Gas-well Testing Research Articles

Research on evaluation method of wellbore hydrate blocking degree during deepwater gas well testing

Although considerable advances have been achieved in recent years, there is still a long way to go for hydrate prediction and prevention during deepwater gas well testing. Well hydrate blockage is a time-dependent continuous process, and evaluation of wellbore blockage is critical for hydrates prevention and control. In this work, the authors presented a novel wellbore blocking degree evaluation model for deepwater gas-well testing. Firstly, a new evaluation model consisting of mass, momentum and energy balance equations considering hydrates formation was proposed. Secondly, considering hydrate phase equilibrium, the finite difference method and iterative technique were used to obtain the model results. Finally, the model was applied to the deepwater gas wells in the South Sea of China for verification and was subjected to sensitivity analysis. The predicted results were in good agreement with field test results. According to the sensitivity analysis, gas output, methanol concentrations, water production rate have different degrees of influence on hydrate blocking. The length of the hydrate stability region (HSR), the position of largest plugging point, and the distribution of wellbore inner diameter are the key to hydrate prevention and control. Meantime, combining the model evaluation results, optimizing the testing process can achieve the purpose of preventing-controlling gas hydrates economically and effectively.

Analytical Treatment of Pressure-Transient Solutions for Gas Wells With Wellbore Storage and Skin Effects by the Green's Functions Method

Summary Even when written in terms of a pseudopressure function, the diffusivity equation for flow of gases through porous media is, rigorously speaking, nonlinear because the viscosity-compressibility product is pseudopressure-dependent. However, several techniques and analysis procedures neglect such nonlinearity. A new methodology for constructing solutions for gas reservoirs through the Green's functions (GF) technique was recently proposed in the literature. Such methodology handles the viscosity-compressibility product variation rigorously, and it was applied to solve several gas-well test problems successfully. However, wellbore storage and skin effects were not considered yet by this new approach. In this work, the GF technique is applied to obtain a new solution for an infinite, homogeneous, isotropic gas reservoir being produced through a single vertical well represented by a line-source with wellbore storage and skin. The solution, however, does not consider non-Darcy flow effects. Even though the wellbore storage introduces a new nonlinearity to an already nonlinear problem, this work presents two accurate approximate solutions compared with the results from a commercial numerical well-testing simulator. This work also shows that the wellbore pseudopressure dimensionless solution is a function of the correlating groups CDexp(2S) and tD/CD, exactly similar to the way that wellbore dimensionless liquid solutions are. Liquid and gas dimensionless solutions under these correlating groups are not equal, though.

Transient Simulation of Wellbore Pressure and Temperature During Gas-Well Testing

An abnormal phenomenon may occur during gas-well testing: the wellhead pressure initially rises and then drops when shutting-in a well; the wellhead pressure initially drops and then rises when opening a well. To determine why and how this phenomenon occurs, a transient nonisothermal wellbore flow model for gas-well testing is developed. Governing equations are based on depth- and time-dependent mass, momentum equations, and the gas state equation. Temperature is predicted using the unsteady-state heat transfer model of Hasan. Boundary conditions include the restriction of formation inflow and wellhead throttling to the flow. The difference equations are established based on the implicit central finite difference method. The model can simulate the influences of temperature and flux (mass velocity). The model also considers the effects of formation inflow and surface throttling on the system. The results indicate wellhead pressure under flowing temperature is higher than that under static temperature, thus causing the abnormal phenomenon. A larger pressure difference makes the abnormal phenomenon more significant. Without considering temperature variation, simulated wellhead pressure would not exhibit the abnormity. Without considering flux variation, simulated pressure curve is not smooth. A new model has thus been validated using a gas field example.

A Variable-Rate Solution to the Nonlinear Diffusivity Gas Equation by Use of Green's-Function Method

SummaryThe hydraulic diffusivity equation that governs the flow of compressible fluids in porous media is nonlinear. Although the gas-well test analysis by means of the pseudopressure function has become a standard field practice, the effect of viscosity and gas-compressibility variation with pressure is often neglected. Moreover, in field operations, the gas well is submitted to a variable rate production to determine well/reservoir properties and an estimation of the absolute open flow (AOF). For slightly compressible fluids, variable rate can be properly handled by superposition in time. Unfortunately, superposition cannot be casually justified for gas reservoirs because of its nonlinear behavior.In this paper, a general solution that properly accounts for both fluid property behavior and variable rate is presented. The proposed solution, which is derived from the Green's-function method by recasting the effect of the viscosity-compressibility product variation as a nonlinear source term, can handle variable gas rate for several well/reservoir geometries of practical interest. From the general solution, an analytical expression for variable-rate tests of a fully penetrating vertical well in an infinite gas reservoir is derived. This expression is applied to a synthetic data set to calculate the pressure response for a buildup test in an infinite homogeneous reservoir. The results compared with a commercial finite-difference numerical simulator show close agreement for both drawdown and buildup periods. It is also shown that the dimensionless pseudopressure converges to the slightly compressible fluid solution for long shut-in times. Thus, during those long times, Horner analysis and log-log derivative plot can be applied to obtain good estimation of reservoir parameters, as discussed previously in literature.

Welltest analysis of hydraulically fractured tight gas reservoirs: a field example from Perth Basin, Western Australia

Welltest interpretation requires the diagnosis of reservoir flow regimes to determine basic reservoir characteristics. In hydraulically fractured tight gas reservoirs, the reservoir flow regimes may not clearly be revealed on diagnostic plots of transient pressure and its derivative due to extensive wellbore storage effect, fracture characteristics, heterogeneity, and complexity of reservoir. Thus, the use of conventional welltest analysis in interpreting the limited acquired data may fail to provide reliable results, causing erroneous outcomes. To overcome such issues, the second derivative of transient pressure may help eliminate a number of uncertainties associated with welltest analysis and provide a better estimate of the reservoir dynamic parameters. This paper describes a new approach regarding welltest interpretation for hydraulically fractured tight gas reservoirs—using the second derivative of transient pressure. Reservoir simulations are run for several cases of non-fractured and hydraulically fractured wells to generate different type curves of pressure second derivative, and for use in welltest analysis. A field example from a Western Australian hydraulically fractured tight gas welltest analysis is shown, in which the radial flow regime could not be identified using standard pressure build-up diagnostic plots; therefore, it was not possible to have a reliable estimate of reservoir permeability. The proposed second derivative of pressure approach was used to predict the radial flow regime trend based on the generated type curves by reservoir simulation, to estimate the reservoir permeability and skin factor. Using this analysis approach, the permeability derived from the welltest was in good agreement with the average core permeability in the well, thus confirming the methodology’s reliability.

Transient Nonisothermal Fully Coupled Wellbore/Reservoir Model for Gas-Well Testing, Part 2: Applications

Summary After the development of a numerical fully implicit nonisothermal wellbore/reservoir simulator in Part 1 of this study (Bahonar et al. 2010), this simulator is implemented for a close and detailed study of gas-well pressure-drawdown (DD) and -buildup (BU) tests. Overall, the developed simulator is an accurate and strong tool for design and analysis of transient gas-well testing, particularly for high- pressure/high-temperature (HP/HT) gas reservoirs. Several numerical results will be presented. This includes demonstration of the behaviour of the wellbore-fluid pressure, temperature, density, and velocity and an overall heat-transfer coefficient during DD or shut-in tests for nonisothermal reservoirs and conceptual comparisons with the isothermal counterparts. Thermal effects on the behaviour of derivative plots and the sandface-flow rate of deep nonisothermal gas reservoirs will be studied. A significant effect of neglecting the heat capacity of tubular and cement materials on the wellhead-temperature simulation, and thus transient well tests, will be demonstrated. A sample case to show that neglecting the thermal effects in the gas-well tests of composite reservoirs leads to unreliable results in well-testing analysis will be presented. Several other numerical experiments, including the presence of a variable wellbore-storage coefficient, gas backflow from the wellbore to the reservoir, and other thermal effects during the gas-well tests, are also presented. Hundreds of millions of dollars are spent every year on well testing around the world (Hawkes et al. 2001). A proper design and truthful interpretation of these tests can be achieved by a reliable coupled wellbore/reservoir simulator, which in turn can save a large portion of the required costs.

Transient Nonisothermal Fully Coupled Wellbore/Reservoir Model for Gas-Well Testing, Part 1: Modelling

Summary A numerical fully implicit nonisothermal wellbore/reservoir simulator is developed. The model entails simultaneous solution of transient coupled mass-, momentum-, and energy-balance equations within the wellbore; energy-balance equations for the tubular and cement materials and the formation surrounding the wellbore; and mass-balance and flow-rate/pressure equations for the reservoir formation. A wellbore heat-loss model that is a strong feature of this study is developed and employed in the model to improve the accuracy of the simulator and to be able to estimate the casing temperature and formation-temperature distribution. The model formulation is completed with an equation of state (EOS) to estimate fluid properties and appropriate friction-factor correlations in the wellbore tubing to compute the frictional pressure drop for different flow regimes. The developed model has several applications in the petroleum industry, particularly in the gas-well testing design and interpretation of both isothermal and nonisothermal gas reservoirs. This nonisothermal simulator is validated through comparisons to both analytical models and an equivalent numerical isothermal coupled wellbore/reservoir simulator that is also developed in this paper. Applications of this simulator to analyzing gas-well testing problems, in addition to several important observations, are extensively studied in Part 2 of this research work (Bahonar et al. 2010). Currently, it has been well accepted that the applicability and significance of a reservoir simulator depend on the behaviour of the wellbore and interaction between the wellbore and reservoir. A robust, accurate coupled wellbore and reservoir simulator is an invaluable tool for the petroleum engineer to help the petroleum industry understand production behaviour, make a meaningful prediction, and make correct decisions in all field-development and production stages.

Technology Focus: Well Testing (February 2009)

Technology Focus Many of us, at one time or another in our careers, have encountered industry colleagues or team members convinced of the need to gather information, sometimes at any cost. In contrast, just as many struggle with advocating even the most rudimentary data gathering to decision makers within their company. A common problem, particularly in small companies, can be a tendency to make quick decisions without considering possible effects by relying on past experience and judgment to expedite decision making for new problems. Planning can seem bureaucratic and unnecessary, particularly to less-experienced or nontechnical professionals. However, when objectives are not established and deliverables are not determined before making the final decision, the result can be a large invoice and information or service that is essentially useless. Modern software, sometimes with less effort than my children put into mastering the latest "Xbox 360" release, will transform any kind of input data into an impressive array of results that can provide compelling justification for a completely wrong conclusion. I am picturing trainloads of investors passing through early oil fields where a strong pump and full tank battery could provide an impressive "gusher" timed by the promoter for just the right moment! And even a cursory look at input and assumptions going into some of our modern analysis can keep us from drawing wrong conclusions. Along with colorful or animated results, the most useful and relevant analysis in our industry will always invest the necessary time to confirm quality of the input data and verify the purpose or application of the results. The papers highlighted on the following pages provide good examples of how unmet needs can provide insight into better analysis methods. New tools result from the need to acquire important information that would otherwise be uneconomic. Thorough engineers and technicians will be interested in the accuracy of their analysis as it relates to the quality of the input data. Solid analysis over time can uncover further insight into the performance of a hydrocarbon asset. I hope that you will find the selected papers of interest and that they will provide stimulation for new approaches in your well-testing and surveillance activities. Well Testing additional reading available at the SPE eLibrary: www.spe.org SPE 116744 • "Equivalent Skin Factors for Nonuniformly Damaged Horizontal and Multilateral Wells" by Turhan Yildiz, SPE, Colorado School of Mines SPE 118940 • "Evaluation of Skin Factor From Single-Rate Gas-Well Test" by K. Aminian, West Virginia University, et al. Additional reading available at OnePetro: www.onepetro.org OTC 19140 • "Oilwell Transient Testing by Injection of Miscible Liquid" by J.J. Voelker, SPE, Chevron

High-Accuracy Wet-Gas Multiphase Well Testing and Production Metering

Summary Dedicated wet-gas flowmeters are now commercially available for the measurement of gas and liquid flow rates and offer a more compact measurement solution than does the traditional separator approach. The interpretation models of traditional multiphase flowmeters emphasize the liquid rate measurements and have been used to well test and meter mostly liquid-rich flow streams. These models were not developed for the measurement of gas flow rates, particularly those of wet gas. A new interpretation is described that allows a traditional multiphase flowmeter to operate in a dual mode either as a multiphase meter or as a wet-gas meter in 90 to 100% gas. The new interpretation model was developed for a commercially available multiphase flowmeter consisting of a venturi and a dual-energy composition meter. This combination results in excellent predictions of the gas flow rate; the liquid rate prediction is made with acceptable accuracy and no additional measurements. The wet gas and low-liquid-volume-fraction interpretation model is described together with the multiphase flowmeter. Examples of applying this model to data collected on flow loops are presented, with comparison to reference flow rates. The data from the Sintef and NEL flow loops show an error (including the reference meter error) in the gas flow rate, better than ±2% reading (95% confidence interval), at line conditions; the absolute error (including the reference meter error) in the measured total liquid flow rate at line conditions was better than ±2 m3/h (<±300 B/D: 95% confidence interval). This new interpretation model offers a significant advance in the metering of wet-gas multiphase flows and yields the possibility of high accuracies to meet the needs of gas-well testing and production allocation applications without the use of separators.

Study on the geochemical thermodynamic nature of CO 2CH 4 and CO 2 rich gas

A marine Paleozoic CO 2 gas field with of CO 2CH 4, CO 2 rich, and other types of gas reservoirs, located in the Yangtze paraplatform, is taken for a case study. By studying the geochemical and thermodynamic nature of these reservoirs on the basis of their geologic settings and origins, the phases and thermodynamic equilibrium conditions have been determined. The results have been used in gas prospecting, resulting in new technical designs for gas-well testing, and providing theoretical bases for the establishment of new geochemical-thermodynamic prospecting techniques.

Gas-Well Testing in the Presence of Desorption for Coalbed Methane and Devonian Shale

Summary Very large potential gas reserves are present in both coalbeds and Devonian shales. In both formation types, long-term gas production is the result of gas desorption from the matrix. Experiments on core samples show that the desorption obeys a Langmuir isotherm. An approximate analytic solution is derived for single-phase gas flow when gas is present as both free gas and gas adsorbed on the matrix. The solution was verified against a finite-difference solution; the agreement between the two methods is excellent. The analysis of test data is simplest if the test is conducted at a constant mass flow rate. It is shown that the effect of desorption cannot be detected in production testing; therefore, analysis can be performed to obtain permeability as if the desorption effect is absent. The procedure to obtain desorption parameters is also given. The effect of desorption shows up in an increased compressibility, of the form cg +(αcm/ϕ)+[pscVLpLTz̅/ϕp̅gTsczsc(pL+pL+p̅g)2], which can be much larger than the gas compressibility. The effect of desorption on total production is substantial. When gases are adsorbed, total production over the life of a well can be an order of magnitude higher.

Non-Darcy Flow and Wellbore Storage Effects in Pressure Build-Up and Drawdown of Gas Wells

Abstract The wellbore acts as a storage tank during drawdown and build-up testing and causes the sand-face flow rate to approach the constant surface flow rate as a function of time. This effect is compounded if non-Darcy flow (turbulent flow) exists near a gas wellbore. Non-Darcy flow can be interpreted as a flow-rate dependent skin effect. A method for determining the non-Darcy flow constant using this concept and the usual skin effect equation is described. Field tests of this method have identified several cases where non-Darcy flow was severe enough that gas wells in a fractured region appeared to be moderately damaged. The combination of wellbore storage and non-Darcy flow can result in erroneous estimates of formation flow capacity for short-time gas well tests. Fortunately, the presence of the wellbore storage effect permits a new analysis which can provide a reasonable estimate of formation flow capacity and the non-Darcy flow constant from a single short-time test. The basis of the Gladfelter, Tracy and Wilsey correction for wellbore storage in pressure build-up was investigated. Results led to extension of the method to drawdown testing. If non-Darcy flow is not important, the method can be used to correct short-time gas well drawdown or build-up data. A method for estimation of the duration of wellbore storage effects was developed. INTRODUCTION In 1953, van Everdingen1 and Hurst2 generalized results published in their previous paper3 concerning wellbore storage effects to include a "skin effect", or a region of altered permeability adjacent to the wellbore. Later, Gladfelter, Tracy and Wilsey4 presented a method for correcting observed oilwell pressure build-up data for wellbore storage in the presence of a skin effect The method depended upon measuring the change in the fluid storage in the well bore by measuring the rise in liquid level. To the author's knowledge, application of the Gladfelter, Tracy and Wilsey storage correction to gas-well build-up has not been discussed in the literature. It is, however, a rather obvious application. Gas storage in the wellbore is a compressibility effect and can be estimated easily from the measured wellbore pressure as a function of time. Several approaches to the wellbore storage problem have been suggested. As summarized by Matthews,5 it is possible to minimize annulus storage volume by using a packer, and to obtain a near sand-face shut-in by use of down-hole tubing plug devices. Matthews5 and Perrine6 have suggested criteria for determining the time when storage effects become negligible. In 1962, Swift and Kiel7 presented a method for determination of the effect of non-Darcy flow (often called turbulent flow) upon gas-well behavior. This paper provided a theoretical basis for peculiar gas-well behavior described previously by Smith8. Recently, Carter, Miller and Riley9 observed disagreement among flow capacity kgh data determined from gas-well drawdown tests conducted at different flow rates for short periods of time (less than six hours flowing time). In the original preprint of their paper, Carter et al.9 proposed that the discrepancy in flow capacity was possibly a result of wellbore storage effects, Results of an analytical study of unloading of the wellbore and nonDarcy flow were recorded by Carter.10 In the final text of their paper, Carter et al.9 stated that they no longer believed wellbore storage was the reason for discrepancy in their kgh estimates. In view of the preceding, this study was performed to establish the importance of non-Darcy flow and wellbore storage for gas-well testing. In the course of the study, a reinspection of the previous work by van Everdingen1 and Hurst2 was made, and the basis for the Gladfelter, Tracy and Wilsey4 wellbore storage correction was investigated and extended to flow testing. WELLBORE STORAGE THEORY As has been shown by Aronofsky and Jenkins,11,12 Matthews,5 and others, flow of gas can often be approximated by an equivalent liquid flow system. The following development will use liquid flow nomenclature to simplify the presentation. Application to gas-well cases will be illustrated later. First, we will use the van Everdingen-Hurst3 treatment of wellbore storage in transient flow to establish (1) the duration of wellbore storage effects, and ( 2) a method to correct flow data for wellbore storage.

The Prediction of Gas-Well Performance Including the Effect of Non-Darcy Flow

Abstract The concept of "a continuous succession of steady-states", which has been applied successfully by Aronofsky and Jenkins to obtain a solution for the nonlinear partial differential equation describing the transient Darcy flow of gas through porous media, is demonstrated to be equally valid for transient non-Darcy flow. A mathematical model, which numerically solves the partial differential equation, is used to check the validity of the succession of steady-states solution. Comparison of sand-face pressure histories computed by the two methods shows excellent agreement. The utility of the succession of steady-states solution in predicting performance of gas wells rests in the fact that no special computation equipment is required. The development of the succession of steady-states solution leads also to a practical method for determining and analyzing field test data. A method for taking gas-well test data under constant-rate conditions is presented. Experimental data obtained in the field by employing the constant-rate method are presented and analyzed in accordance with the succession of steady-states solution. Analysis of data in this fashion is demonstrated to give direct "in situ" information for reservoir permeability, porosity and turbulence or non-Darcy coefficient. Introduction The economics of gas production are dependent upon the transient behavior Of flow within the reservoir. For production from a finite reservoir, the transient flow behavior can be subdivided into two parts. At first, the transient caused by the movement of the pressure "wave" into the reservoir is of importance. Later in the production history, the pressure-wave movement ceases and the second transient stage of material depletion becomes controlling. For reservoirs of relatively high permeability, it can be shown that the pressure wave moves into the reservoir and stabilizes quite rapidly. In the case of relatively impermeable reservoirs, quite the opposite is true. Although it is theoretically possible to compute the production capability of a well from the properties of the reservoir as determined by static tests and core analyses, more reliable information is obtained by conducting flow tests on the well and thereby obtaining some measure of in situ formation properties. For gas wells, there are two basic types of tests in existence: the flow-after-flow method, and the isochronal method. Both of these techniques are tailored to obtain data that can be analyzed in accordance with the empirical performance equation: ...........................(1) In addition, the isochronal method makes provision for the sluggish nature of pressure-wave movement in "tight" formations by requiring pressure build-up between flows and by stipulating that data obtained on successive flows be analyzed at equal elapsed flow times. It can be demonstrated that either test is valid for reservoirs of high permeability. Further, since it has been pointed out that the pressure wave stabilizes rapidly for reservoirs of this type, tests of relatively short duration will give stabilized information on the performance of a well. Further decline of sand-face pressure and/or production rate may be determined by employing material-balance techniques. Cullender points out that for relatively impermeable reservoirs the flow-after-flow method gives invalid results. (See Appendix C.) If the isochronal method of testing is used, there are two alternatives:the tests must be conducted for a sufficient length of time to obtain stabilized information (which may require months to accomplish); orsome method for extrapolating the results of short-term isochronal tests must be employed. The first alternative is impracticable because of manpower, conservation and economic considerations. Recourse to the second alternative requires some assurance regarding the reliability of the extrapolation technique. Poettmann and Schilson present an empirical method for predicting stabilized performance. The present investigation was originally initiated to determine the reliability of this technique. To do this, a mathematical model was developed to simulate the Darcy and non-Darcy flow of gas through porous media. The model consisted of a finite-difference approximation of the nonlinear partial differential equation which was solved on an IBM 7090 computer. Long-term production histories were simulated by the model and compared against predictions obtained from the Poettmann-Schilson method. As the work progressed, it became apparent that a straightforward predictive equation could be developed by utilizing the concept of a succession of steady-states. As a result, the emphasis of the work was redirected to exploit the advantages of the new method. JPT P. 791^

Solutions of Unsteady-State Radial Gas Flow

Abstract Numerical solutions are presented to some problems of unsteady-state radial flow of gas in which Darcy's law holds. These solutions are intended to aid in explaining the observed behavior of gas wells during drawdown tests. The variation of viscosity and Z-factor with pressure is accounted for in the solutions. Results are presented for three types of problems:unfractured wells producing at a constant rate,fractured wells producing at a constant rate andunfractured wells producing through a critical flow prover. In the critical-flow-prover solutions, a correction of the calculated wellbore pressure to account for non-Darcy flow is made. A good correlation is shown between the solutions to the problem of Type 1 and the corresponding diffusivity-equation solution. Using this correlation, the diffusivity solution may be used to obtain gas-well solutions. The solutions to the problems of Type 2 illustrate the difference in drawdown-curve behavior to be expected between fractured and unfractured gas wells. Methods for estimating fracture radius and flow capacity are presented and verified from solutions of Type 2. The solutions to the problems of Type 3 illustrate the differences to be expected between critical-flow-prover drawdown tests and constant-rate tests. A equation is presented for correcting critical-flow-prover data to constant-rate conditions. A description of the methods used in obtaining the solutions appears in the Appendix. Introduction It is necessary for many engineering purposes, such as predicting future deliverability performance, to be able to relate bottom-hole flowing pressure, formation pressure and flow rate for conditions of stabilized flow in gas wells. This information is customarily given as a plot of (p -p ) vs q on logarithmic coordinates. It is desirable to be able to predict stabilized performance curves for gas wells based on data obtained from drawdown tests of short duration. Tests of long duration may be wasteful of gas, and low-permeability wells do not stabilize in a short-duration test. Although it is now commonly recognized that non-Darcy flow plays a significant part in gas-well behavior, it is believed that an understanding of transient behavior of systems obeying Darcy's law is still important in the interpretation of short-term test data, and the prediction of stabilized performance curves there from. In this connection, an equation for correcting Darcy flow solutions for non-Darcy flow effects has existed in the literature for some time. This equation is ......................................(1) It seems likely that solutions of the type presented herein, combined with the non-Darcy flow correction of the type of Eq. 1 and the usual laminar-flow skin factor, could provide a model which would adequately explain transient gas-well behavior for times after the non-Darcy flow region (which should be confined to a small area surrounding the well) has been established. The solutions presented in this paper are for three types of problems:unfractured wells producing at a constant rate,fractured wells producing at a constant rate andunfractured wells producing through a critical flow prover. In the problems of Types 1 and 2, no corrections for non-Darcy flow were made in the results presented, since it was reasoned that the constant-rate boundary condition would facilitate such a correction if it was desired later. For the problems of Type 3, however, a condition of varying flow rate required that a non-Darcy flow correction of the type of Eq. 1 be included. This was done. Based on the numerical results presented, a number of approximations and a correlation are presented which may be useful in analyzing gas-well test data. TYPE 1 PROBLEMS-UNFRACTURED WELL, CONSTANT FLOW RATE It is assumed that the system can be described as a finite cylindrical gas reservoir produced by a single cylindrical well located at the center of the reservoir. Flow is strictly radial. When the flow obeys Darcy's law, the governing differential equation is ..........(2) where M(p) = (2,703 x 10(–6) x 520), and f(R) =. Boundary and initial conditions are: ...................(3) JPT P. 549^

An Approximate Method for Computing Nonsteady-State Flow of Gases in Porous Media

Abstract An approximate method of calculation is developed in this paper which allows duplication of radial unsteady-state gas flow computer results where Darcy's law applies, such as those reported by Aronofsky and Jenkins and Bruce, Peaceman, Rachford, and Rice. Moreover, the new calculation method can be used to obtain results for radial unsteady-state gas flow obeying the quadratic flow law proposed by Duwez and Green. Means are discussed for predicting well behavior at single or superimposed flow rates in finite or infinite reservoirs, determining reservoir rock properties from well-test data, reproducing and interpreting back-pressure test data, and determining the radial extent and reserves of gas reservoirs from well-test data. Example calculations are presented for gas flow following both Darcy's law and the quadratic flow law. Introduction Since the publishing of U. S. Bureau of Mines Monograph 7, most gas-well testing methods have been based on the equation where q= production rate, pf and Pw are formation pressure and sandface pressure, respectively, and y and a are constants to be obtained from test data. These methods, used for predicting both deliverability and "open-flow" capacity of gas wells have been useful and accurate in many cases but unsatisfactory in others. Even at best, however, they do not supply information about the formation or lead to an understanding of nonsteady-state gas flow in porous media. Many theoretically based studies of gas flow obeying Darcy's law have been made. Since the partial differential equations which result from combining Darcy's flow law with the continuity equation are nonlinear, most of the published research consists of either numerical solutions or analytical solutions for linear approximating equations. Such solutions have been of limited value in field work due to their unhandy form and their failure to correlate most field data. There is evidence which indicates that Darcy's law is inadequate to describe gas flow at some flow rates of practical interest. A quadratic flow law, which reduces to Darcy's law at low rates, is more successful in accounting for experimentally observed behavior. This flow law has been applied successfully to a few hypothetical reservoir cases in work which has not yet been published. However, these numerical solutions of the equations involved have been successful only on a special analog computer. Routine applications to field cases would be awkward and have not been attempted. The present paper describes an approximate method for computing nonsteady-state gas flow solutions which has been completely successful in predicting the results for both Darcy flow and quadratic flow obtained by elaborate numerical methods. The new calculation method allows determination of the observable variables in gas-well testing at constant rates. It is similar to the scheme of using a succession of steady states suggested by Muskat in that it makes use of steady-state and material-balance equations. It also is similar to the work of Aronofsky and Jenkins in that the new method includes Aronofsky and Jenkins stabilized flow equation as a special case. It improves upon both of these calculation schemes in that it accurately describes all portions of reservoir history and suggests means of determining reservoir rock properties from well-test data. This paper deals only with production from a reservoir, in which case the rate is defined as being negative. The reservoir studied here is a homogeneous, disk-shaped porous body of uniform thickness, with all boundaries sealed except the inner radial boundary of the well. SPEJ P. 264^