Practical Model for Predicting Pressure in Gas-Storage Reservoirs
- Jean-Eric Molinard (Gaz de France) | Philippe Le Bitoux (Gaz de France) | Veronique Pelce (Gaz de France) | M.R. Tek (U. of New South Wales)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1990
- Document Type
- Journal Paper
- 576 - 580
- 1990. Society of Petroleum Engineers
- 4.1.4 Gas Processing, 5.1 Reservoir Characterisation, 5.10.2 Natural Gas Storage, 4.6 Natural Gas, 5.4.2 Gas Injection Methods, 5.1.1 Exploration, Development, Structural Geology, 5.2.1 Phase Behavior and PVT Measurements
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Summary. optimal planning, design, and operation of gas fields in production or storage critically depends on reliable models for predicting pressures that are often based on sophisticated numerical and mathematical concepts. The pressure that prevails at a given time is a direct function of production/injection schedules, past history, and surface equipment as related to reserves and reservoir characteristics. Maintaining the pressure within prescribed limits is particularly important in underground storage to avoid pressures above a maximum for reasons of safety and migration and below a minimum for surface-Equipment and contracted-deliverability considerations. This paper presents a new way to calculate the convolution integral on which the pressure variations depend. The classic methods were long and costly to run and seldom used. Our method, which identifies this convolution integral with a finite sum of exponential terms, is much quicker and has been implemented in a program called PREPRE, usable on microcomputers. Based on a production/pressure schedule, the model is capable of forecasting the evolution of pressures in main zones of interest, such as wellbores, gathering systems, and surface equipment. The data required for the model-past production/pressure history-are matched by a special algorithm that automatically calculates the main reservoir parameters used as bases for future projections. The quick and easy-to-run program is now in use at all engineering levels at Gaz de France.
Operation of underground storage reservoirs critically depends on models that predict the average pressure with reasonable accuracy and a high degree of reliability.
As the injection/production schedules evolve during the operation of storage reservoirs, the pressure distribution in the gas bubble, in the zone between the gas bubble and surrounding aquifer, and throughout the aquifer changes in a cyclic manner, responding to flow conditions in porous media. The actual relationship between production/injection and pressures is a function of reservoir geometry, reservoir and fluid properties, operating conditions, and the particular storage system design. This paper describes a complete model to represent the storage reservoir and surrounding aquifer, a discretized solution that allows implementation on a personal computer, and practical applications of the method. Specific and typical examples from real-world data are included to demonstrate the relevance of the method to simulation of storage operations.
Description of Storage-Reservoir Model
Production and pressure are basic quantities measured or predicted more frequently than any other variables in the operation of oil or gas reservoirs. In underground storage, the quantity known and actually measured daily is the "inventory evolution." Pressures calculations usually involve the resolution of a mathematical model that includes reservoir description and the effects of operating conditions.Geometry. Fig. 1 is a schematic of a storage bubble in communication with an aquifer through a transition zone. In Fig. 1, the 3D aspect is illustrated through the change in elevation as the radius evolves from that of the gas bubble, rb, to that of the aquifer, rf, Note that the radial model described in Fig. 1 ignores the continual change from mostly gas saturation at left to 100% water saturation at right in the aquifer. The point represented by rf is defined as a point fixed in space where the water saturation is 100% at any moment of the cyclic operation of the storage. It is usually determined as the point where the gas and water gradients intersect.
Assumptions. The physical model developed for analysis involves three main assumptions: (1) pressure drops in the gas bubble are negligible compared with pressure changes in the aquifer, (2) water flow in the transition zone is incompressible, and (3) the pressure prevailing in the aquifer at a radius adjacent to the transition zone is analytically predictable.
Equations Describing Flow. In Fig. 1, in the gas-bubble and the transition zones, the equation giving the volumetric flux is Darcy's law (the case of turbulence in the gas bubble is ignored here):
Integration of Eq. 1 between rb and rf with respect to radius (at a given time) gives, for the case of incompressible flow,
The equation giving the approximate pressure in the gas bubble may also be obtained by integrating Darcy's law between r=r(t) and r=rb(t) (at any time t) as
It must be noted that Eq. 3 is only approximate because it implies two restricting assumptions: (1) that Darcy's law is valid and (2) that p is constant (incompressible flow).
If Eq. 3 is volumetrically integrated between r=0 and r=rb(t), one obtains for the average pressure, p(t)),
At the point r=rf fixed between the transition zone and aquifer boundary, the pressure prevailing as function of time may be expressed as (5)
where the convolution integral represents the cumulative effect of superposing all previous rate history. qw(t) represents the volumetric flow rate of water in Eq. 2 and the volumetric flow rate of gas in Eq. 3. The two are equal simply because of the assumption of incompressible flow in the transition zone. By adding Eqs. 2, 4, and 5, one obtains
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