Multi-scale simulation allows an efficient use of computational resources when one to solve problems that have domains with very different spatial and time scales and can have significantly different discretizations in space and time. This is often the case in coupled reservoir flow-geomechanics simulations. Because of the fluid rock interaction, multi-scale simulation in this context requires a consistent upscaling of pore pressure and downscaling of pore volume change. The significant contrast between characteristic times of the two physical processes also presents opportunities to optimize the amount of coupling required during the time evolution of the simulation via asynchronous (aka loose) coupling
In general, spatial upscaling of the pore pressure will happen over a collection of cells where the change in pressure is not constant. This is to be expected due to diffusive propagation of pressure perturbations in the system. The differences in pressure perturbations will also be enhanced by heterogeneities in porosity or permeability. The impact of heterogeneity in flow properties is also exacerbated when considering variations in the mechanical properties. Clearly any type of upscaling scheme must consider the combination of pressure and mechanical properties at the fine scale to determine their effective properties at the coarse scale. The same is true if you want to downscale volume strain calculated at the coarse scale to the fine scale.
There are also opportunities to have a multi-scale simulation in time. Examination of the governing equations for flow in a deformable porous medium reveals that the variation of the Lagrangian porosity depends implicitly on the variation in pore pressure and also on the variation of the bulk strain . However, the pore compressibility used in reservoir simulation supposes that is a function of only. To accomplish this, past studies have employed chain-ruling where is substituted to find an expression for that is a function of only. In most cases, ¬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 is to its actual value over the time period considered and that the actual value of 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 a solution to this problem is presented which extends the Backus spatial upscaling scheme using the pore pressure - porosity thermodynamic conjugate. A solution is also presented for downscaling the pore volume change from the equivalent anisotropic medium at the coarse scale to the layered, fine scale medium. Validation of this method is given by consideration of a layered Mandel’s problem. The behavior of numerical simulations is shown to be in good agreement with the analytical result using the proposed upscaling method.
Results are shown from simulations that are multi-scale in time where the error in 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.
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