Drawdown Analysis Research Articles

Viscoelastic aquifer model applied to subsidence due to pumping

A viscoelastic aquifer model, proposed earlier by the authors, is applied to analyze and predict the piezometric drawdown and the compaction observed in a major confined aquifer system in the San Joaquin Valley in California. A field approach is used based on a minimal number of ‘bulk’ parameters. These parameters, which are the transmissivity, the compressibilities of primary and secondary consolidation, and the viscosity of the secondary consolidation, were derived by calibration of the model by means of trial and error simulation of 2 years of records of drawdown and compaction. It was found that in the absence of drawdown records the identification may also be accomplished with compaction records in addition to a standard pumping test. Comparison of computed results with available records shows that the viscoelastic model yields a satisfactory prediction of compaction which is far better than that obtainable with the elastic model of the classical Theis‐Jacob approach. The elastic model may be adequate to represent the hydraulic properties of the aquifer in the analysis of the piezometric drawdown; however, it appears to be unsatisfactory to represent the mechanical properties of the aquifer as related to surface subsidence.

Computer Solutions for the Stability of Reservoir Slopes

Prediction of the stability of slopes under various water conditions and for a wide range of loading and soil conditions can be made with the help of a digital computer. Reservoir slopes may reach a very critical state immediately after very rapid drawdown. At present, this condition is analyzed assuming an “instantaneous” drawdown event. This paper presents a method for obtaining the slope stability under a drawdown condition with a defined time-elevation history. The analyses are based on the simplified form of Bishop’s model with circular failure surface and vertical slices. Two options are available for determining the pore pressure. In the time-dependent drawdown analysis, a differential equation equates the volume of water that flows out of the slope in a given time with the product of the velocity of flow, area of flow, and time interval. The solution of this equation yields the position of the phreatic surface with time. This information helps determine the pressures along a trial circular sliding surface, and the variation of the factor of safety with time. The factor of safety decreases rapidly with the lowering of the reservoir level, followed by a slow and nonlinear increase to the steady state value.

Pressure Transient Tests in Formations Having Stress-Sensitive Permeability

In this study of the conventional analysis of gas well tests in stress-sensitive formations it was found that kh and wellbore condition determined from drawdown tests can be significantly in error when permeability is stress-dependent. A similarity in test results suggests permeability is stress-dependent. A similarity in test results suggests that stress sensitivity of the formation and turbulence of gas flow may affect test data in the same manner simultaneously. Introduction Conventional methods of analysis of drawdown and buildup test data are not strictly applicable to gas reservoirs because of the highly variable gas compressibility. However, they have been used with success for that purpose for some time, Conditions for applicability of the pressure, pressure squared, and real-gas potential plots' have been studied and discussed by a number of investigators and further elucidated by Wattenbarger and Ramey. Data taken under conditions of turbulent flow were also analyzed by the same authors and reported in Ref. 8. They concluded that reservoir conductivity determined from drawdown tests run under such conditions is not representative but conductivity determined from buildup tests is nearly correct. A different kind of flow-reducing mechanism has been studied experimentally by a number of investigators, most recently by Thomas and Ward . This mechanism is the reduction in permeability caused by an increase in effective frame stress. In the reservoir an increase in effective frame stress is caused by fluid withdrawal and the accompanying decrease in pore pressure. Since the overburden force on the reservoir pressure. Since the overburden force on the reservoir rock remains the same, the decreasing pore pressure results in an increased effective frame stress. Because low-permeability formations are more affected by stress changes, this effect can be expected to be more significant in deep gas reservoirs. The detrimental effect of stress on rock permeability and gas production has been demonstrated by Vairogs et al. production has been demonstrated by Vairogs et al. This paper presents the results of a theoretical investigation of the use of conventional pressure transient analysis methods in stress-sensitive formations. The great similarity between results from tests under turbulent flow and under stress-sensitive conditions is emphasized. Method of Investigation The numerical model used in the simulation was described in an earlier paper. Briefly, it is a radial gas flow model that uses real gas properties and allows radial variation of permeability and porosity. The latter feature permits the simulation of wellbore damage or stimulation. The model has the additional capability to calculate radial variations in rock frame stress that result from changes in pore pressure near a borehole in an initially stressed pressure near a borehole in an initially stressed medium. Changes in formation permeability at various distances from the wellbore are then calculated from experimentally determined stress/permeability relationships. These relationships are determined in the manner described in Ref. 7. An example of such a relationship is shown in Fig. 1. The initial permeabilities - i.e., permeabilities at a net confining permeabilities - i.e., permeabilities at a net confining stress of 500 psi - are 0.14 md for Sand A and 0.32 md for Sand B. Sand A is twice as sensitive to stress as Sand B, but both exhibit the same proportional permeability reduction. permeability reduction. JPT P. 965

An algorithm for least squares analysis of drawdown in observation wells

An algorithm has been developed for computerized analysis of drawdown measurements in an observation well near a well pumping at constant rate from a semi-infinite, non-leaky aquifer. The algorithm uses the least squares criterium to determine values of T, the transmissivity, S, the storage coefficient, and r i ∗ , the distance between the observation well and the image well, which give the best fit between the observed and theoretical drawdown. An alternate algorithm leads to values of T, S, and r i ∗ which give a least squares fit for data plotted on logarithmic paper. The behaviour of the algorithm is shown for three sets of drawdown data with varying amounts of scatter.

Approximate Solutions For Unsteady LiquidFlow In Composite Reservoirs

Abstract A great deal of interest in composite reservoir flow solutions has been shownby the increasing number of publications dealing with this problem. So far, only a limited number of solutions have appeared. The solutions available donot seem to be of completely general utility. The purpose of this paper is todemonstrate a class of approximate solutions to composite reservoir flowproblems which may be generated readily. These solutions are applicable to welltest analysis and, although approximate, have the advantage of involving simplefunctions only. One interesting result is that secretary approximating formswhich ha""e been presented previously derive readily from this class ofsolutions. Another interesting result is that outer boundary conditions ofeither a closed boundary or a constant pressure boundary may be approximated aslimits of composite reservoir solutions. INTRODUCTION THE TERM ‘COMPOSITE’ was originally used in very early studies of heatconduction in solids composed of two different materials; see Carslaw(l) for example. Despite the well-known analogy between heatconduction and laminar fluid flow through porous media, intense interest influid flow through composite solids has occurred recently. Hurst(2)and Mortada(3) analyzed the interference between oil fields in acommon aquifer of two different permeabilities in 1960. Loucks and Guerrero(4) presented an analysis of drawdown and buildup in radialcomposite systems in 1961. Rowan and Clegg(5) presented approximatesolutions in 1962. Van Poollen and associates have presented several papersconcerning radial and linear discontinuities within porous media and theresultant effects upon well test analysis(6,7). Carter(8)presented an analysis of the depletion of a closed composite radial reservoir.and discussed reservoir limit tests for this class of reservoirs. Adams, etal.(9) employed Hurst's approximate result for radial compositereservoirs to interpret pressure buildup tests in a fractured dolomitereservoir. Recently, Wattenbarger and Ramey(10) employed a radialcomposite reservoir model to simulate the skin effect of a finite storagecapacity. The references given above concerning oil and gas reservoir applications of thecomposite reservoir problem are provided to show the increasing interest inthis class of flow problem. No attempt has been made to provide a completelisting. The bibliographies of the papers cited list other pertinent studies.The main point is that composite reservoir flow problems are significant, andthat the connection between various applications of the solutions is notclearly evident. One interesting fact is that papers concerning composite reservoir problemshave often raised controversies or debates, or left other evident problemsunsettled. Part of the reason must lie in the complex nature of these problems.Fortunately, the approximate solutions which will be discussed here containonly elementary functions. The experimental data of Donnelly and Katz(1) were used for themethane – carbon dioxide system and the data of Price and Kobayashi(10) were used for the methane-ethane-propane system. Theresults of the calculations are shown graphically in Figure 6, where again itis seen that the convergence pressure concept has failed to correlate the K-factors for methane in the presence of the non-hydrocarbon component.

Drawdown in a well of large diameter

A solution for the drawdown in a large‐diameter well discharging at a constant rate from a homogeneous isotropic artesian aquifer, which also takes into consideration the water derived from storage within the well, is presented. A set of type curves computed from this solution permits a determination of the transmissibility of the aquifer by analysis of drawdown observed in the pumped well.

Application of Real Gas Flow Theory to Well Testing and Deliverability Forecasting

Abstract Previous gas well test analyses have been based mainly upon linearizationsof ideal gas flow results, although a method for drawdown analysis based uponreal gas flow results has been proposed. Linearizations based upon ideal gasflow require estimation of gas physical properties at some sort of averagepressure, and implicitly involve the assumption that pressure gradients aresmall everywhere in the reservoir. A new real gas flow equation has beendeveloped by means of a substitution which couples pressure, viscosity and gaslaw deviation factor. This substitution has been called the real gaspseudo-pressure. Use of the real gas pseudo-pressure leads to simple equationsdescribing real gas flow which do not contain pressure-dependent gasproperties, and which do not require the assumption of small pressure gradientseverywhere in the flow system. Equations required to determine flow capacity, well condition and staticformation pressure from pressure drawdown and build-up tests with the real gaspseudo-pressure concept are presented. Also shown are applications of the realgas pseudo-pressure to back-pressure testing, the gas materials balance andrigorous determination of average gas properties for previous gas flowequations. Included are sample calculations for well test analyses.

Pressure Drawdown Analysis, Variable-Rate Case

Abstract A theoretical development is presented which provides a straightforward method of handling the drawdown analysis for both oil and gas wells flowing at variable rates. In the past our inability to analyze variable-rate drawdowns has been a major obstacle in using them for formation evaluation. This method should permit wider use of these tests in the future. Introduction Pressure build-up analyses are widely used to obtain reservoir data such as the effective flow capacity of the formation. On the other hand, although pressure draw- downs have been run for a long time, they have not been used extensively to evaluate reservoirs because they are difficult to analyze. one of the difficulties in analyzing pressure drawdowns has been that the theory developed until now has required the test to be run at a constant rate. For both oil and gas wells this is a difficult requirement to satisfy, especially during the early period of a well's life when drawdown tests are usually run. in spite of this limitation, drawdown tests are often run with build-ups as a check. In this paper a theory, which handles the drawdown analysis for variable rates, is presented for both oil and gas wells. Thus, one of the biggest deterrents to running drawdowns is removed, and they should become a much more useful engineering tool in the future. THEORETICAL CONSIDERATIONS OIL RESERVOIRS The general equation used for describing the unsteady- state radial flow of slightly compressible fluids in homogeneous porous media can be written in cylindrical coordinates as (1) where r is the radial distance in cm, t is the time in seconds, p is the pressure at r and t in atmospheres, phi is the fractional porosity, mu is the viscosity in cp, c is the compressibility in vol/vol/atm, and k, is the permeability in darcies. There are a number of solutions of Eq. I for various boundary conditions. One of these is the so-called "point- source" solution, which approximates the case of an infinite oil reservoir with a well located at r = 0 and produced at a constant rate. The point-source solution of Eq. 1 is (2) and h is the net pay thickness in cm, q is the production rate in cc/sec, p is the original pressure at t = 0, p is the pressure at any t and r, and r, is the well radius. Eq. 2 describes the pressure drawdown of a well where the formation around it is neither damaged nor improved. If a condition of permeability damage or improvement exists, the equation must be corrected for these effects. van Everdingen introduced an additional term: where S is the total flowing time for n constant-rate flow periods, t - t, t - t, ...t - t with rates q, q, ...q . If we use Eq. 4 in its present form to construct a straight-line pressure drawdown plot, k, r, phi, mu, c and S must be known. This is so because for every t a variable factor is left at the right side of the equation. This has been a major difficulty in analyzing a variable-rate pressure draw- down. JPT P. 960ˆ

Advances in Estimating Gas Well Deliverability

Abstract In the past, drawdown tests to estimate well deliverability and diagnose wellbore damage, although simple in themselves, have been of limited value, often because it is inconvenient or costly to obtain the constant flow rate required to analyze them reliably. This paper presents a new drawdown analysis technique developed with the aid of a digital gas-well simulator. The new method does not require a constant rate, is as simple to analyze as the conventional method and is considerably more accurate. It can thus heused in gas wells to analyze the early time data from a back-pressure test when the rate has not yet "stabilized" to a constant value. The new technique takes into account turbulent flow in gas-well systems - a prerequisite to calculatinga gas well's current and potential deliverability. These calculations areeasier if at least two drawdown tests, and preferably, more, can be conducted on the gas well. Fortunately, the practice of running a multiplicity of drawdown tests is already embodied in the four-point isochronal AOF test, where each flow test is separated from the others by a sufficiently long period of shut-in. This type of test lends itself to reliable drawdown analysis. However, in the four-point flow-after flow test, where the flows at each rate are consecutive, the deliverability estimates often are not reliable. The new drawdown technique is not limited to use in gas wells. It can also be used toestimate well productivity and diagnose wellbore damage in oil or water systems. Introduction Gas flow in a porous medium is different from oil or water flow in that itdoes not obey the Darcy Law. In the gas-reservoir-engineering literature (1), this difference is now commonly called "turbulence." Turbulence occurs becausethe low viscosity of gas (compared to that of oil or water) results incorrespondingly higher gas velocities under a given pressure gradient. At these higher velocities, the gas suffers inertial energy losses as well as the usualviscous energy (Darcy) loss.

Rapid Methods for Estimating Reservoir Compressibilities

Ramey Jr., H.J., Member AIME, Mobil Oil Co., Los Angeles, Calif. Abstract Conventional calculation of total system isothermal compressibility for a system containing a free gas phase involves, among other things, evaluation of the change of oil and gas formation volume factors and the gas in solution with pressure. Preferably, this information should be obtained from laboratory measurements made with particular oils and gases. Often, experimental measurements are not available. In this case, it is necessary to obtain pressure- volume- temperature relationships from general correlations such as those of Standing for California oils. In order to speed estimates of compressibility, generalized plots have been prepared of the change of both oil formation volume factors and gas in solution, with pressure from Standing's correlations. A generalized plot for estimating the change in the two-phase (oil and gas) formation volume factor with pressure is also presented. Usually, the effect of gas dissolved in reservoir water upon the total system compressibility is neglected for gas saturated systems, due to the low solubility of gas in water. Results of this study indicate that the increase in total system compressibility caused by solution of gas in water is often as large as the compressibility of water, and can be magnitudes larger for low pressure systems. Generalized results for estimating the change of gas in solution in water with pressure are presented in tabular and graphical form. Introduction During the past decade, pressure build-up and drawdown techniques have gained an important place in reservoir engineering. Build-up and drawdown analyses are only two special applications of the broad field of transient fluid-flow theory. All solutions of transient fluid-flow problems contain a parameter called the total system isothermal compressibility. This property of fluids and porous rock is a measure of the change in volume of the fluid content of porous rock with a change in pressure, and it may vary considerably with pressure. Evaluation of total system isothermal compressibility is not difficult, but it is tedious and time-consuming. Often compressibilities are estimated roughly, or transient flow methods are neglected completely. The benefits of using accurate system compressibility in properly-executed build-up or drawdown analyses are:Better planning of pressure build-ups may be achieved to avoid unnecessary loss of revenue due to excessively long shut-in periods, or to shut-in periods too short to yield useable data.Better and more reliable estimates of static formation pressures for reserves estimates and rate performance estimates.Reliable information for evaluation of well completion effectiveness, and planning and interpretation of well stimulation efforts. The purpose of this paper is to clarify the nature of the total system isothermal compressibility, and to present useful methods for estimation of compressibility, particularly for systems containing a gas phase. DEVELOPMENT Numerous publications have presented solutions to transient single-phase flow of slightly compressible fluids, stressing pressure build-up applications. In transient flow, a compressibility* term arises to permit volume content of fluids in porous rock to change as pressure changes. The basic nature of the compressibility term is usually taken for granted. Problems arise in practical applications of transient fluid theory because most published works consider only one flowing fluid-in an ideal porous system containing only one fluid. In 1956, Perrine presented an intuitive extension of single-phase flow pressure build-up methods to multiphase flow conditions. Later, Martin established conditions under which Perrine's multiphase build-up method had a theoretical foundation. Perrine has shown that improper use of single-phase build-up analysis in certain multiphase flow situations can lead to gross errors in estimated static formation pressure, permeability and well condition. It is likely that much pressure build-up data for Oil wells should be analyzed on the basis of multiphase flow. For both single-phase and multiphase build-up analysis, the isothermal compressibility term in dimensionless time groups often should be interpreted as the total system compressibility. All real reservoirs contain one or more compressible fluid phases. In addition, rock compressibility can contribute in an important way to the total system compressibility. The proper total system compressibility expression may contain terms for compressibility of oil, gas, water, reservoir rock and terms for the change of solubility of gas in liquid phases. JPT P. 447ˆ