Wellbore-Storage Effects in Transient Flow Testing of Gas Wells (includes associated papers 19899 and 20008 )
Authors: Ahluwalia J.S. , Wilkes J.O. 
Published: December 2007

Summary
This paper focuses on the better understanding of and the ability to predict, a priori, from fundamental principles, the early-time transient flow effects encountered in gas wells. In particular, wellbore-storage and unloading effects are analyzed by simulating actual flow conditions, both in the well and in the surrounding reservoir formation. Such a treatment lends confidence to early-time well test data and is used to determine the exact duration of wellbore transients.

Introduction
The expense involved in long-duration gas well testing to determine well performance, reservoir parameters, gas-in-place, and expected recovery is substantial. These tests are conducted and interpreted beyond the time at which transient effects are believed to have died out. Therefore, the trend in modem well testing is toward methods of analysis pertinent to short-time data.
Because wellbore storage and unloading effects are especially important during this early-time period, knowing the exact duration and nature of the transient effects is essential to account for them correctly. Also, early-time transient flow behavior in gas wells is of great interest in underground storage operations related to a constantly changing gas market. A sudden change in surface condition at the wellhead, as in the case of gas withdrawal or injection, results in the generation of a pressure wave, which propagates and undergoes reflection and attenuation within the well. In addition, the gas interacts intricately with its environment. This interaction, correctly described by fundamental equations of energy and motion, defines the pressure, temperature, and velocity variations of the gas with position and time within both the well and the porous formation. Gravitational, frictional, compressibility, turbulence, and heat-transfer aspects and variable gas-property effects are considered.
A finite-difference spatial discretization, in conjunction with the method of characteristics, is used to describe the transient flow behavior over the entire well depth. To describe the flow of gas from the surrounding porous reservoir, through the well perforations, and into the well, a radial-flow finite-element model is used. Care is taken to ensure numerical stability and accuracy with the chosen methods of solution.
When a gas is first made to flow, the gas at the wellhead comes primarily from that stored in the wellbore rather than from the formation. With time, the surface flow then becomes a combination of the flow of gas from the wellbore and from the formation. The flow from the formation eventually equals the flow at the wellhead, signifying the end of the wellbore-storage effect. This storage or unloading effect (as in injection and withdrawal cases, respectively) can result from a wellbore completely filled with a compressible gas or from movement of a gas/liquid interface.
The effect of wellbore storage was first considered by van Everdingen and Hurst. They assumed an exponential-type relationship between the sandface and surface flow rates (translated to standard conditions):
..........................................(1)
where the constant Fd was derived from observations of casing/ tubing and bottomhole pressures. A recent modification was the empirical fitting of certain field data to generate a dependence of this factor on other parameters, such as skin friction and turbulence.
Another way of arriving at the relative approach of one flow rate to the other was to assume the unloading rate of the wellbore per unit pressure difference to be a constant value, CD. The duration of storage effects is determined with this approach. This assumption of a constant storage-constant, however, is not a good assumption because CD is inversely proportional to the pressure, which, during certain large drawdowns, can easily change several-fold. In their works, these authors have made certain simplifying assumptions (e.g., constant fluid and formation properties, laminar flow, and isotherm flow in some cases. More accurate numerical solutions, with fewer simplifying assumptions based on improvisations of these treatments (but with limited and restrictive validity ranges), have been proposed.
The occurrence of the pressure wave, although recognized, has not really been treated thus far. The pressure wave is the transient pressure impulse that travels back and forth within the wellbore, gradually undergoing attenuation because of frictional damping whenever a sudden change takes place in the wellhead conditions.
This study has been undertaken with the hope that these omitted details can be accounted for and better understood by drawing upon fundamentals.

Mathematical Development
The storage and retrieval of natural gas from underground reservoirs involves two basic flow processes. In the case of withdrawal, the gas flows through the porous formation into the well perforations and then to the surface in the well, where it is withdrawn. For in-ection, the same two processes occur, but in reverse order (Fig. 1). In characterizing these time-dependent flows, our developed model determines the variations with time of pressure, temperature, and flow velocity at all defined locations or nodes within the well and the reservoir. Even though we are concerned primarily with short-term transients, the heat transfer between the rock and gas cannot be assumed small and thus be neglected. It is therefore taken into account in both the wellbore and reservoir models. All gas properties are taken as variable and are continually updated at each state of the calculations.
Flow in the Well. Flow in the well is modeled as flow in an inclined pipe (Fig. 2), taking into account friction, compressibility, gravitation, and variable gas properties. Calculations were performed to see whether the heat transfer to the gas in the well from the surrounding rock formation and vice versa was significant enough to warrant inclusion in the wellbore model.
The heat transfer was found to induce temperature changes of several degrees, indicating that exclusion of this transfer term would introduce a sizable error in the evaluation of nodal temperatures within the well region. The heat transfer was therefore incorporated into the model by evaluating appropriate heat transfer coefficients with the Sieder-Tate correlations.
The equations of motion, energy, and continuity are combined with an equation of state (EOS) for the, gas and solved simultaneously for the pressure, temperature, and velocity profiles in the well at each timestep. The continuity equation relates the variations of density and velocity:
..........................................(2)
The equation of motion relates the variations of velocity and pressure:
..........................................(3)
where Efd=frictional dissipation per unit length, defined as
..........................................(4)
JPT
P. 623^