Unsteady-state Fluid Flow
Analysis and Applications to Petroleum Reservoir BehaviorBy
- E.J. Hoffman, P.O. Box 1352, Laramie, WY 82070, USA
The ubiquitous examples of unsteady-state fluid flow pertain to the production or depletion of oil and gas reservoirs. After introductory information about petroleum-bearing formations and fields, reservoirs, and geologic codes, empirical methods for correlating and predicting unsteady-state behavior are presented. This is followed by a more theoretical presentation based on the classical partial differential equations for flow through porous media.
Whereas these equations can be simplified for the flow of (compressible) fluids, and idealized solutions exist in terms of Fourier series for linear flow and Bessel functions for radial flow, the flow of compressible gases requires computer solutions, read approximations. An analysis of computer solutions indicates, fortuitously, that the unsteady-state behavior can be reproduced by steady-state density or pressure profiles at successive times. This will demark draw down and the transition to long-term depletion for reservoirs with closed outer boundaries.
As an alternative, unsteady-state flow may be presented in terms of volume and surface integrals, and the methodology is fully developed with examples furnished. Among other things, permeability and reserves can be estimated from well flow tests.
The foregoing leads to an examination of boundary conditions and degrees of freedom and raises arguments that the classical partial differential equations of mathematical physics may not be allowable representations.
For so-called open petroleum reservoirs where say water-drive exists, the simplifications based on successive steady-state profiles provide a useful means of representation, which is detailed in the form of material balances.
Unsteady-State Fluid Flow provides:
• empirical and classical methods for correlating and predicting the unsteady-state behavior of petroleum reservoirs
• analysis of unsteady-state behavior, both in terms of the classical partial differential equations, and in terms of volume and surface integrals
• simplifications based on successive steady-state profiles which permit application to the depletion of both closed reservoirs and open reservoirs, and serves to distinguish drawdown, transition and long-term depletion performance.
Published: June 1999
- Part I. Reservoir Characteristics.1. Petroleum Reserves and their Estimation.Characterization by unsteady-state behavior.Origins of petroleum.Techniques for estimating reserves.Reservoirs and geologic provinces.2. Pressure/Production Behavior Patterns.Liquids versus gases.Maintenance of production.Reservoir pressures.Reserves and depletion times.3. Pressure/Production Decline Correlations.Reservoir P-V-T behavior.Geometric production decline.Production-time decline.Production loss ratio.Pressure decline.Part II. The Representation of Flow Through Porous Media.4. Concepts of Flow.Unsteady-state flow and compressibility.Flow systems and dissipative effects.Darcy's Law.5. The Classic Differential Equations for Flow Through Porous Media.Continuity equation.Steady-state solutions.Analytic solutions for unsteady-state flow.Computer solutions.6. Integral Forms for Describing Unsteady-State Flow.Volume and surface integrals.The depletion problem.Permeability form.Production period.Prediction of production.Repressurization.7. Two-Phase and Multiphase Flow: Gas, Oil, and Water.Concurrent two-phase flow.Multiphase flow.Immiscible and (partially) miscible drives.Enhanced oil recovery.Part III. Reduction to Practice.8. Steady-State: Productivity Tests.Determination of producing radius.Productivity index.Back-pressure tests.Departure from ideal behavior.9. An Evaluation of Unsteady-State Solutions for Drawdown and Transition.Summary statement.Unsteady-state solutions for drawdown.Experimental comparisons.10. Gaseous Unsteady-State Radial Flow Behavior from the Calculated Results of Bruce et al.Overview.Detailing and analysis of the results of Bruce et al.Closed versus open systems.Determination of reservoir extent and permeability.Back-pressure correlation.Transitional behavior.11. A Critique of Boundary Conditions, Degrees of Freedom and Darcy's Law.Problem and expendiencies.Pressure gradient at the closed boundary.Degrees of freedom.Darcy's Law in radial flow.Systems in chaos.Steady-state profiles.12. The Results of Bruce et al. in Terms of Integral Forms.Review of the derived relationships and correlations.Relation to the results of Bruce et al.13. The Computation of Reserves and Permeability from Stabilized Flow-Test Information (Back-Pressure Tests).Reserves and permeability calculations.Computer applications.Part IV. The Use of Steady-State Profiles for Unsteady-State Flow.14. Approximate Solutions During Drawdown and Long-Term Depletion.Compressible liquids.Compressible gases.Transition (or stabilization) between drawdown and long-term depletion.Estimation of reservoir extent and reserves.15. Representation of Water Drives.Infinite reservoirs (with drive).Finite reservoirs (with drive).Gaseous flow and displacement.Field histories.16. Production-Decline Behavior.Effect of flow up through the well tubing.Decline of production rate.Afterword.Glossary.Symbols.Index.