Sonic Porosity (Raymer-Hunt-Gardner)

Summary

The Raymer-Hunt-Gardner transform gives Sonic porosity (Raymer-Hunt-Gardner) from the ratio of the matrix slowness to the measured slowness, raised to a power. It has a curved response that follows field data better than the straight-line Wyllie relation, especially at high porosity, and it needs no fluid slowness. Use it where Wyllie overestimates.

Inputs and outputs

Equations

The Raymer-Hunt-Gardner relation in the exponent form used in practice is:

limited to the interval 0 to 1. For an exponent of 2 it is the form \(\phi = 1 - \sqrt{\Delta t_{ma}/\Delta t}\), which is what the original two-term relation reduces to when the fluid term is small:

The Wyllie value from the previous page is shown for comparison:

Single-value calculator

Behavior

The curves are concave: porosity rises more slowly than slowness, where the Wyllie line is straight. A lower exponent gives higher porosity. At 90 µs/ft the exponents 1.6, 1.76 and 2.0 give 0.261, 0.240 and 0.215, against 0.258 from Wyllie. At 70 µs/ft the RHG values are all above Wyllie (0.109), and at 110 µs/ft they are all below (Wyllie 0.408), so the two cross near the middle of the range. This is the reason RHG is the safer choice at higher porosity.

Parameter guidance

Matrix slowness is the same as for Wyllie: 55.5, 47.5 and 43.5 µs/ft for sandstone, limestone and dolomite.

Exponent. Values near 1.6 for sandstone, 1.76 for limestone and 2.0 for dolomite are quoted in practice, and 2.0 is the value that follows from the original relation. The exponent is the one parameter to calibrate: fit it so that sonic porosity matches core or the neutron-density porosity in a clean water-bearing interval.

The page's worked example compares the exponent form with the original two-term form and with a commonly quoted chart form, \(\phi = 0.625\,(\Delta t - \Delta t_{ma})/\Delta t\).

Worked example

Slownesses of 70, 90 and 110 µs/ft compared across the sonic transforms, with a matrix of 55.5 and fluid of 189 µs/ft:

dt_ma, dt_f = 55.5, 189.0

def wyllie(dt):
    return (dt - dt_ma) / (dt_f - dt_ma)

def rhg_power(dt, x):
    return 1.0 - (dt_ma / dt) ** (1.0 / x)

def raymer(dt):
    """Raymer-Hunt-Gardner in its original two-term form, solved by bisection:
    1/dt = (1 - phi)^2 / dt_ma + phi / dt_f"""
    lo, hi = 0.0, 0.5
    for _ in range(60):
        mid = (lo + hi) / 2
        if (1 - mid) ** 2 / dt_ma + mid / dt_f > 1.0 / dt:
            lo = mid
        else:
            hi = mid
    return (lo + hi) / 2

def chart_form(dt):
    return 0.625 * (dt - dt_ma) / dt

print(f"{'dt':>4s} {'Wyllie':>7s} {'x=1.6':>7s} {'x=1.76':>7s} {'x=2.0':>7s} {'two-term':>9s} {'0.625':>7s}")
for dt in (70.0, 90.0, 110.0):
    print(f"{dt:4.0f} {wyllie(dt):7.3f} {rhg_power(dt, 1.6):7.3f} {rhg_power(dt, 1.76):7.3f} "
          f"{rhg_power(dt, 2.0):7.3f} {raymer(dt):9.3f} {chart_form(dt):7.3f}")

Output

  dt  Wyllie   x=1.6  x=1.76   x=2.0  two-term   0.625
  70   0.109   0.135   0.124   0.110     0.132   0.129
  90   0.258   0.261   0.240   0.215     0.266   0.240
 110   0.408   0.348   0.322   0.290     0.371   0.310

Assumptions and limitations

• A consolidated, liquid-filled formation with intergranular porosity. Vugs, fractures and gas affect it as they do Wyllie.
• The exponent represents the lithology and compaction of the formation and is constant over the interval.
• The log is free of cycle skips and the matrix slowness is known.
• The exponent form is an empirical simplification of the original two-term relation and is accurate over the usual porosity range only.

QC checks

• Calibrate against core or neutron-density porosity in a clean zone, and check that the residual shows no trend with porosity.
• Compare with the Wyllie porosity. They agree near 0.2 to 0.25 and diverge outside it; a large difference at low slowness points to the matrix slowness.
• Look for isolated spikes (cycle skips) before trusting thin intervals.
• A porosity that is systematically lower than neutron-density in carbonate usually means vuggy porosity the sonic does not see.

Going Deeper

Raymer, Hunt and Gardner proposed their transform in 1980 as an improvement over the time-average equation, which it replaces with a piecewise relation fitted to a large field database. In the original it is a set of velocity-porosity relations for different porosity ranges, and the compact exponent form above is an approximation that is common in software. The chart form with a constant of 0.625 is the version printed in service-company chartbooks. Because the compact form is an approximation, the exponent should be treated as a fitting parameter and not as a physical constant.

References

1. Raymer, L.L., Hunt, E.R. and Gardner, J.S., 1980. An improved sonic transit time-to-porosity transform. Transactions of the SPWLA 21st Annual Logging Symposium, Paper P.
2. Schlumberger, Log Interpretation Charts. Schlumberger, Houston, Texas (updated periodically).
3. Wyllie, M.R.J., Gregory, A.R. and Gardner, L.W., 1956. Elastic wave velocities in heterogeneous and porous media. Geophysics, 21(1), 41–70.

Python reference implementation

Related methods

• Sonic Porosity (Wyllie Time-Average)
• Density Porosity
• Neutron-Density Porosity
• Gas and Organic-Matter Corrections