Radial Flow Equation Research Articles

Radius Of Investigation For Well Tests In Dual Porosity Reserviors

Abstract The reservoir parameters obtained from any transient pressure analysis reflect average values of the characteristics of the area around the well that has experienced a pressure disturbance due to the change of flow rate at the wellbore. This area is described by an associated radius called the radius of investigation. This paper describes a new equation for evaluation of the radius of investigation for well tests in reservoirs under pseudo-steady state interporosity flow regime. The results have shown that the radius of investigation in such reservoirs starts increasing in the fracture network proportional to the square root of the fracture conductivity. When the matrix contribution to the flow of fluids starts, the rate of advance of the radius decreases until its magnitude reaches a maximum value and remains constant until the total system stabilizes. After this time, the radius increases again at a lower rate dependent now on the total system conductivity. Introduction The radius of investigation, also called the radius of influence or radius of drainage, is defined in many ways by several authors(l,2,3,4,5,6). In most definitions this radius defines a circular system with a pseudo-steady state pressure distribution around the wellbore, and takes the form as follows: Equation 1 (available in full paper) where A is a constant and rinv is the radius of investigation. If the start of semi-steady state flow for a homogeneous and symmetrical bounded cylindrical reservoir at a time tDc of 0.3 is used, and the parameters are defined in oil field units where rinv is in feet, t is the duration of flow for a drawdown test or the duration of shut-in when Δt <p for a buildup test in hr., k is the formation permeability in md, Φ is the reservoir porosity in fraction, and ct is the total system compressibility in psi 1, the constant A becomes 0.029. Odeh and Nabor(7), by using an RC (Resistance/Capacitance) analyser, obtained A to be 0.0257, and Kazemi(8), from the numerical finite difference solution, obtained it to be 0.035. Hurst et al.(3), Van Poolen(5) and Slider(9) separately used the concept of unsteady state radial flow to fmd out when to switch from infinite acting solution to finite solution of the homogeneous diffusivity equation. By taking the derivative of the difference between the above solutions with respect to time and putting it equal to zero, the following equation for radius of investigation is obtained: Equation 2 (available in full paper) Matthews and Russell(l0) picked a time tDe of 0.25 intermediate to the two times corresponding to the end of infinite acting and the start of semi-steady state, and obtained the same Equation (2). Muskat(1), Chatas(11) and Craft and Hawkins(12), by equating the volume of fluid produced to the expansion of the fluid contained in the drainage area, and by considering steady state conditions, also obtained the same Equation (2) for rinv. Craft and Hawkins also showed that if the semi-steady state radial flow equation is used to calculate the average reservoir pressure, the same Equation (2) is obtained except that the constant is 0.0459 instead of 0.0325.

Dispersion in consolidated sandstone with radial flow

This paper presents some experimental and theoretical results for dispersion processes occurring in consolidated Berea sandstone with radial flow geometry. A comprehensive review of the derivation and application of several analytical solutions is also presented. The Galerkin finite element method is applied to solve the advection-dispersion equation for unidimensional radial flow.

Well Stimulation Using Acids

Abstract The fundamental aim of well stimulation treatments is to enhance the revenues from oil and gas wells by improving their productivity or, in the case of injection wells, their injectivity, This can be accomplished by removing or bypassing any damage or impairment from around the wellbore, or by creating conductive fractures, emanating from the wellbore, which enhance the formation's ability to flow. Although perhaps a simplification, stimulation treatments can be divided into those using acids and those employing proppants such as sand, sintered bauxite, or ceramics. This article will concentrate on stimulation of well performance using acids. Acid treatments can generally be divided into acid washes, matrix treatments and acid fracs. However before reviewing these techniques, the relevant factors affecting the productivity (or injectivity) of an oil gas (or injection) well need to be understood. The productivity index or PI for oil and gas wells can be expressed by: Equation (1) (Available in full paper) Equation (2) (Available in full paper) Where: Q = production rate re = drainage radius p = average reservoir pressure rw = wellbore radius pwf = flowing bottomhole pressure S = skin factor k = permeability Z = compressibility factor for gas u = viscosity h = formation height Bo = oil formation volume factor CI, C2 = constants for units T = temperature In damage removal/bypass treatments, in which acids are generally used, we increase well productivity by decreasing the parameter (S) - the skin factor. In fracturing treatments, we can increase productivity by increasing the wellbore radius (rw) to some fictitious radius called the effective wellbore radius (rw). FIGURE 1: Effect of skin. (Available in full paper) The skin factor (S) is a term which reflects a zone of altered permeability around a wellbore. Invasion by drilling fluids, cement filtrate and completion fluids together with perforation damage can effectively decrease the permeability of the formation rock near the wellbore. This decrease in permeability results in a pressure drop in excess of that predicted by the radial flow (diffusivity) equation. Thus in the damaged case, the additional pressure drop is accounted for by a 50-called positive skin factor. On the other hand, stimulation techniques such as acidizing will normally increase permeability in the near well bore region, Thus in the stimulated case, we should see less pressure drop than predicted by the radial flow equation. In the stimulated case, a negative skin is sought. These concepts are illustrated in Figure 1. The degree of damage, or lack of it, can be determined by analyzing pressure build-up (PBU) data and deriving a skin factor. If the skin factor is positive and attributable to damage mechanisms, as against turbulence or other limited entry effects stimulation treatments such as acidizing may be required. Increase in Effective Wellbore Radius Fracturing is often used to increase the productivity of low permeability formations. The effect of fracturing can be interpreted as an enlargement of the wellbore radius (rw) to some imaginary radius called the effective wellbore radius (rw).

Implicit flux limiting schemes for petroleum reservoir simulation

Explicit total variation diminishing (TVD) numerical methods have been used in the past to give convergent, high order accurate solutions to hyperbolic conservation equations, such as those governing flow in oil reservoirs. To ensure stability there is a restriction on the size of time step that can be used. Many petroleum reservoir simulation problems have regions of fast flow away from sharp fronts, which means that this time step limitation makes explicit schemes less efficient than the best implicit methods. This work extends the theory of TVD schemes to both fully implicit and partially implicit methods. We use our theoretical results to construct schemes which are stable even for very large time steps. We show how to construct an adaptively implicit scheme which is nearly fully implicit in regions of fast flow, but which may be explicit at sharp fronts which are moving more slowly. In general these schemes are only first-order accurate in time overall, but locally may achieve second-order time accuracy. Results, presented for a one-dimensional Buckley-Leverett problem, demonstrate that these methods are more accurate than conventional implicit algorithms and more efficient than fully explicit methods, for which smaller time steps must be used. The theory is also extended to embrace mixed hyperbolic/parabolic (black oil) systems and example solutions to a radial flow equation are presented. In this case the time step is not limited by the high flow speeds at a small radius, as would be the case for an explicit solution. Moreover, the shock front is resolved more sharply than for a fully implicit method.

Effects of the Quadratic Gradient Term in Steady-State and Semisteady-State Solutions for Reservoir Pressure

Summary Constant-rate analytical solutions of the one-phase radial-flow equation in two dimensions, including effects of the quadratic gradient term, are derived for an oil reservoir with constant diffusivity and compressibility. The combinations of compressibility contributions to the various terms are analyzed. It is shown that the standard condition allowing the quadratic gradient term to be neglected (cp≪1) is incorrect; pressure, p, should be replaced by a function of the production rate and other reservoir parameters except absolute pressure. The effect of the quadratic gradient term, for which quantitative expressions are given for steady-state and semisteady-state flow, may amount to several percent of the pressure drawdown at the wellbore.

Using Finite-System Buildup Analysis To Investigate Fractured, Vugular, Stimulated, and Horizontal Wells (includes associated papers 22059 and 22060)

The rectangular hyperbolic method (RHM) is a steady-state, finite-system concept that has proved to be amazingly accurate in determining final static reservoir pressure at shut-in. The method also calculates slope, skin, percentage of open reservoir thickness contributing to flow, effective drainage radius, and permeability. It apparently also gives reasonable answers in horizontal-well buildups. It has been found that all wellbore reservoir pressure buildups after al wellbore and storage effects have died out follow rectangular hyperbolic curves. The analysis is based solely on Darcy's steady-state radial-flow equation, the basic transient-pressure-buildup equation, and the determination of effective drainage radius from t/sub m//..mu.. in an empirical equation based on field data obtained from the literature. The RHM is not related to any method based on an infinite-system concept.

Flow and Containment of Injected Wastes

AbstractProper design, construction, testing and maintenance of Class 1 (hazardous waste) injection wells can guarantee that all waste is delivered to the injection zone. To assess the effects of waste injection, analytical models were developed which predict waste movement and pressure increases within the injection zone, and describe upward permeation through confining layers.A basic plume model was used to track waste from several injection wells with varied injection history at DuPont's Victoria Texas site. To determine the maxi‐mum distance that any portion of the waste might travel, special purpose models were employed to account for (1) density differences between the waste and the native formation brine, and (2) layered permeability variation within the injection zone. The results were generalized to a “multiplying factor concept,” which facilitates development of a worst‐case scenario.A pressure distribution model based on the Theis (1935) equation for radial flow was applied to the Victoria site, with modifications to account for multiple wells, injection history and geological complexities.Permeation into an intact confining layer was investigated by a new technique based on the Hantush and Jacob (1955) “leaky aquifer” theory. The model defines the maximum permeation distance, taking into account post‐injection pressure decay.Defects within confining layers, such as faults, fractures and abandoned wells, have been considered. Studies to evaluate their detailed characteristics are continuing. Initial results indicate that faults and fractures are not likely to provide conductive pathways in Gulf Coast settings, and site‐specific evaluations are required to assess the impact of abandoned wells.

Exact solution for the compressible flow equations through a medium with triple-porosity

This paper obtains the exact solution for the unsteady radial flow equations of the slightly compressible liquid through a medium with triple-porosity by using the method of decomposition. This solution not only reveals the essential characters of the unsteady flow of liquid through a medium with multiple-porosity, but also comprises the existing primal results.

Design of Micellar/Polymer System for a Wilmington Low-Gravity Oil

Summary This paper describes the design of a micellar/polymer system to displace the heavy crude from a Wilmington Field, California, reservoir. Pattern design, transient testing, and the injectivity test are discussed also. The design problem was complicated by the unconsolidated nature of the reservoir rock. Initial design work was performed in sandpacks with available sand samples. After confirmation that a viable micellar/polymer solution could be designed, fresh cores were obtained. These cores were frozen at the wellsite and thawed only after placement in a laboratory core holder. System optimization work then was conducted. This work was complicated by some unusual problems. Introduction The HXa subzone of the Upper Terminal of the Wilmington field produces a 35-cp (0.035-Pa.s) oil with an API gravity of 18 degrees (0.95 g/cm3). This is considered the heaviest oil for which a micellar/polymer process has been designed. A good system design requires extensive laboratory core flood testing using actual reservoir fluids and actual reservoir rock. The latter requirement was complicated by the fact that the upper terminal sand is unconsolidated.This paper describes problems and procedures used to overcome those problems in preparation for a test [approximately 10 acres (40 500 m2)] of Marathon Oil Co.'s Maraflood process in this field. The City of Long Beach operates this property with Long Beach Oil Development Co. as its sub-contractor. This project is cosponsored by the U.S. DOE. More details may be found in the first and second annual reports published by the U.S. DOE on this project and the pending third annual report. Core Flooding Procedure The core floods were conducted with a radial core flood apparatus designed at Marathon's Research Center in which actual reservoir core material was used. Fig. 1 shows how a reservoir core disk is flooded. The core holder was adapted to take a 5 1/2 in. (14-cm) diameter core. Flooding was from the center to the outer perimeter. Overburden pressures were maintained by axial loading. The Phase A core floods used available unconsolidated upper terminal and Ranger sand samples. The samples were (1) cold cleaned with a series of solvents, (2) vacuum dried and (3) packed into the core holder. The sand was packed in synthetic produced water and then dried. It then was (1) saturated with the synthetic produced water, (2) flooded with reservoir oil to residual water and (3) flooded with water to residual oil. At this point, the micellar/polymer fluids were injected. The core floods were conducted in a constant-temperature bath maintained at the mean reservoir temperature of 125 degrees F (52 degrees C). Overburden pressure of 1,400 psi (9653 kPa) was maintained. Core pore volume and water permeability were determined during the water saturation phase of the operation.A constant-rate nonpulsating positive displacement pump was used to displace fluid. Each core was flooded with 1.5 PV of fluid from the center to the outer perimeter. Pressures were measured at the injection point and at the two locations across the top of the core as shown in Fig. 1. Produced fluids were collected and measured to determine oil recovery.Pressure data gave indication of the mobility control achieved in the core floods. The ring between the wellbore and the first tap includes 7.7% of the core pore volume. The first and second taps encompass the pore volume from 7.7% to 32.5%. The second tap to perimeter encompasses a pore volume from 32.5 to 100%. A reciprocal relative mobility was obtained by rearrangement of the Darcy radial flow equation for radial conditions. JPT P. 1606^

Utilizing Log-derived Permeability to Predict Rates of Gas Production

Introduction Log data have been utilized to identify hydrocarbon-bearing reservoirs and identify hydrocarbon types. To date, limited use has been made of this data to evaluate how a well will actually perform in terms of flow rates. Log-derived permeability and standard reservoir fluid flow equations can be used to calculate absolute open flow for a natural flowing well. From this, various rates can be established for different draw-down pressures. The various rates can then be used to monitor completion efficiency. The technique has been tested in the Morrow Springer, a gas sand in Oklahoma and the Texas Panhandle. Test results indicate this to be a useful method for predicting the rate at which a well will produce. Introduction Reservoir modelling and computer processing of well logs can be used to predict the absolute flow potential of natural gas reservoirs. The standard induction, density and neutron suite and computer processing can give flow rate potentials and volumetric reserves in addition to the usual porosity, water saturation and correlation. This is an important step in totally utilizing the information gained from well logs. The technique presented here is to calculate permeability from open-hole log data and combine it with physical reservoir parameters and Darcy's radial flow equation to predict a well's performance. This is done in much the same way that a reservoir engineer would take data from a four-point test and back calculate the average permeability. This paper will suggest a more useful sequence, beginning with permeability and working to a synthetic four-point test. The ability to predict a well's potential has several advantages. A standard is set to judge completion efficiencies an economic evaluation of the well can be done before or during the pipe-setting operation, and the date of payout can be more closely estimated. In addition, knowing the magnitude of the well potential, an operator can order equipment of the proper size, thus saving waiting time and getting the well on line as soon as possible. Calculation of flow potential allows an operator to monitor his well during the critical completion stages and to detect abnormalities before further work is done. This ensures the best possible completion and the most gas the well can deliver. Computer integration makes it possible to determine accurately the downhole hydrocarbon type and volume. Again, by using accepted engineering practices, the actual recoverable gas reserves can be obtained with very little error, save for the true drainage radius. Reserve estimates done by a computer take the human bias out of the calculation and give the reserves credibility and repeatability. Any interpretation is very dependent upon the model or models used. By providing a direct link between log data and well performance, an erroneous interpretation model can be detected and corrected rapidly, thus making the information more useful to everyone concerned. Theory Darcy's equation for radial. steady-state flow of a compressible fluid in the region of laminar flow is used to describe the well's behaviour in the text of this paper.

A Finite Element Calculation for Determining Thermal Conductivity

AbstractA new simple method for in situ measurement of thermal conductivity of porous materials was developed. A heated or cooled Al or glass probe was placed in soil, and its temperature was monitored over time. Thermal conductivity was determined using a finite element method to solve the radial heat flow equation for the temperature of the heated or cooled probe. The probe temperature, as a function of time, was modeled, and model thermal conductivity was adjusted until predicted probe temperatures agreed with measured values. Conductivities for sand, silt loam, and forest litter were measured over a range of water contents. Values obtained using a glass probe were not significantly different from those obtained using an Al probe. Conductivities were in good agreement with those obtained using a water bath method or computed from the deVries equation. There were significant differences between values obtained with heated and cooled probes in sand at intermediate water content.

An Analytical Solution to Solute Transport Near Root Surfaces for Low Initial Concentration: I. Equations Development

AbstractIn Part I of this two part paper the governing differential equation for radial flow to a root with constant moisture properties, is transformed into a nondimensional and more useful form. An analytical solution to the differential equation with two appropriate sets of boundary conditions is developed. The solution can be expressed as an infinite series of Bessel functions, powers of nondimensional distance and an exponential.Equations for total and diffusive nutrient uptake are developed for both growing and nongrowing roots. Nondimensional equations are also presented for the various components of the nutrient flux (diffusive, convective, and total).The equation for total nutrient uptake for a root growing at a rate f(t) such that f(o) = Lo (initial length), is examined in detail when f(t) is an exponential.

Flow to a Nonpenetrating Well in a Leaky Artesian Aquifer with Variable Dischargea

ABSTRACTA differential equation for axisymmetric radial transient flow to a nonpenetrating well in an infinite leaky aquifer in spherical coordinate system has been formulated. A general solution for nonsteady‐state flow was obtained for nonpenetrating wells in leaky artesian aquifers with a general time variable discharge function, as well as for the specific case of exponential decay discharge function. The solution is expressed in terms of standard mathematical functions.

Steady Flow Equations for Wells in Partial Water-Drive Reservoirs

This paper presents steady-state radial flow and pressure distribution equations for wells under partial water drive or fluid injection. A parameter f is introduced to characterize the strength of water drive or fluid injection and to show its influence on well productivity. Introduction No convenient method exists in petroleum literature to establish the well producing rate and pressure-distribution behavior for wells in partial water drive reservoirs and in fluid injection projects under steady conditions. Such behavior is understood for wells in closed or depletiontype circular reservoirs and in reservoirs with constant pressure at the outer boundary. The latter situation is encountered in an active water drive reservoir or in fully developed five-spot injection patterns. For most wells, a realistic condition at the reservoir drainage boundary is one of partial flow because of limited aquifer or fluid injection in the aquifer or both. Brownscombe and Collins presented a producing rate equation for a well at steady flow in a closed reservoir. The corresponding flow equation for constant pressure is well known. The derivation of those equations is presented by Craft and Hawkins. Dietz suggested a method presented by Craft and Hawkins. Dietz suggested a method of incorporating the effect of partial water drive on drainage area mean pressure of a well by changing shape factor values. Steady radial flow equations for a well in a circular partial water drive reservoir are developed in the partial water drive reservoir are developed in the following discussion. Theory Consider a well producing at a constant rate from a circular oil reservoir. The formation is assumed to be uniform in thickness, porosity, and permeability. The fluid has a constant viscosity and compressibility at reservoir temperature. After the initial transients have died down, a well reaches steady flow conditions, where the time rate of change of pressure, dp/dt, is constant with time. It is normally termed pseudosteady or quasisteady state for nonzero values of this constant and "true" steady state when the constant is zero. Under true steady state, pressure at a point acquires a constant value with respect to pressure at a point acquires a constant value with respect to time. Under pseudosteady conditions the absolute value of pressure at a point is changing with time, but the time rate of change of pressure is constant. Let us consider the flow rate across a radius r under the influence of the following conditions at the outer reservoir boundary at r : 1. No-Flow at r : In a closed or depletion-type reservoir, there is no flow across the outer reservoir boundary, implying that the pressure gradient (dp/dr) at r is zero. The fluid volume at the well (q) is produced solely by expansion of fluid in a cylindrical shell between the radii r and r . In other words: (1) (2) as rw less than less than re. The term [pi r h phi c (dp/dt)] represents the expansibility of the fluid contained between the radii re and rw . JPT P. 1654

Productivity of Perforated Completions in Formations With or Without Damage

The productivity of perforated completions situated in either homogeneous reservoirs or zones of permeability damage is described. Calculations for both types of reservoirs require use of a perforation skin factor; calculations for a damaged reservoir additionally require a damage skin factor. Nomograms are presented for calculating these factors. Introduction In perforated completions, fluids enter the wellbore through tunnels made by bullets or jets that penetrate the casing, cement sheath, and part of the producing formation. The productivity of such completions has been studied by various investigators, with either computer models or electrolytic analog models. Harris recently published a series of curves for calculating skin factors published a series of curves for calculating skin factors in terms of predetermined dimensionless well and perforation parameters. These skin factors are used in computing parameters. These skin factors are used in computing the productivity ratio (the productivity of a cased and perforated completion vs that of an equivalent open-hole perforated completion vs that of an equivalent open-hole completion). Harris' graphs, however, are difficult to use because they are based on dimensionless parameters that must be predetermined, and they require extensive extrapolation predetermined, and they require extensive extrapolation for well diameters larger than 6 in. In addition, Harris' model covers only the simple, regular patterns, and requires that the perforations in a horizontal plane be directly above or below those in adjacent planes. Application of Harris' model is also restricted by the assumption that the formation permeability has not been damaged by the normal drilling and perforating processes. processes. The study reported here describes the productivity of perforated completions situated in either homogeneous perforated completions situated in either homogeneous reservoirs or zones of permeability damage. Calculations for both types of reservoirs require use of a perforation skin factor; calculations for a damaged reservoir additionally require a damage skin factor. Nomograms are presented for calculating these factors. presented for calculating these factors. Separate nomograms are provided for the two basically different types of perforation patterns: simple and staggered. In the latter, perforations in a horizontal plane can be 90 degrees or 180 degrees out of phase with those in adjacent horizontal planes (Table 1). Three samples illustrate the use of these nomograms for calculating skin factors and productivity ratios. The effects on productivity of perforation parameters such as size, patterns, penetration, and density are shown by graphs relating them to the productivity ratio. Development of Nomograms Well in a Homogeneous Reservoir The usual relationship of flow rate and pressure drop for quasi steady-state flow is(1)7.08 krh (p-pw) qr = ln (0.47 re/rw) which is for one-dimensional radial flow into an open hole. When flow is into perforations that penetrate the formation, the flow geometry takes on three-dimensional aspects. The difference in pressure drop between convergent flow to perforations and radial now to an open hole is the basis for the perforation skin factor, Sp. This factor is used in a similar radial flow equation,(2)7.08 krh(p-pw) qr = [sp+ ln (0.47 re/rw)] JPT P. 1027

The Solution of Nonlinear Equations

Abstract This paper presents procedures for treating problems involving transient flow of gases in porous problems involving transient flow of gases in porous media. The methods involve stepwise calculations using linearized equations derived from the nonlinear, second-order equations that describe transient flow of gas. The distribution of pressure around a gas well at various times can be readily calculated with a desk calculator or a small computer. Equations and procedures are offered for both infinite and limited reservoirs. Solutions by these new technique's are shown to be in good agreement with computer solutions available in the literature. Also discussed is a procedure using relatively few image wells for treating problems in reservoirs with curved, irregular boundaries. Introduction This paper is concerned with solving problems involving transient flow of gas in porous systems. The main contribution of this paper is the development of a technique that permits the effective use of a desk calculator for the computations. The methods presented here will permit individual engineers not having access to one of the larger computers to solve many practical problems that heretofore would have been intractable. problems that heretofore would have been intractable. The equation describing unsteady-state flow of a perfect gas in a horizontal reservoir is reported by perfect gas in a horizontal reservoir is reported by Katz to be (1) This equation is obtained by combining the continuity equation, Darcy's law, and the following density relationship for the gas. =...............................(2) The intractableness of Eq. 1 stems from the fact that it is a nonlinear, second-order differential equation for which no analytical solution is known. In 1953, a pioneering use of computers was presented by Bruce et al., who approximated the presented by Bruce et al., who approximated the differential equation with difference equations and solved these numerically. Their solutions developed for horizontal flow of a perfect gas for circular reservoirs were presented in terms of dimensionless parameters in plots of pressure vs radius for various parameters in plots of pressure vs radius for various times and flow rates. Later, Aronofsky and Porter, using computers, solved radial flow problems for nonideal gases, permitting gas properties to vary as linear functions of pressure. Recently, advances in solving problems involving the flow of nonideal gas have been made by Ramey and his colleagues, who have developed equations and computer solutions for real gases in terms of pseudo-reduced pressures. In the present paper, a method of solving the flow equations for gas is developed for both infinite and limited reservoirs. The methods, which make use of the analytical solutions for the corresponding linear differential equation for radial flow, can be used with a desk calculator or a small computer to solve problems characterized by nonradial as well as by radial reservoir geometry. An example solution is given to illustrate the method for the flow of a perfect gas in a circular, horizontal reservoir; the results have been compared with those of Bruce. The technique can also be applied to the flow of nonideal gases in nonuniform systems, as well as to oil reservoirs above the bubble-point pressure and to aquifers. In all cases, the results of the example problems obtained by the procedures presented in the text are compared with procedures presented in the text are compared with available published and accepted numerical or analytical results. Finally, a discussion is offered with the intent of guiding the reader toward successful application of the technique in solving practical reservoir problems.

Some Effects of Contaminants on Real Gas Flow

In gas well testing and deliverability forecasting, failure to make corrections for contaminants can lead to intolerable errors. In general, where impurities exceed ten percent by volume, corrected values of the real gas pseudopressure function should be used. Introduction Isothermal gas flow through porous media has been the object of considerable study - the work, in general, deals with the flow of ideal gases and is therefore not generally applicable to gas reservoirs having low permeabilities or high reservoir pressures. This is because ideal gas flow treatment implicity assumes that pressure gradients in the system are small. Furthermore, the pressure dependence of gas viscosity and the gas law deviation factor are usually neglected. Recently, Al-Hussainy et al. presented a theory for real gas flow. Unlike previous work, theirs demonstrated that the pressure dependence of viscosity and the gas law deviation factor could be taken into account by expressing the gas flow equation in terms of a pseudopressure function. A nonlinear flow equation similar to the diffusivity equation was developed that did not require the assumption of small pressure gradients. In addition, a table of the real gas pseudopressure function was presented for gases containing pseudopressure function was presented for gases containing small amounts of contaminants. In a companion paper Al-Hussainy and Ramey showed how the real paper Al-Hussainy and Ramey showed how the real gas flow theory could be applied to standard gas well tests. Again, this study was limited to gases containing small amounts of contaminants. The purpose of our paper is to investigate some of the effects of gas contamination on real gas flow. Particular attention is devoted to constructing tables of Particular attention is devoted to constructing tables of the real gas pseudopressure function corrected for various amounts of nitrogen, carbon dioxide and hydrogen sulfide. Comparisons are made with the uncorrected results of Al-Hussainy et al. Finally, several example calculations are presented for some typical well tests, showing the errors that can occur in failing to make adequate corrections for gas contaminants. Real Gas Flow Equations For completeness, some of the development of Al-Hussainy and Ramey is included here. Principal assumptions are:Flow is isothermal and laminar.Gas composition is constant.The gas law is pV = znRT.The reservoir is homogeneous and isotropic.The formation has constant thickness and no dip. The equation for radial flow of a real gas is (1) Eq. 1 can be expressed in terms of the real gas pseudopressure function defined by pseudopressure function defined by (2) which yields (3) where cg(p)is the isothermal compressibility of gas given by (4) The pressure Pm is an arbitrary base pressure. JPT P. 1157

Transient Flow Characteristics of Low Permeability Gas Reservoirs and Improvement of Deliverability by Nuclear Explosion

Abstract A program for computing the transient radial flow of natural gas, taking into account significant variable properties of the fluid and of the porous medium was used in an investigation of flow characteristics of formations for which the permeability-thickness product kh is 1 to 100 md-ft. A graphic summary of the results obtained may be used with well flow-test data to estimate kh for the formation in which the well is completed. Assuming that a nuclear explosion in a gas-containing formation would create a rubble-filled chimney with a diameter of 170 ft and fissures radiating from the chimney wall out to a distance of 255 ft, formation pressure gradients were computed for the recovery of gas with a well drilled into the chimney. These computations for 640-and 160-acre spacing indicate that diminution of the radius segment re - rf over which flow occurs in unaltered formation provides for markedly greater ultimate recovery of gas in place than is possible with conventional well completions. A formation drilled with conventional wells 1.0 mile apart must have a productivity product of kh = 197 md-ft to deliver for 20 years 1.0 MMscf/D of gas into a pipeline operated at 300 psi. Aided by nuclear stimulation, one well can meet the same performance requirements with a kh of only 49 md-ft for the native formation. With a spacing of 160 acres and nuclear stimulation, kh need be only 9 md-ft. Introduction Project Gasbuggy of the El Paso Natural Gas Co. and government agencies is concerned with the creation by nuclear explosion of a chimney and radiating fissures in a low-permeability formation containing natural gas. A well is to be drilled and completed in this chimney and tests will be made on this well and other wells in the vicinity to determine the economic feasibility of nuclear stimulation for recovery of the gas. Engineering studies of the Bureau of Mines pertaining to parts of this project have been in progress since 1963. Because of the complexity of the experiment, its cost, and possible economic significance, it is desirable to define the reservoir flow problem, determine by computing the relative importance of the parameters involved, and obtain a measure of the benefits that may be expected. The results should be generalized for application to other projects. Two parts of this problem have been investigated: - determination of average properties of the native formation from flow tests of wells, and - evaluation of the transient flow performance of a low-permeability formation in which a relatively high-permeability area has been created to receive a recovery well. BASIC EQUATIONS The partial differential equation for transient radial flow of a gas phase, (1) was integrated in the range rd r re. With rd/re = 0.005 and 640-acre spacing rd = 14.9 ft, so that flow over most of the radius is computed by means of Eq. 1. A program previously described by the authors was used for this purpose.

Physical Changes of Reservoir Properties Caused By Subsidence and Repressuring Operations

Abstract The surface of the Wilmington oil field has now subsided as much as 29 ft at the center of an elongated depression roughly coinciding with the field productive area. To determine the compacting intervals, a method was developed to detect changes in length of individual casing joints. Both reservoir estimates and casing joint measurements indicated a pore volume loss of at least 3 porosity percent. Casing joint measurements, also were used to defect casing elongation in zones of water injection. The vertical expansion of the reservoir is also seen on the surface as uplift that now amounts to, as much as 8 or 9 in. in the areas of heaviest injection. Rate-pressure data from injection wells in Wilmington plot as a transitional curve on coordinate paper rather than showing a sharp change of slope as is normally seen when overburden pressure is exceeded. Engineers working in the field and familiar with the unconsolidated nature of the formations could not reconcile the rate-pressure performance with formation fracturing. By interrelating the zonal expansion and surface uplift with the rate-pressure curves and the radial flow equation, it was concluded that the formations experience a change in pore volume and permeability as pressures are increased. Introduction Wilmington field is located near the southwestern edge of the Los Angeles sedimentary basin. The structure forms a major part of the anticlinal trend that extends about 20 miles from Torrance field to the Huntington Beach offshore pool (Fig. 1). Wilmington produces from seven zones of Pliocene and Miocene age, spanning depths from 2,000 to 6,000 ft. The upper four zones, extending to a depth of 4,000 ft, are the intervals of primary interest (Fig. 2). They consist of arkosic sands, siltstones and shales with sand percentages varying from 25 to 65. The sands are unconsolidated and usually contain varying amounts of interspersed shale and silt material. Porosities range from 33 to 37 percent and permeabilities average 500 to 2,000 md in the various zones. The problem of surface subsidence over Wilmington field has been well publicized and has attracted widespread interest. The surface now has subsided as much as 29 ft at the center of an elongated depression roughly coinciding with the field productive area (Fig. 3). Much of the field is overlain by valuable industrial property in the Long Beach harbor district and by the City Civic Center. Surface subsidence in these areas caused millions of dollars of damage and posed a threat of inundation to the Long Beach Naval Shipyard, the harbor and the civic center. Concurrent with the construction of dikes and sea walls, the city and the other operators and landholders in the area of subsidence launched a program to determine the causes and possible remedies. The result is the largest waterflooding program in the world. Current field injection is about 800,000 B/D with ultimate plans calling for 1.5 million B/D. With this program in effect, subsidence now has been either stopped or reduced to about 1 in./year. In the areas of maximum repressuring the surface has regained some of the former loss of elevation. Problems Related to Reservoir Changes During both early field development and later waterflooding stages, it was evident that not only were surface elevations affected, but also physical changes were occurring in the reservoir. Changes of reservoir characteristics were indicated by measurements of subsurface pressures, plots of the relative permeabilities of gas and oil, and water injection rates vs injection pressure. During early development, when the surface was subsiding rapidly, many studies were conducted to determine the reservoir mechanism of compaction. Most observers studying the problem related subsidence to compaction in the oil zones-compaction caused by a pressure reduction concurrent with the withdrawal of reservoir fluids.

Determination of Unconfined Aquifer Characteristics

The various methods that have been proposed to solve the nonlinear differential equation for unsteady radial flow toward a well in an unconfined aquifer are divided into three categories. Each category is characterized by a particular assumption made on the h multiplier of the equation. A comparison is made as to the origin, characteristic assumption, and applicability in solving for the hydraulic conductivity and specific yield of an unconfined aquifer. The methods of Category I linearize the nonlinear differential equation and thus should be utilized for drawdown ratios of approximately 5%, or less. Adjustment of the field data or the confined aquifer type curve is characteristic of methods described in Category II. These methods should be restricted to drawdown ratios of 25%, or less. The reduction in flow thickness in the aquifer is directly taken into account by the method in Category III. Therefore, this method applies to all drawdown ratios of less than approximately 75%.