Practical Control of Timestep Selection in Thermal Simulation

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Summary We present a new technique for estimating accumulating time truncation error in thermal reservoir simulation. The calculation uses only current reservoir information, thereby avoiding difficulties encountered by some past formulations, and is integrated easily into fully implicit reservoir simulators. We describe an error-controlled timestep selector based on this estimate as well as provisions for limiting the calculated timestep based on the imminent appearance or disappearance of a gas phase. We also provide a new technique for estimating the solution of certain Newtonian iterations. The algorithm is capable of calculating timesteps at the beginning of a simulation and when major well rate changes are specified. The resulting timestep selector enhances simulator accuracy without increasing computational work and affords greater reliability through the reduction of failed timesteps. Introduction Recently, interest in the development of timestep selection schemes that monitor time truncation error for fully implicit thermal reservoir simulators has increased. If timestep size is based on truncation error, which measures the deviation of the reservoir simulator's equations from simple linear behavior in time, more efficient and reliable simulations should result. Efficiency follows from the reservoir simulator's ability to take large, stable timesteps when little nonlinear behavior is present, and reliability follows from the appropriate reduction in timestep size calculated when nonlinear transient behavior appears, avoiding failure of the nonlinear equation-solving technique (Newton's method). Such error-controlled approaches improve on methods that use linear information and limit primary variable changes over a timestep. These latter techniques are parameter sensitive and monitor nonlinear behavior only indirectly. Mehra et al. show some advantages of basing the actual timestep calculation on methods that account for rapidly decaying transient behavior-such as those presented for stiff equations by Lindberg. These presented for stiff equations by Lindberg. These transients occur in reservoir simulation when well rates change substantially or when certain phases (particularly gas or steam) appear or disappear. Unfortunately, previous truncation-error estimates were not reliable when transient behavior was initiated. For instance, timesteps are required to increase in size in Mehra et al., precluding the timestep selection method from responding to some transient phenomena. These difficulties could result from the use of truncation-error estimates that rely on pretransient reservoir states in their calculations. We have pretransient reservoir states in their calculations. We have developed a new technique that estimates truncation error based only on the current reservoir state. It responds properly to transients by reducing the timestep level for a brief properly to transients by reducing the timestep level for a brief period, which aids convergence of Newton's method, and period, which aids convergence of Newton's method, and then quickly builds up the timestep level. Thus transient behavior is handled efficiently. We present the uses for an error-controlled timestep selection scheme in a frilly implicit thermal simulator, and describe some additional features. A provision for limiting the timestep size, based on prediction of steam appearance or collapse, is useful particularly for thermal reservoir simulation because it reduces the need for repeated timesteps. A new extrapolation procedure that obtains the first Newtonian cycle of the next timestep when the timestep level is relatively constant can reduce computational costs up to 20%. Also, the timestep selector can calculate a timestep at the beginning of a simulation or at specified times when major well rate changes are made, thus relieving the user from estimating suitable values. The overall timestep calculation introduces little simulator overhead and is easily integrated into any fully implicit reservoir simulator. The user controls the timestep selector by setting a single relative-error parameter, while the simulator internally calculates certain maximum pressure and temperature parameters. pressure and temperature parameters. Error-controlled timestep selection is a robust technique for simulating thermal processes efficiently. Additionally, errors in the calculated results (pressures, saturations, and temperatures) lie within the error limits specified by the relative-error parameter and are smaller than those exhibited by simulations performed with specified variable changes. Thus these techniques enhance simulation accuracy, one of the original goals of truncation-error techniques.

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