Authors: Da Zhu (RGL Reservoir Management Inc.) | Jacky Wang (University of Calgary) | Yi Su (University of Calgary) | Ian D. Gates (University of Calgary)
Numerical simulators have been extensively used in reservoir engineering for several decades. These simulators, based on energy, material, and momentum (multiphase Darcy law) balances and thermodynamic equilibrium of components between phases, solve a coupled set of nonlinear partial differential equations. We have observed multiple states for simulation of Steam-Assisted Gravity Drainage (SAGD) with multiple steam-to-oil ratios resulting at the same steam injection rate. The existence of multiple solutions and potentially limit cycle behavior and its associated bifurcation branching in the operation parameter space inspires us to consider a dynamical approach to reservoir simulation. There are four dominant states of stability: absolutely stable; neutrally stable; unstable subject to infinitesimal perturbation; and unstable subject to finite amplitude perturbation. In essence, instability is a process that releases potential energy stored in the base state to the perturbation state. In a reservoir simulation, if we impose a perturbation with a certain magnitude to a quasi-steady state, linear stability theory predicts that once the system becomes unstable, the magnitude of the perturbation grows with time infinitely. However, in reality, due to nonlinearity the system causes it to evolve to a new quasi-steady state. The questions that we are going to address in this paper are: How can we use a transient reservoir simulator to detect instability of the system that may lead to different and multiple operating states? As a case study, we will use a 2D homogeneous SAGD model. Once the model reaches a quasi-steady state, we will call it our base state. Then we impose different steam injection rate perturbations on the system and see how the system responds to these changes. Different behaviors result –for finite amplitude perturbations, the state evolves to a new state (Hopf bifurcation) (Strogatz 2014). Our goal is to use an existing commercial simulator to construct multiple operating states and describe an approach to detect them. Multiple operating states could have significant implications for process control and risk/uncertainty management of reservoir operations.
