Cyclic 1D matching of time-lapse seismic data sets: A case study of the Norne Field

Velocity, path-length and travel time changes

Depletion-related strain and velocity changes will perturb the travel path and length of a normally-incident ray. The distance $dz$ travelled by the ray in time $dt$ at velocity $v$ is given by

$$dz = v dt.$$ (1)

For small perturbations $\delta z\ll dz$ , $\delta v\ll dv$ and $\delta t\ll dt$ , we can write

$$\frac {d(\delta z)}{dz} = \frac{\delta v}{v} + \frac {d(\delta t)}{dt},$$ (2)

where, $d(\delta z)/dz=\epsilon_{zz}$ is the vertical ``depth'' strain and $d(\delta t)/dt=\epsilon_{tt}$ is the apparent ``time'' strain. The $R$ -factor (Hatchell and Bourne, 2005a) relates the fractional change in velocity ${\delta v}/{v}$ to the depth strain $\epsilon_{zz}$ and time strain $\epsilon_{tt}$ :

$$\frac{\delta v}{v} = -R \epsilon_{zz} = -\frac{R}{1+R} \epsilon_{tt}.$$ (3)

The $R$ -factor can be measured from core samples in the laboratory (Hatchell and Bourne, 2005a; Bathija et al., 2009), derived from the estimated time-shifts (Hawkins et al., 2007), or derived from other production-related data (Hodgson et al., 2007). For most reservoir rocks, $R$ values of $4$ to $6$ have been shown to be reasonable (Hodgson et al., 2007; Hatchell and Bourne, 2005b). In this paper, a constant $R$ value of $5$ is assumed.

Figure: Apparent displacements by cyclic 1D search

Cyclic 1D matching of time-lapse seismic data sets: A case study of the Norne Field