Material Balance Errors Research Articles

Effect of Solvent Concentration on Scaling Butane Solvent Enhanced Oil Recovery Processes Using Reservoir Simulation

Understanding scaling of enhanced oil/bitumen recovery processes is essential in moving laboratory scale experimental results to field scale. Scaling theory for thermal processes is well understood and has been applied to steam processes. However, scaling of hybrid steam (thermal) /solvent (mass transfer) processes is still not well defined nor well understood. This paper investigates the scaling behavior of hybrid steam/butane gravity drainage processes using reservoir simulation (commercial thermal compositional simulator CMG STARSTM). Previous research has used reservoir simulation to confirm scaling groups for waterflooding. A similar strategy was used in this study whereby the scaling of a hybrid (steam) solvent oil recovery process was examined using reservoir simulations at three different reservoir scales: lab scale, semi-field scale and field scale to examine the influence of the mass transfer mechanisms of diffusion and dispersion on the scalability of the process. In particular, the influence of butane solvent concentration on scaling a steam/butane gravity drainage process was investigated by considering several butane mole fraction concentrations injected with steam (1%, 2%, 5%, 7%, 10%, 15%, 21%, 25% and 50%). Temperature contours, and mole fraction contours of butane in both the oil and gas phases were examined for various solvent injection concentrations to examine scalability. Numerical results are provided with no diffusion and dispersion, diffusion only, dispersion only and with both diffusion and dispersion added to the simulations. Results confirmed scalability of the process with no capillary effects when the simulation results were non-dimensionalized, although there were some issues with material balance errors in some of the simulation results particularly at high solvent concentrations. For low injection concentrations, the contours were almost identical (indicating scalability) for the three scales for the operating condition studied. In addition, capillary effects were also studied, and similar to scaling thermal processes, the capillarity effects influenced scalability of the process under the conditions studied particularly at higher injection concentrations. Scalability using reservoir simulation was generally preserved with low injection concentrations, but unusual behavior was observed at higher injection concentrations (>5%). Oil recovery curves were non-dimensionalized to make comparisons amongst the three scales. The oil recovery curves displayed an unusual S-shaped behavior at higher injection concentrations when capillary effects were included especially for the lab and semi-field scales. In all cases when all of the mechanisms are included (diffusion, dispersion and capillary effects), Scale 1 shows a much faster recovery than Scale 3 which suggests that the lab scale might temporally overestimate the field scale recovery for this particular process scenario.

Non-isothermal numerical simulations of dual reflux pressure swing adsorption cycles for separating N2 + CH4

A non-isothermal model of dual reflux pressure swing adsorption (DR-PSA) was developed using a commercially available software package to numerically solve the dynamics of the unit operation. Importantly the model includes a full energy balance, which is a feature not reported previously in the literature for simulations of DR-PSA cycles even though bed temperature swings of 10–20K have been observed in experimental studies. The simulation allowed solution of the pressure-flow network for cycles in the PL-A configuration, where feed gas enters the middle of the low pressure column and where pressure inversion is conducted by transferring gas rich in the more adsorbed component between beds. At cyclic steady state the simulations contained material balance errors comparable in magnitude to those reported previously for isothermal DR-PSA simulations; however, these were accounted for using a robust correction scheme. Predictions of the pressure, flow and temperature profiles within the beds for a range N2+CH4 mixtures being separated using activated carbon were in good agreement with the corresponding results of 24 DR-PSA experiments recently reported by Saleman et al. (2015). The root mean square deviation of the predicted methane mole fractions from the experimental values were 0.003 and 0.024 for the light (N2-rich) and heavy (CH4-rich) product streams, respectively. Parametric studies conducted with the model show how the cycle design can be optimised with respect to reflux flow and the feed-purge step duration, and also illustrate the need for reliable values of the sorption mass transfer coefficient.

Partition coefficients for a mixture of two lubricant oligomers

AbstractUniversal base oils that remain pourable over wide temperature ranges would have important advantages for lubrication applications. The model system used in this project was a poly(α‐olefin) synthetic base oil modified with polydimethylsiloxane (PDMS) to lower the pour‐point temperature. Although the blend was miscible at room temperature, phase separation occurred at temperatures lower than 258 K. Partition coefficients of such nonideal oligomer mixtures can (1) help define operating temperature ranges and (2) provide a basis for designing molecular weight distributions of each lubricant that control or prevent phase separation. The poly(α‐olefin) base oil family is branched oligomers with two to five n‐mers at levels greater than 1 wt %, whereas PDMS additives are linear oligomers having between 10 and 50 sequential n‐mers at levels greater than 0.5 wt %. In this study, Fourier transform infrared measurements of the poly(α‐olefin) and PDMS compositions in each phase provided an overall material balance. Poly(α‐olefin) oligomers were detected with size exclusion chromatography with a differential refractive‐index detector, and PDMS oligomers were detected with matrix‐assisted laser desorption/ionization–time‐of‐flight mass spectrometry. The best sets of measurements for the individual oligomers in each phase were selected by minimization of the overall material balance errors. For both oligomers, components with high molecular weights were preferentially excluded from the phase rich in the other polymer and were relatively independent of temperature. The partition coefficients of poly(α‐olefin) components increased with increasing oligomer length, whereas the partition coefficients of the PDMS components decreased with increasing oligomer length. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011

Iterative coupling reservoir simulation on high performance computers

In this paper, the iterative coupling approach is proposed for applications to solving multiphase flow equation systems in reservoir simulation, as it provides a more flexible time-stepping strategy than existing approaches. The iterative method decouples the whole equation systems into pressure and saturation/concentration equations, and then solves them in sequence, implicitly and semi-implicitly. At each time step, a series of iterations are computed, which involve solving linearized equations using specifi c tolerances that are iteration dependent. Following convergence of subproblems, material balance is checked. Convergence of time steps is based on material balance errors. Key components of the iterative method include phase scaling for deriving a pressure equation and use of several advanced numerical techniques. The iterative model is implemented for parallel computing platforms and shows high parallel effi ciency and scalability.

Improvements to Reservoir Material-Balance Methods

Summary Experience with material-balance data sets from the field and from simulation has revealed some procedures that can be used to improve analysis of both oil and gas reservoirs: Failure to account for a weak waterdrive can result in significant material-balance errors. The assertion of previous authors that weak waterdrive exhibits a negative slope on the Cole (gas) and Campbell (oil) plots has been confirmed. A weak waterdrive is much more unambiguous on these plots than on commonly used plots, such as the p/z plot for gas. A modified version of the Cole plot is proposed to account for formation compressibility. The reservoir drive indices are a useful tool for determining the correctness of the material-balance solution because they must sum to unity. The drive indices should never be normalized to sum to unity because this obscures their usefulness and leads to a false sense of security. A modified version of the Roach plot (for gas) is proposed that improves interpretation in some waterdrive situations. Material balance has not been replaced by reservoir simulation; rather, it is complementary to simulation and can provide valuable insights to reservoir performance that cannot be obtained by simulation.

A finite element solution of a fully coupled implicit formulation for reservoir simulation

AbstractAn iterative method is presented for solving a fully coupled and implicit formulation of fluid flow in a porous medium. The mathematical model describes a set of fully coupled three‐phase flow of compressible and immiscible fluids in a saturated oil reservoir. The finite element method is applied to obtain the simultaneous solution (SS) for the resulting highly non‐linear partial differential equations where fluid pressures are the primary unknowns. The final discretized equations are solved iteratively by using a fully implicit numerical scheme. Several examples, illustrating the use of the present model, are described. The increased stability achieved with this scheme has permitted the use of larger time steps with smaller material balance errors.

A modified method of characteristics technique and mixed finite elements method for simulation of groundwater solute transport

A comprehensive groundwater solute transport simulator is developed based on the modified method of characteristics (MMOC) combined with the Galerkin finite element method for the transport equation and the mixed finite element (MFE) method for the groundwater flow equation. The preconditioned conjugate gradient algorithm is used to solve the two large sparse algebraic system of equations arising from the MMOC and MFE discretizations. The MMOC takes time steps in the direction of flow, along the characteristics of the velocity field of the total fluid. The physical diffusion and dispersion terms are treated by a standard finite element scheme. The crucial aspect of the MMOC technique is that it looks backward in time, along an approximate flow path, instead of forward in time as in many method of characteristics or moving mesh techniques. The MFE procedure involves solving for both the hydraulic head and the specific discharge simultaneously. One order of convergence is gained by the MFE method, as compared with other standard finite element methods, and therefore more accurate velocity fields are simulated. The overall advantages of the MMOC‐MFE method include minimum numerical oscillation or grid orientation problems under steep concentration gradient simulations, and material balance errors are greatly reduced due to a very accurate velocity simulation by the MFE method. In addition, much larger time steps with Courant number well in excess of 1, as compared with the standard Galerkin finite element method, can be taken on a fixed spatial grid system without significant loss of accuracy.

A model for wetland surface water dynamics

A mathematical model for overland flow through vegetated areas has been developed. Input information includes site topography and porosities, power law parameters for the velocity correlation, and the driving forces: precipitation, evapotranspiration, stream flows, groundwater recharge or discharge, and pumped additions. Depths and flow rates are then calculated as functions of position and time. A one‐dimensional algorithm has been developed to execute these calculations. Numerical solutions to the nonlinear partial differential equations are accomplished using special finite difference approximations. The method permits long simulation times (several years), without significant material balance errors. The model and simulator have been validated using data from the Porter Ranch peatland located near Houghton Lake, Michigan.

Gas Phase Appearance and Disappearance in Fully Implicit Black Oil Simulation

AbstractThis paper describes the pseudogas and variable substitution methods for handling gas phase appearance/disappearance in fully implicit black oil simulation. The results of several test runs indicate that there is virtually no difference between these two methods in terms of efficiency. The pseudogas method is easier to program than variable substitution but requires some crucial reallocations of saturations and shows greater material balance errors.

A Small-Volume Continuous Oil/Water Separator for Laboratory Displacement Experiments

Introduction An experimental task commonly encountered in laboratory core displacement experiments is the separation of oil and water phases. While such physical separations are not inherently difficult, they can be time-consuming and may lead to material-balance errors if numerous small samples are collected. In this paper, we describe an oil/water separator of low volume that relies on the wetting characteristics of commercially available filter membranes. Because a flowing stream containing oil and water can be separated continuously without a large-volume mixing cell, the separator allows on-line measurement of the compositions of oil and brine streams. It might also prove useful in relative permeability measurements. permeability measurements. Description and Operation Fig. 1 is a schematic of the device. The two-phase fluid enters the chamber between two porous membranes having different wetting behaviors. Oil, the less-dense phase, rises and contacts the upper, hydrophobic phase, rises and contacts the upper, hydrophobic membrane. Water settles and contacts the lower, water-wet membrane. Because it is hydrophobic, the upper membrane resists passage of water through its pores but readily admits the passage of water through its pores but readily admits the nonpolar oil. The lower membrane can be wet by either oil or water but, if first wet by water, surface tension effects will prevent oil from entering the pores. Therefore. the first step in operating the separator is to prime it with water to establish a water-wet condition, which must be maintained during operation. Before the separator is assembled, we saturate the water-wet membrane using a vacuum to eliminate air from the pore spaces. After assembly, we fill the separator with water to the level of the vent of the siphon-breaker. JPT P. 487

Three-Dimensional Geothermal Reservoir Simulation

Abstract This paper presents the development of a three-dimensional, finite-difference model for the simulation of geothermal reservoirs. The model is designed to simulate geothermal reservoirs that contain water in any of its vapor or liquid states and provisions are included for properly treating changes provisions are included for properly treating changes of state during a time step. This logic provides for a stable calculation of state change and eliminates pressure, heat balance, and material-balance errors. pressure, heat balance, and material-balance errors. Mass and energy balances are solved simultaneously using an implicit pressure-explicit saturation (IMPES) formulation. An implicit treatment of production rates, capillary pressure, and transmissibilities is included as an option. Thus, entire field, cross-sectional, or individual well studies can be performed efficiently. Example problems are presented to demonstrate the model's utility and to provide insight into the nature of geothermal production under various conditions. In particular, an example is presented of a reservoir initially containing subcooled liquid where fluid conditions near the production well range from subcooled liquid to saturated steam and then to superheated steam. Introduction Geothermal energy represents an essentially untapped, alternate source of energy world-wide. Exploration for this type of energy has increased, however, and should lead to the discovery and development of new geothermal areas. Notable producing geothermal fields include Wairekei in producing geothermal fields include Wairekei in New Zealand, Cero Prieto in Mexico, Matsukawa in Japan, Larderello in Italy, and The Geysers in California. Recently, industry literature has dealt with zero-(one-cell), one-, and two-dimensional, geothermal reservoir simulation. Most studies treat steam in only one state or mention the difficulty of simulating blocks that change from one state to another. Whiting and Ramey presented a zero-dimensional, geothermal model that included water influx. They discussed reservoir performance during the various states that steam can assume. Their study also included a successful match of the pressure/ production performance of the Wairekei geothermal production performance of the Wairekei geothermal reservoir in New Zealand from 1956 to 1966. Coats developed a three-dimensional, finite-difference, steam-flooding model that simultaneously solved the mass and energy equations. An IMPES formulation was used to solve for pressures implicitly followed by a solution for temperatures, saturations, and so forth. Saturations calculated using explicit rates, capillary pressures, and transmissibilities were recalculated at the end of each time interval, using an implicit formulation for these variables. Brigham and Morrow discussed the P/Z behavior of geothermal steam reservoirs. Their model was similar to Whiting and Ramey's model, but they allowed for the presence of a vapor and liquid zone. Mass and energy balances were written for both zones, and logic for a constant or falling liquid level was included. Martin analyzed internal steam drive in geothermal reservoirs by assuming that temperature, pressure, and fluid saturation gradients as well as capillary pressure and gravity were negligible. Gould described the development of a vertical two-phase, pressure-drop calculation for flow in geothermal wells. He discussed the effect of heat transfer to the surrounding formation and the effect of steam quality on production. Mercer and Faust developed a two-dimensional (areal), two-phase geothermal model using a Galerkin finite-element formulation in space and finite-difference formulation for time. They chose pressure and enthalpy as dependent variables and pressure and enthalpy as dependent variables and solved for these two variables simultaneously. They presented a 10-month simulation of a hypothetical hot-water reservoir with initial conditions similar to those in the Wairekei reservoir. No provisions were included during the course of a run for changing states - for example, from subcooled liquid to saturated steam. SPEJ P. 151

Reservoir Simulation of Variable Bubble-Point Problems

Abstract This paper presents the development of a general beta reservoir simulator that will model conventional (fixed bubble-point) problems as well as problems involving a variable bubble point, such as gas injection projects above the original bubble point and water injection projects resulting in a collapsing gas saturation, Provisions are included for allowing the pressure to cross the bubble point with the same relative ease as in a conventional simulator. Example problems are presented to demonstrate the utility of the model for gas and water injection problems. problems Introduction Many reservoir simulation problems involve treating a variable bubble point throughout the reservoir. For example, when gas is injected into an undersaturated reservoir, gas will go into solution, increasing the bubble point of the oil. As this oil moves away from the injector, the bubble point of surrounding areas also may increase point of surrounding areas also may increase because of mixing. During waterfloods of saturated reservoirs, the gas saturations in regions near the injectors frequently reduce to zero at pressures below the original bubble point. Thus, the bubble point will vary areally throughout the field. point will vary areally throughout the field.Recent publications have discussed certain aspects of the variable bubble-point problem. Most of these papers contain only a brief discussion of this problem. Ridings discusses the need for allowing saturation pressure to vary continuously throughout the reservoir as long as there is free gas associated with the oil. In the model presented by Cook et al., free gas saturation is monitored for saturated systems and the bubble point is set equal to the prevailing reservoir pressure when the gas saturation in a cell disappears. Bubble points for undersaturated cells are allowed to change because of the entrance of free gas and mixing. Spilletta et al. assume that a cell that is saturated or undersaturated at the beginning of a time step will remain so throughout the time step. They then solve their saturation equations for water and gas saturations. The bubble point of any undersaturated cell is adjusted to account for nonzero gas saturation, and the water saturation is modified to conserve oil. Steffensen and Sheffield devote their paper to the reservoir simulation of a collapsing gas saturation during waterflooding. In their model, blocks that have a free gas saturation at the beginning of a time step and have zero or negative gas saturations at the end of a time step are detected, and the bubble points for these cells are set equal to the estimated pressure where Sg reduced to zero. Gas saturation for these blocks is set equal to zero and S is set to 1 - S . The oil saturation in adjacent saturated grid blocks is then adjusted so that oil material balance is maintained. Mixing caused by flow between undersaturated blocks is neglected. This paper presents a comprehensive analysis of modeling variable bubble-point problems.* It treats the specific problems of simulating gas injection above the bubble point as well as waterflooding depleted reservoirs. It differs from previously reported work by accounting for the effect of bubble-point change on computed pressure change during a time step. Also, provisions are included that allow the pressure to cross the bubble point with the same relative ease as in a conventions! simulator In regard to waterflooding, mis paper differs from the work of Steffensen and Sheffield in mat it allows for bubble-point changes caused by mixing. DEVELOPMENT OF FLOW EQUATIONS To simulate the variable bubble-point problem, the expansion of the flow equations above the bubble point must include the effects of pressure and bubble point on fluid properties. Also, special consideration must be given to cells passing through the bubble point if both pressure and material-balance errors are to be eliminated. SPEJ P. 10

Some Practical Considerations in the Consruction of a Semi-Implicit Simulator

Introduction The semi-implicit reservoir simulator has become a very important part of the total simulation package necessary for the practicing reservoir engineer. Any multiphase simulation of a single well tom, problem (e.g., a study of water-oil coning) is very problem (e.g., a study of water-oil coning) is very expensive unless such a simulator is available. With a "standard" reservoir simulator, this difficulty arises because the relative permeabilities and capillary pressures, which depend upon saturations, lag one time step behind the pressure calculation. For this reason such a simulator is said to be implicit in pressure and explicit in saturation (abbreviated as IMPES). When "new" saturations are obtained, they are formed from "old" relative permeabilities and capillary pressures. The mathematical form of the equations is pressures. The mathematical form of the equations is such that an uncontrolled oscillation in the saturation values develops if the time step is too large. Only by taking smaller time steps can this oscillation be suppressed in an IMPES simulator, and very small time steps are then necessary for simulating coning behavior. The same problem can also appear in any production well model (in an IMPES simulator) that production well model (in an IMPES simulator) that distributes fluid production proportionally to phase mobilities. These saturation oscillations can be eliminated by making the well model "implicit in saturation." To overcome this instability, Blair and Weinaug developed a fully implicit simulator. All coefficients were updated iteratively until convergence occurred. It was necessary for them to stabilize their solution technique by the use of Newtonian iteration. Then it was found that a time-step limitation occurred because of nonlinearities, since the Newtonian iteration would not converge without a good initial estimate. Coats and MacDonald proposed an effective solution to this problem. They suggested estimating the relative permeabilities and capillary pressures by an extrapolation; e.g., pressures by an extrapolation; e.g.,(1) A mathematical investigation showed that since Sn + 1, the saturation at be new time step, is found simultaneously with the pressures as part of the solution, the mathematical time-step limitation inherent in the IMPES technique as a result of using "old" relative permeabilities would not occur. They also suggested that the equations be linearized by dropping products of (Sn+1 - Sn) and (pn+1 - pn). The equations are then more nearly linear. Hence, the difficulties in convergence of the solution technique are greatly reduced (Newtonian iteration is not needed). Nolen and Berry showed that linearization of the accumulation terms was not necessarily the best strategy (in problems that have solution gas), because material-balance errors would result. They felt that linearization of the flux terms made little difference. Many questions still remain unanswered by these papers. Nonlinearities remain in the equations, papers. Nonlinearities remain in the equations, particularly when a phase is near its immobile particularly when a phase is near its immobile saturation. Because of the use of upstream weighting of the relative permeabilities, another type of nonlinearity (potential reversal) can occur. Furthermore, the question of a practical procedure for selecting the time step must be settled. Finally, there are nonlinearities in the well model, which can cause slow convergence, or failure to converge, especially when dealing with a well completed in several layers or with a well that changes constraints. The problems just mentioned are all more severe if large time steps are used. Reducing time-step size is expensive, and in many cases difficult to automate. In this paper we present our experience in treating or circumventing these problems. We have felt that an important principle to follow is to eliminate time-step limitations due to mathematical instabilities. Thus, one should be able to run steady-state problems with essentially unlimited time-step size. For transient problems, it is expected that time truncation errors would normally govern the time-step size. One final, practical goal was to avoid running problems that would be annoying and mysterious to the field reservoir engineer. SPEJ P. 216

Comparative Evaluation of the Strongly Implicit Procedure (SIP) For Gas Reservoir Simulation

Abstract The present work was principally concerned with an evaluation of the new SIP algorithm for solving the matrix problem arising from the application of the various implicit difference techniques to the nonlinear partial differential equation for the flow of a real gas in two dimensions, expressed ill two different forms, utilizing the original and the transformed variables, respective. A number of con,- emotional difference methods were also tested for comparison purposes. The equations resulting from the application of the completely implicit scheme were solved by both SIP and SOR (successive over-relaxation), in order to evaluate the speed of convergence using the two methods. In addition, a series of numerical experiments was carried out using the SIP algorithm in order to study the effect of the number of iteration parameters on the number of iterations required for convergence, the rate of convergence and the material balance error induced. It was found that the explicit schemes tested suffered from time step size limitations and increasing material balance errors with time. The implicit schemes, in particular the modified implicit scheme, is therefore recommended. Computational results show that the SIP algorithm is three times as fast as SOR. The optimized SIP algorithm, although involving more computational work per iteration requires fewer iterations to converge than SOR, for a given accuracy. The number of SIP iteration parameters should be optimized where feasible. INTRODUCTION NUMEROUS AUTHORS have discussed gas reservoir simulation schemes, involving both explicit and implicit differencing methods. The present study gives a comparison of some of the schemes, and was principally concerned with an evaluation of the Strongly Implicit Procedure (SIP) recently proposed by Weinstein, Stone and Kwan(1) for solving the equations resulting from the conventional as well as from the completely implicit scheme due to Blair and Weinaug(2). Mathematical Model The equation describing the flow of a gas in a two dimensional porous medium can be derived by a consideration of the continuity and Darcy's equations, and is as follows: (Equation Available In Full Paper) where -q represents the production term; the remaining symbols are explained in the Nomenclature. Equation (1) can be rendered quasilinear by use of the gas potential, P*, proposed by AI-Hussainy, Ramey and Crawford(3) and by Russell, Goodrich, Perry and Bruskotter(4): (Equation Available In Full Paper) By use of Eq. (2), Eq. (1) can be transformed to the following form (later referred to as Model 1) : (Equation Available In Full Paper) and may be expressed as a function of P*. In the same manner, Eq. (4) can be used to transform Eq. (1) as follows (later referred to as Model 2): (Equation Available In Full Paper) Eqs. (3) and (5) were solved in the present study by use of a number of explicit and implicit differencing schemes. The hypothetical reservoir employed for numerical computations was a 320-acre square (side = 3,734 ft) with a formation thickness of 20 ft, porosity of 25 per cent and permeability of 100 md (in the homogeneous case;

Simultaneous flow and temperature correction in the equilibrium stage problem

AbstractIterative methods for the solution of the steady state equilibrium stage problem are studied. Equations are first developed for calculating the effect of a change in temperature or flow rate on all energy balances, and then it is shown that these equations can be used in the multiple variable form of the Newton‐Raphson correction process to correct either the temperatures or the flow rates when only energy balance errors are used. Similar equations for material balance errors which have been developed previously are then combined with the energy balance equations to provide a method for simultaneous correction of the temperatures and the flow rates. Because first‐order interactions between flow rates and temperatures are included the method is applicable to a wide range of equilibrium stage problems. Sample problems are presented, and it is shown that quadratic convergence is obtained for the simultaneous correction process.

Methods for Predicting Gas Well Performance

Abstract The depletion performance of gas wells has been investigated bymathematical simulation techniques. The gas well model which was studiedconsisted of a single well located in the center of a bounded, cylindrical, homogeneous reservoir. Dependency of gas compressibility and viscosity onpressure was considered in studies of well performance at both constant massflow rate and constant flowing pressure conditions. To carry out theinvestigation, the nonlinear, second-order, partial differential equation whichdescribes Darcy flow of a nonideal gas through porous media was solvednumerically. Some of the previous investigations of gas well performance have been oflimited general use, because assumptions were introduced to simplify either thegas properties or the basic differential equation. Other studies have beenrigorous in these respects but have presented a very limited set of calculatedresults. The present study was attempted to present a rigorous theoreticalmodel and sufficient numerical results to permit meaningful conclusions to bedrawn. It was found that all terms must be retained in the partial differentialequation to make accurate predictions. The neglect of higher-order terms, e.g., terms of the order of the "gradient squared", leads to serious material balanceerrors at large times and to conservative estimates of gas well performance. The higher the gas flow rate and/or the lower the permeability-thicknessproduct of the formation, the more pronounced are these deviations. Forexample, in a well draining 640 acres in a 25-md-ft formation (8,120 MMcf gasin place) at a constant rate of 993 Mcf/D, the rigorous solution predicts abottom-hole pressure decline from 4,000 to 1,000 psia in 8.7 years. Ifhigher-order terms are neglected in the differential equation, this decline inpressure is predicted to occur in 5.3 years. With the results of the numerical solution of the differential equationas a basis, simple, easy-to-use approximations for predicting gas wellperformance for Darcy flow conditions have been developed. These simpleapproximations are based on the familiar equations for flow of a single, slightly compressible fluid. The approximate methods possess a high degree ofaccuracy and enable the prediction of long-term gas well performance to be madequickly and accurately without the use of a digital computer. Both transientand stable flow period approximations were developed. Introduction In recent years income from the sale of natural gas and associated productshas represented an ever-increasing fraction of the industry's total revenuefrom operations. To meet the surge in demand for natural gas, the industry hasdepended heavily upon established reserves and has actively pursued developmentof new reserves. This search has progressively led to reservoirs whichyesterday were too tight and/or deep to yield the desired return on investedcapital. More than ever before, evaluation accuracy is now required to forecastthe criteria upon which engineering recommendations and management decisionsare based. Considerable effort has been expended by both research and operationspersonnel on the development and application of methods for analyzing andpredicting the performance of gas wells. Fundamentally, the problem is thefamiliar one of extracting data during the drilling, testing and earlyproduction life of a well and applying these data within an accurate simulationmodel to predict long-term behavior. During the past 30 or more years avoluminous literature dealing specifically with gas field problems has beengenerated. A recent book1 lists a comprehensive bibliography of publishedmaterial through 1959. Over 1,200 references are cited. Since then 39additional articles on natural gas technology have been published inTransactions volumes of the Society of Petroleum Engineers of AIME. Most existing theory for predicting gas well performance requires that oneor more idealizations (e.g., steady-state flow, ideal gas of constantviscosity, small and constant compressibility and constant-viscosity fluid) beapplied. Although existing theory may apply directly or be adapted by variousartifices to describe specific gas well and reservoir behavior, no widelyapplicable method is available, and existing methods appear to be subject toappreciable error unless better limits of applicability are defined.