Rock-physics templates: reading rocks in elastic space

Why impedance and $V_{\mathrm{P}}/V_{\mathrm{S}}$

• $Z_{\mathrm{P}}=\rho V_{\mathrm{P}}$. The acoustic impedance controls the normal-incidence reflection coefficient (Section 5.1), and it is what a post-stack inversion delivers (Section 7.3). A soft, porous or light rock has a low $Z_{\mathrm{P}}$ and sits on the left of the template; a stiff, dense, tight rock sits on the right.
• $V_{\mathrm{P}}/V_{\mathrm{S}}$. For an isotropic rock $(V_{\mathrm{P}}/V_{\mathrm{S}})^2 = K_{\mathrm{sat}}/\mu + \tfrac{4}{3}$, so the ratio follows the bulk modulus against the shear modulus, and the density cancels. A pore fluid changes $K_{\mathrm{sat}}$ but not $\mu$ (Section 5.3), so gas pulls the ratio down. Estimating it needs $V_{\mathrm{S}}$ as well, from a shear log or a pre-stack inversion.

Together the two axes separate what the fluid does, mostly down, from what porosity and stiffness do, mostly sideways. They do not separate everything: Goodway’s $\lambda\rho$ and $\mu\rho$ of Section 5.2 are the same information in other coordinates, and no choice of axes can split rocks whose $K$, $\mu$ and $\rho$ are all the same.

The grain frame: friable and cemented sand

A clean sand starts as a loose pack of grains at about the critical porosity, $\phi_{\mathrm{c}} \approx 0.40$, above which it would be a suspension. Hertz–Mindlin contact theory (Mindlin 1949) gives the moduli of a random pack of identical spheres under an effective pressure $P$, from the grain’s shear modulus $\mu$ and Poisson’s ratio $\nu$ and the number of contacts per grain $n$:

For quartz ($K$ = 36.6 GPa, $\mu$ = 45.0 GPa) at 2 km, with 25 MPa of effective pressure and Murphy’s (1982) $n = 8.64$ at $\phi_{\mathrm{c}} = 0.40$, the pack has $K_{\mathrm{HM}}$ = 2.06 GPa and $\mu_{\mathrm{HM}}$ = 3.03 GPa, less than a tenth of the mineral’s. Two ways of losing porosity then give two frames, the cemented one built in two steps:

• Friable sand (Dvorkin and Nur 1996): finer grains fill the pores of the pack, with no cement. The modified lower Hashin–Shtrikman bound joins the pack at $\phi_{\mathrm{c}}$ to the mineral at zero porosity; the frame stays soft until the porosity is low.
• Contact cement (Dvorkin and Nur 1996): quartz cement coats the grains evenly and so welds their contacts. The first few percent stiffen the pack steeply: with 2 % quartz cement the pack at 38 % porosity has $K$ = 3.75 GPa and $\mu$ = 5.21 GPa.
• Constant cement (Avseth, Dvorkin, Mavko and Rykkje 2000): from that cemented pack, porosity falls by poorer sorting at a fixed cement content, by the same lower bound. It describes sands cemented a little and sorted variously, as Avseth and co-authors found in North Sea reservoirs.

Pressure, coordination number, mineralogy and the choice of frame each move every line of the template. Clay lowers the mineral’s moduli and raises $V_{\mathrm{P}}/V_{\mathrm{S}}$; deeper burial stiffens the pack. A template is therefore built for the rocks it will read, and checked against their logs before it reads anything.

The fluids, and the shale trend

Gassmann’s equation (Section 5.3) saturates each frame. Figure 7.2 uses the fluids of Figure 5.3 at 2 km (Batzle and Wang 1992): brine of 2.98 GPa and 1.054 g/cm³, gas of 0.041 GPa and 0.132 g/cm³, and the book’s 40 API oil, 0.95 GPa and 0.83 g/cm³. Partial gas saturation mixes brine and gas uniformly, the Wood average, the lower limit of Section 5.3. Each porosity gives a curve from the brine line to the gas line, and the template is the grid of those curves and the fluid lines.

Shales are drawn as a trend, not modelled: the shale of Figure 5.5, with $V_{\mathrm{S}}$ from Castagna’s mudrock line and density from Castagna, Batzle and Kan (1993), as $V_{\mathrm{P}}$ rises with compaction from 2.7 to 4.6 km/s. It runs from high $V_{\mathrm{P}}/V_{\mathrm{S}}$ at low impedance down towards the sands at high impedance.

Reading Figure 7.2

• The figure opens on a friable brine sand of 22 % porosity at $Z_{\mathrm{P}}$ = 7.21 km/s·g/cm³ and $V_{\mathrm{P}}/V_{\mathrm{S}}$ = 1.83. Gas in the same pores would put it at 5.60 and 1.49. With 3 % scatter, 56 of the 60 samples read as brine on its own template, at porosities of 21 to 23 %; the other 4 fall nearest the shale trend.
• Porosity moves a brine sand left and up. From 10 % to 35 % porosity $Z_{\mathrm{P}}$ falls from 10.02 to 5.45 while $V_{\mathrm{P}}/V_{\mathrm{S}}$ rises from 1.69 to 1.95. As the frame loosens, the brine carries a growing share of the stiffness against compression and none against shear, so $V_{\mathrm{P}}$ falls more slowly than $V_{\mathrm{S}}$. The gas line stays nearly flat, 1.50 to 1.45, so brine and gas sands part as porosity rises.
• A little gas does most of it. The dotted line of 10 % gas lies close to the full gas line: at 25 % porosity the first tenth of gas covers 86 % of the fall in $V_{\mathrm{P}}/V_{\mathrm{S}}$ from brine to gas. This is the fizz-water problem of Section 5.3, seen on the template.
• Cement looks like hydrocarbon. A brine sand of 20 % porosity with 2 % contact cement plots at 8.54 and 1.67, to the right of and below a friable brine sand of the same porosity (7.57 and 1.81). Read on the friable template it lies nearest the oil line at 12 % porosity, and only 25 of the 60 samples read as brine.
• Tight rocks crowd together. At 6 % porosity the brine and gas lines are 0.13 apart in $V_{\mathrm{P}}/V_{\mathrm{S}}$, against 0.44 at 30 %; with 5 % scatter, 8 of 60 samples of a tight gas sand read as brine and 11 as oil. A tight brine sand there runs close to the shale trend, and with 3 % scatter 20 of its 60 samples read as shale.

Where a template is ambiguous

• Porosity against fluid, on impedance alone. A friable gas sand of 22 % porosity has the impedance of a friable brine sand of 34 % porosity. Only $V_{\mathrm{P}}/V_{\mathrm{S}}$ (1.49 against 1.94) tells them apart, which is why the template needs pre-stack data.
• Cement against fluid. Cement and hydrocarbon both lower $V_{\mathrm{P}}/V_{\mathrm{S}}$ at a given impedance, so a cemented brine sand read on a friable template looks like an oil sand of lower porosity. The frame has to be chosen from the wells, the burial history (quartz cement grows mostly above about 70 to 80 °C) and the density, before the fluid is read.
• Saturation. Under uniform mixing the lines of 10 % and 100 % gas are close; the template cannot tell a commercial gas column from a few percent of gas. Patchy saturation spaces them more evenly, and which applies is itself uncertain.
• Lithology at high impedance. Tight sands, cemented sands and compacted shales converge towards the right of the template, so small errors in the data carry a sample from one to another.

Building and checking a template

1. Choose the template’s conditions: mineralogy, depth, effective pressure, temperature, and the fluids at those conditions (Batzle and Wang).
2. Compute the dry frame for each candidate model (friable, contact cement, constant cement) and saturate it with Gassmann for brine, oil and gas.
3. Plot the well logs, $Z_{\mathrm{P}}$ and $V_{\mathrm{P}}/V_{\mathrm{S}}$ from $V_{\mathrm{P}}$, $V_{\mathrm{S}}$ and $\rho$ at every sample, coloured by gamma ray or by the petrophysical interpretation.
4. Check which model the brine-filled sands follow, and adjust the cement content, pressure or mineral moduli until the model fits the wet sands; then check that the hydrocarbon sands fall where the model puts them.
5. Only then read inversion results on the template, with the scatter of the inversion in mind: a point is a probability over the lines, not a position on one of them (Section 7.5).

Pitfalls

• A template for the wrong depth. Effective pressure stiffens the pack, and the gas modulus rises with depth; a template built at 2 km misreads a sand at 3 km.
• Too stiff in shear. Hertz–Mindlin with perfect adhesion overestimates the shear stiffness of loose sands, so real unconsolidated sands plot at higher $V_{\mathrm{P}}/V_{\mathrm{S}}$ than the friable line of Figure 7.2. Practical templates reduce the contact shear stiffness, calibrated on the logs.
• Clay and mineralogy. A shaly sand has a softer mineral and a higher $V_{\mathrm{P}}/V_{\mathrm{S}}$; a template for clean quartz reads it as wetter or more porous than it is.
• Logs and seismic are not the same measurement. Sonic logs sample centimetres at kilohertz; inversion averages tens of metres at tens of hertz. Upscale the logs before comparing, and expect inversion points to scatter more.
• Two coordinates are not everything. Points that coincide on $Z_{\mathrm{P}}$ and $V_{\mathrm{P}}/V_{\mathrm{S}}$ may separate in density, which a far-angle inversion can estimate.