Drainage Boundary Research Articles

Mathematical model for one‐dimensional desiccation and consolidation of sedimented soils

AbstractA mathematical model has been developed to represent the physical phenomena that occur during the desiccation and one‐dimensional consolidation of successive layers of hydraulically transported sediments as they are periodically deposited in a containment area. The governing boundary value problem, defined in terms of pore water pressures, consists of two field equations (one for the saturated domain and one for the unsaturated domain), a drainage boundary condition, an evapotranspiration boundary condition and a series of continuity conditions at the interfaces between different layers. A number of simplifying assumptions were made to render the field equations tractable, and a step‐by‐step numerical procedure was used to solve the linearized boundary value problem; at the end of each step, the errors introduced by the simplifying assumptions were corrected. Based on a thorough study of the convergence and stability conditions associated with the numerical approximation employed, a system of automatic corrections was incorporated into the computer program to reduce the time increment if stability problems originate during the solution. Based on the results of a parameter study, it was found that, although drainage conditions at the bottom of the, layer do exert some influence on the consolidation rate during deposition of the dredgings, the evapotranspiration potential renders this effect almost negligible when desiccation takes place at the surface. In contrast, transpiration plays an important, role on the consolidation rate during the early stages of desiccation, but its effect is reduced considerably as the water table approaches an equilibrium position.

Use of satellite infrared data for geomorphology studies

Enhanced nighttime thermal infrared imagery and digital data from a NOAA polar orbiting satellite were used to map drainage patterns and landforms in North and South Dakota. Features delineated include the Missouri Escarpment from Saskatchewan to the Nebraska border, the Manitoba Escarpment, Coteau des Prairies, recessional moraines on the Coteau du Missouri and partial drainage boundaries for the following rivers: James, Big Sioux, Minnesota, Red of the North, Souris, and the main stem of the Missouri plus its western tributaries in the Dakotas. In several instances drainage from gentle slopes in the Midwest was discerned and correlated with local relief. Analyses of satellite digital thermal data for western tributaries of the Missouri River revealed north-facing slopes to be warmer than south-facing slopes by an average of 1.5°C. Comparisons of ground and satellite temperatures for 11 stations in South Dakota showed good agreement.

Experimental evaluation of theoretical solutions for subsurface drainage and irrigation

The Richards equation for two‐dimensional saturated‐unsaturated water movement was solved for drainage and subirrigation boundary conditions by using finite difference methods. The solutions were compared to results of experiments conducted for various initial and boundary conditions in a large laboratory soil tank. The soil water characteristic and hydraulic conductivity function were measured by conventional methods and used as inputs in the numerical solution to the Richards equation. Although predicted water table drawdown was somewhat slower than was observed, agreement between the numerical solutions and experimental observations was generally good. Approximate methods were also used to characterize drainage and subirrigation and were found to be in good agreement with solutions to the more exact Richards equation and with experimental results when a variable drainable porosity is considered. In view of significant field variability of soil properties and the difficulty of measuring those properties, required by the Richards equation, it appears that simpler, approximate methods will continue to be more useful than the exact approach for predicting water table movement under drainage and subirrigation conditions.

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

Strength of Water Drive or Fluid Injection From Transient Well Test Data

This paper presents a method for determining the strength of water drive or fluid injection from transient well test data. Graphs are presented for interpreting drawdown and buildup data and for determining drainage-area mean pressure and drainage boundary pressures. Introduction Three possibilities exist for time variation of pressure at the drainage boundary of a well or reservoir:pressure decreases with time,pressure increases with time, andpressure remains constant with time. In closed or depletion reservoirs and reservoirs with water drive and partial fluid injection, the pressure at the drainage boundary declines with producing time. Partial fluid injection occurs when fluid volume introduced into a reservoir system from outside, such as in aquifer injection, is less than the volume of fluids withdrawn from a system. In contrast, the reservoir boundary pressure increases with time in reservoirs with excess fluid injection. The boundary pressure remains constant with time under full injection, which requires that the fluid volumes entering a system be equal to fluid volumes withdrawn, such as occurs in a balanced five-spot fluid injection system. Strictly speaking, a mass balance is required instead of a volume balance as proposed above. However, this approximation is used commonly when using reservoir material-balance equations. It has been demonstrated recently that well test behavior under a constant pressure condition at the reservoir boundary is quite distinct from that under usual depletion or closed reservoir boundary systems. In practice, one seldom encounters either a truly closed or a truly constant pressure boundary condition, but finds that most reservoirs fall somewhere between these two conditions; that is, pressure and fluid influx is changing with time at the reservoir or well drainage boundaries. Several important questions are of concern to a practicing reservoir engineer: Is there a way to estimate the strength of water drive early in the life of a reservoir? How effective is a given fluid injection program for pressure maintenance? What kind of well drawdown and buildup pressure behavior would be obtained under these conditions? This paper aims to answer these questions for wells under water drive and under partial, full, or excess fluid injection programs. To characterize the well pressure behavior in water drive reservoirs Miller et al. (MDH) considered the idealized case of a well in the center of a finite circular reservoir with constant pressure at the drainage boundary. Perrine reviewed the work of MDH and presented a curve to estimate the mean pressure that, in fact, yields the initial pressure in such systems. Hazebroek et al. studied pressure falloff in water injection wells located in isolated five-spot patterns. Dietz proposed a method to incorporate the strength of water drive in mean pressure determination by changing shape-factor values. Earlougher et al. used an infinite-array superposition scheme to show the influence of the constant pressure condition on the MDH-type buildup graph. Ramey et al. and Kumar and Ramey used this superposition scheme to determine the drawdown and buildup behavior for a well in a constant-pressure square. Well test behavior for wells in closed reservoirs of almost any geometrical shape has been studied in detail and is summarized by Matthews and Russell. JPT P. 1497^

Closure to “Exit of Ground Waters on Drainage Boundaries”

Closure to “Exit of Ground Waters on Drainage Boundaries”

Discussion of “Exit of Ground Waters on Drainage Boundaries”

Discussion of “Exit of Ground Waters on Drainage Boundaries”

Reservoir Limit Tests in a Naturally Fractured Reservoir - A Field Case Study Using Type Curves

Pressure buildup, interference, and pulse tests in a naturally fractured Pressure buildup, interference, and pulse tests in a naturally fractured dry gas reservoir are influenced by reservoir limits. Type curves are matched to test data to estimate drainage area and to compute porosity and permeability. Calculated porosity and permeability values compare well permeability. Calculated porosity and permeability values compare well with published data for natural fracture systems. Introduction The case studied is a dry gas reservoir in which three wells are completed. The wells are spaced 2 and 8 miles apart in a 10-mile line along the crest of an anticline with about 100 sq miles of closure (Fig. 1). The dashed contour in Fig. 1 is the drainage boundary that was initially estimated from geologic and production test data assuming a uniform gas-water contact. This drainage area is about 18 miles long and 3 miles wide. Only one productive stratigraphic unit is common to all three productive stratigraphic unit is common to all three wells. This is a naturally fractured zone of thinly bedded, clean orthoquartzites that accounts for 90 percent of deliverability at Well 1, 95 percent at Well 2, and 100 percent at Well 3. Type of completion, fractured zone thickness, and other reservoir data are presented in Table 1. No cores were taken directly from the naturally fractured orthoquartzite zone, but cores from other orthoquartzites had 2.5-percent average porosity and less. than 0.1-md permeability to air. Test data studied in this field case history have two chronological groupings:data recorded when Well 2 was completed, consisting of one pressure drawdown and four pressure buildup tests at Well 2; anddata obtained 4 years later, consisting of pressure interference at Wells 3 and 1 caused by flowing Well 2 for 450 hours, pressure buildup at Well 2 immediately following the interference test, and pulse response at Well 3 caused by pulsing Well 2. The field was never on production except to conduct pressure transient and production except to conduct pressure transient and deliverability tests. Analysis of the field tests data is organized into four sections:(1)general discussion of the pressure drawdown and buildup behavior in light of recently published well-test theory;(2)computation of porosity and published well-test theory;(2)computation of porosity and estimation of drainage area by matching the buildup data to type curves;(3)computation of porosity, permeability, and drainage area by matching the permeability, and drainage area by matching the interference data to type curves; and(4)analysis of pulse behavior in the presence of reservoir limits. General Pressure-Buildup Behavior Buildup Tests 1 through 4, recorded at completion of Well 2, are presented in Tables 2 through 5. The pressure drawdown corresponding to Buildup Test 4 is also pressure drawdown corresponding to Buildup Test 4 is also shown in Table 5. Fig. 2 is a graph of pressure as a function of the logarithm of time for the drawdown test. All four buildup tests are plotted in Fig. 3, using the technique of Horner. Pressure buildup during Test 1 becomes a linear function of the logarithm of the Horner time ratio, and extrapolates to initial pressure at infinite shut-in time. Each of the other tests plotted in Fig. 3 has an early period in which pressure is a linear function of the period in which pressure is a linear function of the logarithm of the Horner time ratio and a late period in which pressure bends upward. JPT P. 1097

Exit of ground waters on drainage boundaries : BROWZIN, BS EBASCO SERVICES, NEW YORK, USA 4F, 6T, 15R. J. GEOTECH. ENGNG DIV. ASCE, V102, GT2, FEB. 1976, P159–173

Exit of ground waters on drainage boundaries : BROWZIN, BS EBASCO SERVICES, NEW YORK, USA 4F, 6T, 15R. J. GEOTECH. ENGNG DIV. ASCE, V102, GT2, FEB. 1976, P159–173

Exit of Ground Waters on Drainage Boundaries

A known analytical solution provides the location of the exit point of the free surface of ground waters as a function of the slope angle and the reduced discharge in a form of a finite integral. This integral was evaluated numerically and a substitute polynomial equation provided to determine the function. Based on the exact position of the exit point, a new discharge formula free from any preliminary assumptions is derived. It covers the range of slope angles from near 0° to 180° for infinitely long aquifers. It was found that the same formula is applicable to certain finite lengths. Applicability of the function for the location of the exit point and the discharge formula for earth dams is considered. It is explained that a hydraulically equivalent rectangle, which substitutes the upstream prism, must have the width of the base substantially smaller than that used in older theories.

Well-Test Analysis for a Well in a Constant-Pressure Square

Abstract Very little information exists for analyzing well tests wherein a part of the drainage boundary is under pressure support from water influx or fluid injection. An idealization is the behavior of a well in the center of a square whose outer boundary remains at constant pressure. A study of this system indicated important differences from the behavior of a well in a square with a closed outer boundary, the conventional system. At infinite shut-in, the well with a constant-pressure boundary will reach the initial pressure of the system, rather than a mean pressure resulting from depletion. It is possible to compute the mean pressure in the constant-pressure case at any time during shut-in. Interpretative graphs for analyzing drawdown and buildup pressures are presented and discussed. This case is also of interest in analyzing well tests obtained from developed five-spot fluid-injection patterns. Introduction Moore at. first demonstrated the application of transient flow theory to individual well behavior in 1931. Classic studies by Muskat, Elkins, and Arps in the 1930's and 1940's set the stage for two important papers in 1950 that clearly elucidated the basics of modern well-test analysis. One paper by Horner 5 summarized methods for analyzing transient pressure data from wells in infinite reservoirs (new wells in large reservoirs), and a well in a closed, circular reservoir under depletion (fully developed fields). The second paper by Miller, Dyes, and Hutchinsons considered two cases for wells assumed to have produced a long time before shut-in for pressure buildup. One case assumed a closed circular drainage boundary, and the other case assumed a circular drainage boundary at constant pressure. The former would represent annual well tests for fully developed fields, and the latter would represent wells under full water drive in single-well reservoirs. Since 1950, several hundred papers and a monograph have developed the behavior of a constant-rate well in a closed drainage shape of almost any geometry. Key in this development was a classic study by Matthews, Brons, and Hazebroek. The constant-pressure outer-boundary drainage region problem introduced by Miller-Dyes-Hutchinson was reviewed by Perrine in 1955, discussed by Hazekoek el al. in 1958 in connection with five-spot injection patterns, and mentioned briefly by Dietz in 1965. The only other studies dealing with water-drive conditions (constant-pressure outer boundaries) appear in Ref. 7 (Page 44) and in papers by Earlougher et al., published in 1968. It is clear that this case was eitherconsidered totally unimportant, orstudiously avoided. Almost all effort was expended on studying closed outer boundary (depletion) systems.Another problem concerned the conventional assumptions involved in developing well-test analysis method. Even for the common closed (depletion) systems, field applications raised the question of the importance of assumptions. Homer method of graphing assumed the well had been produced a short time, whereas the Miller-Dyes-Hutchinson method assumed that production was long enough to reach pseudosteady state -a long time in many cases. Engineers involved in applications were further confused by differences in methods, as well as by the importance of the assumptions required for analytical solutions that established welltest methods. Recently, Ramey and Cobb showed that an empirical approach could be used to avoid assumptions (which were sufficient but unnecessary) inherent in many previous analytical studies. It was decided to apply this method to the limiting case of a well in a full-water-drive, single-well reservoir - a well in a constant-pressure square. This case is a rarity not often seen in practice. It is closely approached by either an injector or a producer in a developed fluid-injection pattern, by a single injector in an aquifer gas-injection storage test, or by some single-well reservoirs in extensive aquifers.The main point is that a well in a constant-pressure square sets a limiting condition similar to a full water drive. The more common case of a well in a partial-water-drive reservoir should lie between this behavior and that of a closed square. SPEJ P. 107^

One‐dimensional consolidation of a clay layer with impeded drainage boundaries

In many field transient water flow problems of practical interest the main source of settlement is a single compressible layer bounded above and below by layers that contribute little to total settlement but that have coefficients of permeabilities and thicknesses such that they can be considered neither freely draining nor impervious. Furthermore, the top and bottom drainage layers can have unequal permeabilities and thicknesses. An analytical solution to the one‐dimensional consolidation of a clay layer with unlike drainage boundaries is developed, and the results are presented in the form of percent consolidation versus time factor to facilitate their use by engineers.

Pressure Buildup in Reservoir Limit Testing of Stratified Systems

If a reservoir is characterized by a special type of heterogeneity such as stratification with no crossflow, the usual methods of determining reservoir limits are either not applicable or not practical. Proposed here is a way of estimating the drainage radius from pressure buildup data - a method especially useful whenever the drawdown data are not reliable. Introduction The reservoir limit test is a method of determining either the proximity of the boundaries of a reservoir to a flowing well or the size of drainage volume associated with the well. This method makes use of the usual equations for pressure drawdown and was first introduced by Park J. Jones. Other types of flow and pressure buildup tests that yield similar results, or pressure buildup tests that yield similar results, or additional improvements, can also be considered as reservoir limit tests; this is a natural extension of the basic ideas stated by Jones and others. If the reservoir is either homogeneous or heterogeneous with crossflow between various components, it has been shown that almost identical sets of pressure drawdown equations can be used for reservoir pressure drawdown equations can be used for reservoir limit calculations. Examples should be found in stratified reservoirs with crossflow and in some naturally fractured reservoirs. If, on the contrary, there prevail special types of heterogeneity such as stratification with no crossflow, then the usual methods are either not applicable or impractical. This latter type of reservoir is the subject of this work. The basic principles of the reservoir limit calculations can be summarized as follows. 1. During the early transient period of a drawdown or a buildup test, the wellbore pressure is linear with the logarithm of time, and the onset of non-linearity is related to the proximity of the reservoir boundaries. The distance to the nearest boundary can be approximated by Eq. 1 for drawdown and by Eq. 2 for buildup: (1) (2) For an ideal circular reservoir with a centrally located well, Odeh and Nabor reported to be about 0.4, and to be about 0.25. Their work was based on an RC analyzer. Our approach, based on a numerical finite difference solution, yielded a value of 0.214 for, which will be used as a norm in this paper. The value of will be taken as 0.25. In paper. The value of will be taken as 0.25. In fact, no precise value can be assigned to and . Their value depends mostly on the nature of the reservoir and the accuracy of the pressure measurement.2. During the semisteady period the wellbore pressure of a drawdown test is linear with time, and the pressure of a drawdown test is linear with time, and the onset of this linearity is given by Eq. 3: (3) where for an ideal circular reservoir is about 0.1. Nonsymmetrical drainage boundaries give rise to larger values of . For instance, its value is 0.7 for a well in the center of a 4 × 1 rectangular reservoir. For a crossflow system is still about the same as that of its equivalent homogeneous reservoir of the same geometry. For no-crossflow stratified systems is much larger than the aforementioned figures. JPT P. 503

A Mathematical Model of Repeated Steam Soaks of Thick Gravity Drainage Reservoirs

Certain equations have been used to predict the performance and heat requirements of steam soaks applied to gravity drainage reservoirs. Good agreement has been obtained between calculated and observed performance. Introduction The steam soak process is the most widely applied and most successful thermal supplemental recovery process in use today. This process, which consists of injection of steam in various quantities into a well that is subsequently used as a producer, can be repeatedly applied to the same well. The treatment, if successful, will increase the production rate of the well for a given period. For this reason, the steam soak process should be considered primarily as an acceleration method rather than as a method for producing oil that could not be recovered by primary means. However, application of this process to reservoirs containing viscous crude may produce oil that otherwise would have zero present value or would be produced at such low rates by primary methods that it would not be profitable. A major necessity for the successful application of the steam soak process is a source of natural energy. This source of energy may exist in one or more of the following forms: expansion of fluids, compaction, gravity drainage. This study was undertaken to determine the effect of gravity drainage on steam soak production and recovery and to develop a method for predicting production rates, ultimate recovery, and heat requirements when applying this process to a reservoir in which the primary driving mechanism is gravity drainage. (For additional discussion of some of the factors that affect recovery of viscous oils by the steam soak process, see Ref. 5.) Model The model developed for analyzing the performance of the steam soak process in gravity drainage reservoirs is a modified form of that developed by Matthews and Lefkovits. The model used in this study consists of two concentric cylindrical volumes of productive formation with cap and base rock of impermeable material. The inner cylinder, extending from the radius of the wellbore to some predetermined radius, is referred to as the hot zone; and the outer cylinder extending from the exterior boundary of the hot zone to the outer drainage boundary will be referred to as the partially heated zone. Fig. 1 is a schematic cross-section of the model, indicating its thermal properties. Horizontally the reservoir is divided into two regions, the upper region having a high gas saturation and residual liquid saturation, and the lower region containing a high liquid saturation. The boundary between these two regions is referred to as the free surface, which drops and changes shape as fluid is produced from the reservoir. The model was used to analyze the steam soak process by applying the following assumptions: The horizontal formation is composed of one ormore homogeneous noninterconnected layerscontaining incompressible fluids. Each layer is treatedseparately. The hot zone is heated by the initial soak and ismaintained at the steam temperature correspondingto atmospheric pressure by resoaking. The height of the free surface at the producingwell is at all times zero (producing well is pumpedoff). JPT

Extensions of Pressure Build-Up Analysis Methods

Extensions of Pressure Build-Up Analysis Methods

The Effect of Vertical Fractures on Transient Pressure Behavior of Wells

Abstract Transient pressure behavior of wells intersected by a single vertical fracture has been examined by means of a heat flow analogy. Results are correlated in terms of dimensionless pressure change and dimensionless transient time. The resulting curves are found to lie between the dimensionless curves for a cylindrical well and a line source when effective well radius is expressed as one-fourth of the total fracture length. Results are also shown for various degrees of afterflow. Use of the curves provides a means of determining reservoir permeability, wellbore radius, and thereby fracture length, even when the final slope of the transient curve is not obtained. Superposition of curves may be used to confirm existence of vertical fractures in the reservoir adjacent to the well or to indicate a damaged zone on the fracture face. Introduction Hydraulic fracturing has become a standard well completion practice for the purpose of increasing flow. Results are normally gauged in terms of some sort of "stabilized" flow rate established before and after the treatment. This procedure may be misleading because pressures in or around the well do not become "stabilized" during the test period. Transient pressure tests on fractured wells can overcome the objections to stabilized flow tests by indicating when a stable pressure gradient is reached. In addition, the transient pressure tests can be used to determine formation flow capacity and effective wellbore radius. The effective wellbore radius may be used as a measure of the success of the fracturing operation. The slope and position of the transient pressure-log time curve may be used to determine the flow capacity and wellbore size after sufficient lapse of time so that the pressure gradient in the vicinity of the fractures becomes constant. However, in tight reservoirs or in short tests this condition may not be reached. Presence of the fractures in tight reservoirs may cause a constantly increasing slope to the transient pressure curve. If this slope is picked too soon, formation flow capacity will appear high and the fracture size will appear too small. This paper is intended to clarify the early transient behavior of vertically fractured wells. It shows how formation flow capacity and wellbore size can be determined even though the maximum slope of the transient pressure curve is not obtained during the test. BASIC THEORY Development of the techniques of transient pressure analysis for wells has been summarized by Perrine. He reviews the various definitions of wellbore effects such as skin effect, productivity ratio, damage factor and condition ratio.In the present paper, use of these terms is avoided by defining an effective wellbore size. For the fractured well this effective wellbore size is larger than the actual wellbore. Assumptions made are constant flow rate at the well, constant fluid compressibility, constant fluid mobility and uniform porosity and formation thickness. From the basic differential equations of flow, two different solutions may be obtained. The first, the line source solution, is commonly used in transient pressure analysis. It expresses the change in pressure at any radius due to withdrawal or injection of fluid at the line source from an infinite reservoir at uniform initial pressure. This equation, in oil field units, is (1) or, after a long time: (2) The second solution frequently used in transient pressure analysis is based on a finite reservoir. In this case, flow between the reservoir boundary and the well has occurred long enough for a pressure gradient constant in time to exist. Shut-in of the well results in pressure change in the well which, after a sufficiently long time, exhibits a straight-line behavior when plotted against the log of time. For a constant pressure at a circular, concentric drainage boundary with radius, the difference in well pressure and drainage pressure in the straight-line portion is approximately: (3) Expressing the initial pressure difference at steady flow as (4) the resulting change in well pressure by combining Eqs. 3 and 4 is (5) Comparison of Eq. 5 with Eq. 2 shows that they are identical if in either case represents the change in well pressure following the interruption of the stabilized reservoir pressure condition. JPT P. 1365^

Studies on Pressure Distribution in Bounded Reservoirs at Steady State

Published in Petroleum Transactions, AIME, Volume 204, 1955, pages 182–189. Abstract The purposes of this study areto determine the accuracy of a previously proposed method for calculating average reservoir pressure andto find a method for estimating the shape of the drainage area of each well in a bounded reservoir. Both numerical and experimental methods are used to determine the pressure distribution and the location of drainage boundaries in idealized reservoirs at steady state. A number of photographs are included which show flowlines and drainage areas in reservoir models of various shapes. Results from these studies are then compared with those from the previously proposed method for calculating average reservoir pressure. It is concluded that the previously proposed method is sufficiently accurate for most reservoir engineering purposes. In addition, some simple methods are given for estimation of shapes of drainage boundaries. Introduction A method for the determination of the average pressure in a bounded reservoir was previously developed by Matthews, Brons, and Hazebroek, and presented at the Petroleum Branch Fall Meeting in 1953. In applying this method, the reservoir was first divided into drainage volumes; second, the average pressure in each drainage volume was calculated; and third, these drainage volume pressures were averaged volumetric ally to give the average pressure in the reservoir. It was shown that each well's individual drainage volume, at steady state, is proportional to the well's production rate; however, the question of how to estimate accurately the shape of these drainage volumes could not be answered. Furthermore, in calculating the average pressure for each drainage area, the actual drainage areas, which are in general quite asymmetric in shape, are replaced with one of the symmetric prototypes given in T.P. 3876. A question may then be raised as to how accurately the symmetrical figures depict the pressure behavior of the actual unsymmetrical drainage areas. The study reported on in the present paper was made in an attempt to shed light on the above questions.

The Effect of Slant Hole, Drainhole, and Lateral Hole Drilling on Well Productivity

Introduction Methods have been developed for drilling (a) slant holes and (b) one or more curved holes from a common central hole in producing formations. These wells will have productivities exceeding that of a single hole drilled normal to and fully penetrating the producing stratum. other factors being equal. This increase is due to the decrease in resistance to flow in the vicinity of the wellbore by an increase in the cross section exposed to flow with increasing footage drilled in the formation and due to the geometrical arrangement of the holes with respect to the drainage radius or boundary. Where other factors are not equal, for example, where zonal damage exists in the single, fully penetrating hole but not in the slant or curved holes, additional increases in productivity will accrue. Many of these wells have been drilled in the past and are currently being drilled with various results reported. While the authors are aware of some of the practical aspects of drilling and completing these multiple. curved holes, it is their hope to provide some basic data on the improvement in productivity to be anticipated in these wells for a number of hole arrangements or patterns. Model Studies Electric analogue or model studies have been used for solving some reservoir fluid flow problems in which the mathematical solutions are unknown, too approximate or too complex. For example, recent studies have used this method to determine the effect of shot density, diameter, and depth of penetration of gun perforations on well productivity. The success of these studies depends upon the analogy between Ohm's Law for electrical flow and Darcy's Law for incompressible fluid flow in homogeneous rock. Where a geometrical scale reduction is desired, a single scale factor is applied to all dimensions.

Pressure Distribution in Oil and Gas Reservoirs by Membrane, Analogy

The pressure distribution in a producing oil or gas reservoir has beenobtained mathematically in a limited number of special cases where the boundaryof the reservoirs are simple geometric figures with the wells convenientlylocated within them. Using some of the previous work as a basis, it can beproved that a number of additional problems of pressure distribution can besolved by use of soap film. This method is particularly applicable to areservoir drained by a number of wells, or to a reservoir of irregular shape, or to both. Since this method obtains the pressure distribution, it alsoestablishes the drainage boundaries of the wells. This paper derives the analogy between the equation representing the pressuredistribution in a reservoir and the equation of the deflected membrane. It alsodescribes the apparatus and the method of operation. The experimental method isemployed to solve a problem whose mathematical solution is known and theresults compared. These are found to be in good agreement. The method is applicable to any homogeneous liquid, compressible orincompressible, or to a gas, flowing in a reservoir, of uniform thickness, permeability and porosity. The flow state may be either the steady state or thetransient state wherein the density or pressure of the fluid in the reservoirdeclines at a constant rate. Pressure-Distribution Problem Fluids encountered in the oil reservoir are compressible and expand and movetoward the drilled hole where the pressure is lowest. The loss of fluid fromthe reservoir lowers the pressure in the more distant parts of the reservoir, permitting the fluids in these parts to expand and move toward the well. T.P. 919