Analyzing Injectivity of Polymer Solutions With the Hall Plot
- R.S. Buell (Chevron U.S.A.) | H. Kazemi (Marathon Oil Co.) | F.H. Poettmann (Colorado School of Mines)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- February 1990
- Document Type
- Journal Paper
- 41 - 46
- 1990. Society of Petroleum Engineers
- 4.1.9 Tanks and storage systems, 5.3.2 Multiphase Flow, 5.4.1 Waterflooding, 5.5.8 History Matching, 5.5 Reservoir Simulation, 1.8 Formation Damage, 5.6.4 Drillstem/Well Testing, 4.1.2 Separation and Treating, 6.5.2 Water use, produced water discharge and disposal, 5.3.4 Reduction of Residual Oil Saturation, 5.4.7 Chemical Flooding Methods (e.g., Polymer, Solvent, Nitrogen, Immiscible CO2, Surfactant, Vapex), 4.1.5 Processing Equipment
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Summary. The Hall plot was originally used to analyze water-injection wells. This paper demonstrates that the Hall plot can also be used to analyze injection of polymer solutions. In particular, it is possible to determine the in-situ and residual resistance factors of a polymer solution from the Hall plot. The analysis methods developed are used to examine two field injection tests and one hypothetical example. The analytical results are verified with a reservoir simulator.
Polymer floods, micellar/polymer floods, and injectivity- or productivity-profile-modification treatments are the most common applications of polymer solutions. The interpretation of injection pressures and rates associated with polymer solution injection is important to the efficient application of the solutions. The Hall plot is a useful tool for evaluating performance of injection wells.
The Hall plot was originally developed for single-phase, steady-state, radial flow of Newtonian liquids. Since the advent of polymer and micellar solutions for EOR, it has also been applied to the injection of these solutions. Moffitt and Menzie used the Hall plot to evaluate injection of polymer solutions but did not verify the validity of the Hall plot for this application. This paper verifies the validity of the Hall plot for evaluating polymer solution injection.
Because of the complex nature of polymer solution flow through porous media, exact analytical solutions are generally not possible. However, some relatively simple approximate analytical solutions can be developed. To verify the analytical solutions for polymer solution injection, a two-phase, radial, numerical reservoir simulator was developed. The simulator is designed to consider the more important phenomena and effects that occur when polyacrylamide or polyacrylamide polymer solutions are injected into porous media. The simulator has the following characteristics: slightly compressible flow, two-phase flow, non-Newtonian rheology, adsorption/retention with permeability reduction, concentration effects, skin, and wellbore storage. It was used to history match two field injectivity data sets.
Development of the Hall Plot
The Hall plot was originally proposed to analyze the performance of waterflood injection wells. Hail simply used Darcy's law for single-phase, steady-state, Newtonian flow of a well centered in a circular reservoir: (1)
Hall integrated both sides with respect to time to obtain
Separating the integral of Eq. 2, Hall then rearranged to obtain
The relation between surface and bottomhole pressures for steady-state vertical flow is given by
Hall substituted Eq. 4 into Eq. 3 to arrive at
Hall simply dropped the second term on the right side of Eq. 5 and plotted the integral of wellhead pressures with respect to time vs. cumulative injection, which came to be known as the "Hall plot." By plotting in this format, Hall observed that if an injection well was stimulated, the slope decreased, and if a well was damaged, the slope increased. While Hall's conclusions regarding changes in slope are valid, the second term on the right side of Eq. 5 is often not negligible in comparison with the other terms and therefore usually cannot be dropped.
In industry applications, the Hall integrals dt and f frequently are used. The slopes calculated from these integrals should not be used for quantitative calculations unless a correction procedure is applied. Fig. 1 is a Hall plot based on the data for Well A, where the integral dt has been plotted vs. cumulative injection. Several changes in slope can be seen on the plot, but there has been no change in transmissibility or skin. The changes in slope are caused by changes in rate, which occur because the integral dt has been neglected. Fig. 2 is a Hall plot based on data for Well C. The three most common forms of the Hall integral have been plotted for the same data. For each integration method, the slopes of the curves are quite different.
Injection data must be plotted in the form of Eq. 2 to make valid quantitative calculations; i.e., cumulative injection should be plotted vs. (Pwf-pe)dt. The slope of the Hall plot from Eq. 2 is then given by (6)
Eq. 6 will not be appropriate when multiple fluid banks with significantly different properties exist in the reservoir.
Advantages and Disadvantages
The Hall plot is a steady-state analysis method, whereas falloff tests, injection tests, and type-curve analysis are transient methods. Transient pressure analysis methods determine the reservoir properties at essentially one point in time. The Hall plot is a continuous monitoring method; i.e., reservoir properties are measured over a period of weeks and months. The Hall plot, therefore, can help identify changes in injection characteristics that occur over an extended period.
Hall's method has several advantages. Integrating the pressure data with the Hall integral [ (pwf-pe)dt] has a smoothing effect on the data. Data acquisition for the Hall plot is inexpensive because only the recording of cumulative injection and surface pressures is required.
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