In a coupled reservoir fluid flow-Geomechanics simulation, the rate of change in the pore compressibility can be orders of magnitude different from pressure diffusion rate. The significant contrast between characteristic times of the two physical processes presents opportunities to optimize the amount of coupling required during the time evolution of the simulation via asynchronous (aka loose) coupling.
Examination of the governing equations for flow in a deformable porous medium reveals that the variation of the Lagrangian porosity (δφ) depends implicitly on the variation in pore pressure (δp) and also on the variation of the bulk strain (δε). However, the pore compressibility used in reservoir simulation supposes that (δφ) is a function of (δp) only. To accomplish this, past studies have employed chain-ruling where ∂ε/∂p is substituted to find an expression for (δφ) that is a function of (δp) only. In most cases, ∂ε/∂p is not known analytically and must be derived numerically using changes in strain and pressure seen at previous coupling stages during the time evolution of the simulation. Because of the reliance on previous history, the determination of when to take the next coupling step is usually based on an arbitrary condition like requiring a fixed number of time steps to occur in the flow simulation between coupling steps.
When a coupling step occurs, a posteriori error estimate of the pore volume change is made. The simulation continues if the error is sufficiently small. If the error is large, the simulation reverts to the state at the previously coupling step and a smaller time interval elapses before the next step is taken. Obviously, reverting to a previous step is inefficient and should be avoided. Given that the error is governed by how closely the numerical determination of ∂ε/∂p is to its actual value over the time period considered and that the actual value of ∂ε/∂p depends on the heterogeneity in the subsurface as well as the rate of hydrocarbon production, a physically based coupling criterion is required for maximal efficiency.
Here, results are shown from simulations where the error in ∂ε/∂p is estimated a priori based on the rate of change of pressure. This information is then used to estimate when a significant error in porosity has accumulated which triggers a coupling step. Additionally, in each coupled time step, the actual error is calculated and use of this information results in an automated coupling method that is self-corrective and unconditionally stable. This scheme also exhibits a significant performance gain compared to other common schemes utilizing arbitrary or time based conditions for coupling when accuracy is taken into consideration.