Abstract Geertsma's analytical method is often used to estimate changes in stress and strain in the rock around compacting reservoirs, and then to estimate seismic traveltime shifts. The present work extends Geertsma's analytical solution to the case where a hard rock layer is located beneath the compacting reservoir. The analytical solution for the three components of displacement has been derived using both thermo-elastic and poro-elastic theory. The impact of the hard rock layer on the seismic traveltime shifts are estimated by applying the dilation factor method for strain sensitivity of the seismic velocity, and then comparing the results for our model (Rigid Basement Model) to the results for the original Geertsma model.
The rigid layer prevents any displacements below a given depth below the reservoir. This has consequences for the displacements above the rigid layer. In our model example, these results in a larger stretching of the overburden and correspondingly larger time-shifts compared to the Geertsma model. Thus the strain field due to a hard rock layer may explain the high time-shifts observed in deeper layers, as experienced in Shearwater (Staples et al., 2007; Cox et al., 2008).
The theory used for the derivation of the method is known for the users of the Geertsma's model, thus the extension proposed in this work can easily be implemented in a code based on Geertsma's analytical solution.
Introduction Reservoir compaction due to production depletion causes changes in displacement, stress and strain fields in the subsurface. When the compaction propagates to the surface, it may result in visible subsidence and changes in elastic properties of the rocks (Landrø and Stammeijer, 2004). Monitoring both surface deformation and time-shifts from time-lapse seismic is an important tool for dynamic reservoir characterization. Examples of some monitored compacting reservoirs are the Ekofisk field (Guilbot and Smith, 2002), the Valhall field (Barkved et al., 2005), the Shearwater field (Staples et al., 2007), and the Dan field (Hatchell et al., 2007), all located in the North Sea.
A forward model is needed for predicting the changes of the subsurface due to a pore pressure drop. In general, the forward model integrates geomechanical modeling and synthetic seismic modeling trough a suitable rock-physics model. In 2005, Hatchell and Bourne (2005a) proposed a simple and useful model, which uses Geertsma's (1973) analytical model for computation of the vertical strains in the half space, and then approximates the zero offset time-shifts through the empirical dilation factor. Alternatively, we may use more extensive numerical modeling both for the geomechanical and for seismic side. There are available, for instance, finite element codes for geomechanical modeling, and ray tracing or finite difference codes for synthetic seismic modeling. We can even couple flow simulation and geomechanical modeling, as proposed by Minkoff (1999); or Settari (2001). The rock-physics model is generally an empirical relation (Hatchell et al., 2005b; Røste et al., 2006; Bauer et al., 2008). Recently, Fuck et al. (2008) proposed a forward model, which relate geomechanics to seismic through nonlinear elastic theory.
Compaction and subsidence may be modeled both with analytical and numerical methods. Analytical models are based on several simplifying assumptions, but give the solutions quickly. On the other hand, the numerical codes are more flexible with respect to complex geometry and heterogeneities of the subsurface. However, this complexity causes an increase in the time needed for set up and computation. Thus, the analytical models are generally preferred in the initial stage of production to get a test of compaction and subsidence; they also enter preferentially in the loop for testing new rock-physics models (Cox, 2008), or to build inverse models for history matching of the pressure distribution in the reservoir (Hodgson, 2007).