Identifying reservoir depletion patterns from production-induced deformations with applications to seismic imaging |

We begin by formulating a *closed* system of four equations that describes a homogeneous *quasi-static* linear poroelastic medium (Segall, 2010):

and

In the above equations is a spatially-distributed

where in equation 3 is

The most ``mathematically accurate'' way of computing the displacement field and associated pore pressure change is to solve a boundary-value problem for 1,2 with known data (e.g., known pressure evolution within existing wells, measured earth displacements or estimated stresses) used as *boundary* or *initial* conditions. However, even in the simplest cases of a homogeneous medium, analytical solutions of boundary-value problems for equations 1,2 is challenging. *Uncoupling* equations 1 and 2, where permissible, could result in more tractable problems, both analytically and numerically. For example, assuming a known pore pressure change, we can solve equation 1 for the displacement field
, using
in the right-hand side as a ``body force'' distribution (Geertsma, 1973),(Segall, 1992).

In our approach we use the elastostatic Green's tensor for the pure elastic equilibrium equation in the left-hand side of equation 1 to compute the displacement as

assuming (including body forces is trivial). The elastostatic tensor in equation 4 has the meaning of the displacement along axis at point due to a concentrated force along axis at point (Wang, 2000),(Segall, 2010). From equation 4 we can see that the divergence of the elastostatic tensor has the meaning of deformation due to a concentrated dilatational force.

In order to apply equation 4 to practical reservoir models and computation of *surface subsidence*, the corresponding Green's function should be constructed for a *half-space* with the free-boundary condition imposed on its bounding plane (Segall, 2010). We use the analytical expression for the Green's function obtained by Mindlin (Mindlin, 1936) - see Appendix A for the details. The integral in the right-hand side of 4 is taken over the reservoir domain and hence singularities corresponding to
do not appear. However, the terms in non-diagonal tensor components that contain
in the denominator blow up at locations directly above (or below) the reservoir and must be truncated in a numerical quadrature. Another important aspect of using an analytical expression for the Green's function is that the divergence in the right-hand side of equation 4 can be calculated analytically. However, in our implementation we compute the divergence using central differences of the second-order of accuracy.

Identifying reservoir depletion patterns from production-induced deformations with applications to seismic imaging |

2012-05-10