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Summary

This paper presents a semi-analytical wellbore/reservoir model¹ that can describe general wellbore effects, especially the thermal effect, on high-temperature gas well pressure-buildup tests. A numerical simulator has been developed from the model. Using different combinations of wellbore and reservoir parameters, the simulator generated curve shapes that differed with wellbore thermal effects. Many of the curve shapes have been observed in the field.^2,3 Using the results from this paper, engineers can distinguish between general wellbore effects and reservoir behavior in the pressure data, which will make the interpretation more accurate. Also, with the help of the simulator developed from the model, engineers can effectively design gas well pressure-buildup tests by running the simulator to determine the minimum time required to obtain the data not distorted by the wellbore effects.⁴

The governing equations of the wellbore model are based on mass, momentum, and energy balances for single-phase gas in one-dimensional space. The gas pressure/volume/temperature correlation was also used. Different flow regimes (laminar, transitional, and turbulent) inside the wellbore are modeled for calculating the friction factor. As one boundary condition, a simple analytical reservoir model was connected to the wellbore model at the bottomhole using Duhamel’s principle. Heat-loss effects account for forced-convectional heat transfer inside the tubing, heat conduction between tubing and formation, natural convection and radiation heat transfer of annular fluid, and transient heat flow in the formation. Pressure, temperature, velocity, and gas properties inside the wellbore can be predicted at any depth during an entire pressure-buildup test. Variable wellbore storage, momentum, and thermal effects can be simulated.

Introduction

The concept of wellbore storage was introduced when pressure transient analysis was first established as a viable method for evaluating well and reservoir performance.^5,6 Much effort has been spent on minimizing wellbore storage effects on well test data. Traditional type-curve matching ^7-9 is the first attempt to interpret the pressure data with constant wellbore storage. Semilog analysis methods^10,11 are very useful when the true semilog straight line can be identified.

Analyzing buildup tests with changing wellbore storage due to phase redistribution has been discussed in the literature.^12,13 The results can mimic field data. However, they lack physical justification and cannot be used for forward modeling. Pseudopressure and pseudotime functions have been used to account for variable wellbore storage for gas wells. ^14-17 Different approaches to modeling transient two-phase flow in the wellbore^18,19 describe the phase redistribution phenomena physically by applying mass and momentum conservation equations. But no thermal effects are included. The above methods all assume that the temperature of the fluid in the wellbore is a constant.

The various aspects of heat transfer between a wellbore fluid and the formation have been studied by many authors over the last few decades. ^20-28 Most of them are steady-state models, which assume that fluid properties and flow rate are not functions of time in the wellbore. A few wellbore models, mainly for geothermal wells with two-phase transient flow inside the wellbore, exist in the literature.^29-33 Generally, these models do not work for single-phase gas wells. Hasan³⁴ and Kabir ³⁵ presented a transient wellbore/reservoir model for estimating bottomhole pressure and temperature from measured wellhead pressure and temperature in high-pressure and high-temperature reservoirs.

Gas well pressure-buildup test data do not match traditional liquid type curves mainly because the gas properties are strong functions of pressure and temperature. Most of the wellbore models available are constructed assuming isothermal conditions in both wellbore and reservoir. However, it has been noticed that gas temperature changes inside the wellbore caused by heat loss to the surrounding formation can change the curve shapes of pressure transient data at early times. This causes problems in both the interpretation and design phases of gas well pressure transient tests, especially buildup tests.

To improve design and analysis of gas well pressure-buildup tests, a wellbore model that can evaluate or generalize wellbore effects during a gas well buildup test is necessary. The model can be used in the forward mode to predict pressure transient behavior, which can help in designing a gas well pressure-buildup test much more effectively and economically.

Model Development

We developed a one-dimensional, transient gas flow wellbore model in this study. The model was constructed using mass balance, momentum balance, and energy balance equations. Real gas pressure/volume/temperature (PVT) correlations were used to complete the model equation system. A homogeneous, radial flow, single-phase, line-source reservoir model was chosen and connected to the wellbore model at the bottomhole by applying Duhamel’s principle, which combines the sandface flow rate with the bottomhole pressure to form a boundary condition. In this simple reservoir model, the dimensionless pseudopressure function is used to allow the use of the liquid solution for the gas case.

Gas flow in the pipe is quite different from gas flow in the reservoir. The wellbore-pressure transient propagation is dominated by inertial effects, as well as gravity and friction effects. During a gas well pressure-buildup test, a large disturbance to fluid motion is initiated. Therefore, an unsteady-state viscous model is required to simulate the entire buildup test. The net effect of large changes during the buildup test is a wavelike response in the wellbore, which is dampened quickly over time and is governed by Navier-Stokes-like equations. In the reservoir, however, the fluid flow is subjected to relatively moderate pressure disturbances, and is governed by diffusion-like equations.

Model Equations.

The general one-dimensional mass balance equation can be written as {\partial \rho \over \partial t}+{\partial (\rho v)\over \partial x}=0.\eqno ({\rm 1}) The general one-dimensional momentum balance equation with a constant cross-sectional area can be expressed as {\partial \over \partial t}(\rho v)+{\partial \over \partial x}(\rho v^{2})+144g {c}{\partial p\over \partial x}+\rho \bar {g}+\tau =0.\eqno ({\rm 2})