Stoneley waves are dispersive (velocity varies with frequency), and are generated from the interaction between borehole and formation. At low frequency, it is called a tube-wave. A Stoneley waveform has a frequency content of 0.1 to 3 KHz, with most of the energy in the late arrivals at about 500 Hz.

Stoneley wave is affected by porous media fluid and by shear modulus at low frequency. Stoneley wave slowness can be modeled in non-permeable zones as follows:
      1: DTST^2 = (DENSfluidf * DTS^2 / DENS) + DTCfluid^2

Where:
  DTST = stoneley slowness (μs/ft)
  DTS = shear slowness (μs/ft)
  DENS = bulk density (g/cc)
       calculated from probabilistic analysis  by summing the density contriburion from all minerals and fluids
  DENSfluid = mud filtrate sensity (g/cc)
  DTCfluid = mud filtrate slowness (μs/ft)

By cross-plotting DTST^2 versus DTS^2 / DENS across a zero permeability zone, the slope of the straight line is DENSfluid and Y-intercept is DTCfluid^2. There is one condition on the linear fit:  all data on thecross plot should be above or on the fitted line.

The Stoneley permeability index is estimated by taking the ratio of actual measured Stoneley slowness and modeled slowness as per above model. The formula can be written as follows:
      2: STI = DTST / (((DENSfluid * DTS^2 / DENS) + DTCfluid^2)^0.5)

Where:
  STI = Permeability index from Stoneley wave travel time (fractional  - range = 0 to 1)

Flow Zone Index (FZI)
This STI model is still affected by  variations in mud cake and formation fluid type. In addition, it does not provide a means to estimate mobility or permeability magnitude directly.

Stoneley permeability index is not a permeability estimation, but it is an index of fluid movement in porous media around the borehole. Since fluid movement is a function of pore throat distribution, pore shape, and pore size, the Stoneley permeability index is a tortuosity index only. These factors can be combined in a concept called Flow Zone Index (FZI).

The Stoneley permeability index Kist is a direct measurement of FZI. Since FZI approaches to zero when Stoneley permeability index approaches to 1 in non-permeable zones, and both of them approach to infinity when permeability approaches to infinity, then a simple relationship can be derived between FZI and STI as following:
      3: FZI = IMF * (STI – 1)

Where:
  FZI = flow zone index
  IMF = flow zone index matching factor

With this equation, the only empirical factor to match actual permeability profile is IMF. Since the grain modulus has an effect on Stoneley slowness, IMF can be computed in  the probabilistic model by summing the volume weighted IMF for eaach individual mineral in the model. Note: the Schlumberger report does not indicate how the individual IMF parameters were derived. Values quoted were:
  Zone "A" Calcite = 12, Illite = 0.001, Kaolinite = 0.001, Quartz = 1.0, Siderite = 0.0
  Zone "B" Calcite = 10, Illite = 0.001, Kaolinite = 0.001, Quartz = 130, Siderite = 0.0

Stoneley permeability can be computed by using effective porosity and FZI with the following: equations:
    4: PERMst = MPERM * FZI^2 * (PHIe^3 / (1- PHIe)^2)

Where:
  PERMst = permeability from Stonely method (md)
  PHIe = effective porosity
  FZI = flow zone index
  MPERM = permeability calibration factor (default = 1014)

This method is valid for the following conditions:

This method is limited to the DSI vertical resolution of 3.5 ft. The volume measured by the DSI is 0.5 to 1 ft thick with 3.5 ft long cylinder.