Sonic Log

Key Takeaways

• Porosity determination from the sonic log uses the Wyllie time-average equation (derived empirically by M.R.J. Wyllie and colleagues at Gulf Research in 1956): DT_log = phi * DT_fluid + (1 - phi) * DT_matrix, which can be rearranged to solve for porosity: phi = (DT_log - DT_matrix) / (DT_fluid - DT_matrix), where DT_log is the measured transit time in the formation, DT_matrix is the transit time of the mineral matrix (55.5 us/ft for calcite/limestone, 43.5 us/ft for dolomite, 47.5-51.0 us/ft for quartz/sandstone, depending on the silica content), and DT_fluid is the transit time of the pore fluid (185-189 us/ft for fresh water, 179-185 us/ft for salt water, approximately 617 us/ft for gas at reservoir conditions); the Wyllie equation assumes that the rock is a simple two-component system (matrix mineral plus pore fluid) with no cement or dispersed clay, and it works reasonably well for consolidated, clean sandstones and carbonates in liquid-saturated conditions; the Hunt-Raymer-Gardner equation (Hunt et al., 1981) provides a better fit for unconsolidated sands and over-pressured formations where the Wyllie equation over-predicts porosity because the assumption of a simple matrix-fluid mixture breaks down; for gas-bearing formations, the low DT_fluid of gas relative to water causes the Wyllie equation to significantly over-predict porosity (the "gas effect" or "gas crossover" observed when the density-derived porosity is substantially lower than the sonic-derived porosity in the same interval, a classic gas indicator in log interpretation).
• Borehole compensation in the sonic tool is achieved by using a long-spacing transmitter-receiver configuration (the borehole-compensated sonic, or BCS, tool introduced by Schlumberger in the mid-1960s) that averages the up-going and down-going measurements to eliminate the effect of tool tilt and formation dip: the standard BCS tool has two transmitters (upper and lower) and two pairs of receivers at 3 and 5 feet above each transmitter; by averaging the DT measurements from the upper and lower transmitter-receiver pairs, the borehole effects (including any tilt of the tool in the borehole, variable standoff from the borehole wall, and borehole fluid variation) are cancelled from the measurement; the long-spacing sonic (LSS) tool (with transmitter-to-receiver spacings of 8 and 10 feet instead of the standard 3 and 5 feet) was developed to improve signal quality in slower, noisier formations and to provide deeper investigation depth (sensing the formation farther from the borehole where drilling-induced damage is less severe); the dipole sonic tool (introduced in the 1990s) generates a flexural wave in addition to the compressional wave, allowing simultaneous measurement of the shear wave velocity (Vs) as well as the compressional velocity (Vp), which together with the formation density provide a complete set of elastic moduli for mechanical properties calculations.
• Seismic-to-well tie (also called synthetic seismogram generation) is one of the most critical applications of the sonic log: the seismic reflection time at a given geological boundary in the subsurface is determined by the two-way traveltime of a seismic wave from the surface to the reflector and back, which depends on the interval transit times of all the formations above the reflector; by integrating the sonic log from the surface to each formation boundary, the seismic two-way traveltime at each depth in the well can be calculated, creating a depth-to-time conversion that allows the geological formations identified in the well (from core, cuttings, and wireline logs) to be located on the seismic section at the correct seismic reflection time; combining the sonic log (which provides the velocity) with the density log (which provides the formation density) gives the acoustic impedance (Z = density * velocity) of each formation, and the reflectivity between two formation layers (the seismic amplitude of the reflection between them) is calculated as R = (Z2 - Z1)/(Z2 + Z1); a synthetic seismogram is generated by convolving the reflectivity series with the seismic wavelet estimated from the recorded seismic data, producing a simulated seismic trace that can be directly compared to the actual seismic trace at the well location to validate the geological interpretation and calibrate the depth-to-time conversion.
• Cycle-skipping is a common artifact in sonic log data that occurs when the first arrival of the compressional wave at the receiver is too weak for the detection circuit to trigger and the tool instead measures the arrival of a later cycle of the wave train, producing a DT value that is too large by a multiple of the wave period (typically 1 or 2 cycle-periods): cycle-skipping is most common in poor cement bond zones (where the compressional wave energy leaks from the casing into the annular fluid rather than continuing to the formation), in gas-bearing intervals (where the gas saturation attenuates the compressional wave), in unconsolidated formations (where the wave amplitude is low and the first-cycle signal is below the detector threshold), and in highly deviated boreholes (where the standoff between the tool and the borehole wall is large and variable, reducing the signal amplitude); cycle-skipped data appears as a characteristic "spike and cycle" pattern on the DT log, with the DT values jumping erratically to anomalously large values and then returning to baseline; cycle-skipped data must be identified and excluded from quantitative applications (porosity calculation, seismic-to-well tie) because the erroneous DT values will introduce errors in the calculated porosity and the synthetic seismogram; quality control of the sonic log specifically for cycle-skipping is a mandatory step in log interpretation workflows.
• Mechanical properties estimation from the sonic log is an application that has grown in importance with the development of hydraulic fracturing and wellbore stability engineering: the compressional wave velocity (Vp, derived from DT = 1/Vp) and the shear wave velocity (Vs, measured directly by the dipole sonic tool or estimated from Vp using empirical relationships) are combined with the formation bulk density (from the density log) to calculate the dynamic elastic moduli of the formation: Young's modulus E = rho * Vs^2 * (3Vp^2 - 4Vs^2)/(Vp^2 - Vs^2), Poisson's ratio nu = (Vp^2 - 2Vs^2)/(2*(Vp^2 - Vs^2)), and the bulk modulus K and shear modulus G from the Lame parameters; these dynamic moduli (measured from sonic velocities) must be converted to static moduli (the moduli that control actual formation deformation under applied stress) using calibration relationships established from core testing in the laboratory, because dynamic moduli are typically 2 to 5 times larger than static moduli for the same formation; the static Young's modulus and Poisson's ratio are the primary inputs to hydraulic fracture design models (which calculate the fracture width, height growth, and net pressure for a given treatment design), to wellbore stability models (which calculate the mud weight window between shear failure and tensile fracture at the borehole wall), and to geomechanical models (which predict subsurface deformation from reservoir pressure depletion or fluid injection).