,Q_v/F^{*} = V_{sh}/R_{sh}$, chosen because the shale volume and the shale's resistivity are the only clay a log carries, so the models meet the same clay (Section 8.6 takes it the same way). Over the shaly oil sand it averages 0.75 meq/cm3. Read half foot by half foot as Section 7.6 reads it, the bed gives clean Archie 19.6 %, the 19.6 % built, and Waxman-Smits with the shale line's $Q_v$ reads 13.9 %, 5.7 points too little water: it takes out a clay conduction the well does not have. On Ogbon-1 the model's correction is an over-correction; Section 8.6 runs the shaly-sand models down the whole well with that in mind. Section 8.5 tells the same physics as two waters, bound and free.

References

• Waxman, M. H. and Smits, L. J. M. (1968). Electrical conductivities in oil-bearing shaly sands. SPE Journal, June 1968, 107-122. SPE-1863-A. doi:10.2118/1863-A.
• Waxman, M. H. and Thomas, E. C. (1974). Electrical conductivities in shaly sands: I. The relation between hydrocarbon saturation and resistivity index; II. The temperature coefficient of electrical conductivity. Journal of Petroleum Technology, 26(2), 213-225. doi:10.2118/4094-PA.