Average Velocity

Average velocity is the velocity from surface described by the standard equation:

V = Z / T

where Z is the vertical depth and T is the one-way time (OWT) in seconds

Interval Velocity

Interval velocity is the average velocity over a given interval:

Vint = deltaZ / deltaT

where deltaT is the OWT over the interval in seconds

Interval velocity is constant at any depth within the interval.

Instantaneous Velocity

Instantaneous velocity is the velocity at a specific point at depth Z within a given interval. Over this interval Vinst changes as Z changes in relation to a gradient K. The Vo+Kz function describes how Vinst varies linearly over the interval. Using this equation we can estimate instantaneous velocity at a specific depth point within the interval.

Vinst = Vo + Kz

where Vo is the instantaneous velocity at zero meters
Vinst can also described as the rate of change of depth with time over a given interval:

Vinst = deltaZ / deltaT

Reference Velocity at MSL (Vo)

 The reference velocity is referred to as Vo because we calculate the instantaneous velocity at MSL (0 meters). Where the interval is deep and the gradient is large ths can result in a negative reference velocity. Commonly, the reference velocity is taken from the top of the interval to obtain a more meaningfull reference velocity (Vref).

Vo = Kz / [e (K.t) - 1]

Where t is OWT in seconds from surface and Vo is the instantaneous velocity at zero meters
Calculating Depth from Reference Velocity (Vo)

 The classic calculation relating depth Z to Vo and time is:

Z = (Vo / K ) * [(e K.t) -1]

Where Z is depth
Vo is reference velocity at MSL
K is gradient
t is OWT in seconds from surface

For layer-cake models this equation only describes the first layer. Subsequent layers are described using:

deltaZ = (Vref / K ) * [(e K.deltaT) -1]

where,
deltaZ is the depth from the top of the interval (Zref - Z)
deltaT is the OWT corresponding to deltaZ
Vref is the instantaneous velocity at the top of the interval
Calculating Vref at the Top of an Interval (sic. Vo from Vint)

 If you know K and have time and depth values from the well, you can calculate the Vref at the top of the layer as follows:

(Equation 1)

Vref = K . deltaZ  /  (e (K . deltaT) -1)  

Where, Vref is instantaneous velocity at the top of the interval
K is the gradient
deltaZ is interval thickness in m (deltaZ = deltaT . Vint)
deltaT is interval thickness in OWT in seconds

Similary, if you know interval velocity Vint and want to calculate Vref using a known K. You obtain DeltaT or DeltaZ from:

Vint = deltaZ / deltaT

(Note this is the same equation as above for Vinst!)

Calculating Depth (Isopach) When Vo and DeltaT are Used

 When the Vo is measured from surface (0 meters) you can calculate the depth using the following formula:

Z2 = [((Vo/K) + Z1) . e (K . deltaT)] - (Vo/K)

Where, Z2 is the depth at the bottom of the interval
Z1 is the depth of the previous horizon
deltaT is the OWT over the interval in seconds
Vo is reference velocity at zero meters
K is gradient

Calculating Vref For a Different Reference Depth (Zref)

 Following from Equation 1 above, we obtain Reference Velocity (Vref) for a reference depth Zref at a different depth level:

Vref = V1 - K * (Z1 - Zref)

Where, Vref is the instantaneous velocity at Zref
Zref is the reference depth
V1 is the instantaneous velocity at Z1, i.e. it would be Vref at the top of our layer
Z1 is the depth of the top of the layer
K is the gradient

Note: on a scientific calculator 'e' is the inverse ln