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By P. van Meurs

This paper describes a technique which permits visual observation of oil displacement processes througlrout the interior of a porous structure as thick as 2 in. A model having glass walls is filled with finely powdered glass. Such a model becomes completely transparent if the powdered glass pack is saturated with an oil having the same refractive index as the glass. When water or gar, with a refractive index different from that of the oil (glass) is injected, the model becomes opaque in the space occupied by the water (gar). visual observation and photography of these displacement processes are thus rendered possible. The following processes have been studied: (I) displacement of oil by water (linear drive) in a homo-geneous formation and in a stratified formation, (2) water flooding (five-spot well pattern), (3) solution gas-drive process, and (4) oil-bank formation. INTRODUCTION Most of the scaled model experiments heretofore carried out in the Koninklijke/Shell Laboratorium, Amsterdam, have been done in steel containers, in which oil-saturated sand packs represented the oil-bearing formation. In such experiments, for instance those on the displacement of oil by water, the cumulative oil production was measured as a function of the cumulative water injection. In order to interpret the measured production data, a need has for some time been felt for a means of observing visually what happens in the course of such experiments. Some information of this kind was obtained earlier by observing the water-oil distribution in various cross sections of the sand after the conclusion of the experiment.' However, the technique used did not guarantee that the fluid distribution remained undisturbed. Moreover, the procedure was somewhat elaborate and the information obtained limited. Therefore a new technique has been developed, which is described below. DESCRIPTION OF EXPERIMENTAL TECHNIQUE AND APPARATUS Glass models simulating part of an oil reservoir are filled with ground Pyrex glass of a specific grain size, representing the porous structure of the prototype. Before an experiment is started, the porous formation has to be saturated completely with an oil having exactly the same refractive index as the Pyrex glass powder. All light-scattering surfaces of the glass grains become invisible and the formation is then transparent. If the oil is displaced by another fluid (e.g. by water in a water-drive experiment, or by gas liberated from the oil in a solution gas-drive experiment) the formation becomes opaque in the region where the oil is expelled from the rock.* The back of the apparatus is painted black and illumination is from the front; therefore, in the photographs of the experiments, the oil-bearing formation appears black, whereas the water- or gas-invaded areas are white.

Jan 1, 1958

By M. R. J. Wyllie, Michael A. Torcaso

With the continued deep drilling of today, increasing numbers of high pressure and high temperature gas-condensate reservoirs are being discovered. Correspondingly, the ranges of properties of gas-condensate reservoir fluids are also being increased. The literature1*'." contains several excellent reviews of the property ranges in presently known condensate reservoirs. When one examines these property ranges, it becomes apparent that the need for reliable data is great if the engineer is to evaluate properly or appraise a gas-condensate reservoir. It is the purpose of this paper to present phase-behavior data that will point out some of the important variables that influence recovery from gas-condensate reservoirs produced by pressure depletion. All these variables may be resolved into two major categories: the hydroc:arbon composition of the reservoir fluid and the physical properties of the reservoir. But the variables which we will consider are those which are usually determined from well tests shortly after discovery and are, consequently, readily available to the engineer. They are as follows: (1) the producing separator gas-stock-tank liquid ratio, (2) stock-tank liquid gravity, and (3) reservoir temperature. Laboratory depletion studies have been performed on several reservoir fluids to illustrate the effect of changes in the cronsidered variables upon ultimate recoveries. The deviation factors, Z, of a series of reservoir fluids were determined experimentally to illustrate their order of magnitude and their effect upon the calculation of in-place reserves existing at high pressurcs and temperatures. The procedures followed in conjunction with the performance of these studies and their associated calculations are illustrated in Fig. 1. For the purpose of this paper, the depletion recoveries are expressed only in terms of pcr cent of initial stock-tank liquid, with the abandonment pressure assumed to be 500 psig. VARIATION OF RECOVERY WITH INITIAL PRODUCING SEPARATOR GAS-STOCK-TANK LIQUID RATIO The producing separator gas-stock-tank liquid ratio has long been one of the principal criteria in assumptions and generalized correlations' used for the estimation of ultimate recovery from gas-condensate reservoirs. Fig. 2 presents a plot of the cumulative recovery of stock-tank liquid at abandonment as a function of the initial producing separator gas-stock-tank liquid ratio. The graphical illustration represents the results of 44 laboratory depletion studies performed on actual gas-condensate fluids. These data are presented not only to illustrate the general trend of ultimate recovery increase with initial producing ratio increase but to emphasize the considerable variation in possible recovery at any given producing ratio. Table 1 presents the properties of two actual reservoir fluids chosen to illustrate the approximate extremities of producing ratios for gas-condensate systems. The fluids designated as A and B exhibited initial producing ratios of 3,400 and 114,000 scf/bbl, respectively. Fig. 3 presents a graphical comparison of the cumulative stock-tank liquid recoveries, calculated from the well-stream compositions and produced volumes for each fluid during depletion. It is to be noted that Fluid A exhibited a recovery of only 16.5 per cent when produced to an assumed abandonment pressure of 500 psig while the corresponding recovery for Fluid B was calculated to be 62.2 per cent. This difference in recovery, which exemplifies the general trend of recovery increase with initial producing ratio increase, is readily explained by consid-

By J. S. Aronofsky, J. P. Heller

This paper presents a mathematical analysis of the fluid mixing which occurs during flow through porous media. The analysis is based on the well-known diffusion equation with mass transfer term. It is pointed out that the use of this equation is justified by a single general assumption which does not specify any particular mechanism for the mixing process. Formulas are given for tracer fluid concentrations for two different boundary and initial conditions. Calculated numerical values compare closely with some published results of experiments in which no viscosity or density difference existed between displaced and displacing fluids. INTRODUCTION A phenomenological theory of the mixing and diffusion process is presented for the flow in a porous medium of two miscible liquids of equal viscosity and density. This is based on the classical diffusion equation which has been used to explain such processes as Brownian motion and heat conduction. More recently, this same differential equation has been derived (in a manner analogous to the work in this paper) for the problem of heat transfer during fluid flow in porous media.' The purpose of this paper is to show that the available experimental evidence justifies the use of this simple diffusion equation. The recent data of Koch and Slobod' and of von Rosenberg' are used to test this diffusion model by comparing concentration curve shapes and calculating "effective diffusion coefficients". The same differential equation for miscible displacement, as well as some experimental confirmation, has been published in Japan by K. Yuhara,4 and recently in this country by Day.5 Yuhara arrives at his con-clusions through analogy with turbulent diffusion, de- scribing the microscopic velocities in terms of "eddy motion" — although unlike the ordinary eddies of hydraulics. Day proceeds directly from Scheidegger's6 concept of the dispersion of velocities. After a very clear description of Scheidegger's theory, he applies it to calculate the motion first of a drop and then of a thin layer of brine into a bed of sand. For the latter case, Day derives and solves the same differential equation as Eq. 1. In our presentation no detailed assumptions are made as to the mechanism of displacement. In fact, several microscopic processes may well be involved—the only general assumption made is that their cumulative microscopic effect may be described by the addition of a second-order term to the differentirl equation of continuity. DERIVATION OF EQUATIONS Let us consider the displacement in one spatial dimension of Liquid A from a sample of porous media by Liquid B. A and B are assumed completely miscible and of the same viscosity and density. Both fluids are also assumed to be incompressible and it is specified that the total flow is the same for any cross section. It is evident that the rate at which Liquid B would be carried across such a section would be given by the product of the fluid velocity, u, and the concentration, C, of B in the fluid mixture assuming the constancy of u and C over the cross section. Since the volume of Fluid B is conserved, this leads to the one-way wave equation - u ?C/?x = ? ?C/?l where x increases in the direction of u—the "Darcy velocity", Q/A (or velocity flux per unit area), and 9 is the fractional porosity. Consider the case where fluid B is displacing Fluid A and the concentration distribution is represented by a step funcLon (with a vertical discontinuity). The solution of Eq. 1 predicts no mixing

Jan 1, 1958

By R. G. Bentsen, D. A. T. Donohue

The cyclic steam injection process has become the most widely applied and most successful thermal recovery technique in use today.1-4 Normally, steam stimulation is repeated several times during the life of a project. The problem is to select the optimum number and size of steam treatments so that the profits realized over the entire life of the project are maximum. Obviously, operating costs rise directly with the frequency and amount of steam injection. Operating efficiency, on the other hand, can be adversely affected if the amount and frequency of stimulation are insufficient. Thus arises a multistage decision problem; that is, a decision must be made at each sub-period (day, for example), as to whether steam should be injected, and, if so, how much. It is desirable that these decisions be made in such a way as to secure maximum net profit before taxes. This can be achieved by dynamic programming. To use dynamic programming, we must be able to predict not only the production rate-time curve resulting from a given stimulation, but also how changes in stimulation policy will affect this curve. In this study, the effect of steam injection on well performance is estimated on the basis of a simple model that takes into account both steady-state and transient flow. Oil production rate is calculated for both types of flow, and the larger of the two values is used. This allows the calculation to change to radial steady-state flow at such time as this mechanism becomes more effective than the transient mechanism. We propose to show how dynamic programming can be used to optimize the steam soak process with respect to net profit. For this, a mathematical representation of the steam soak technique is necessary. A further objective, therefore, is the formulation of a mathematical model capable of realistically simulating the physical process. Oil Production Response Model The advantage of using a mathematical model is its generality and ease of manipulation. However, the model must be an accurate representation of the physical process, for a solution derived from a model can be no better than the model itself. The question now arises as to how well the model should be expected to simulate reality. In general, the simplest model should be tried first, for additional accuracy is apt to require additional cost and time. Also, the search for a high degree of accuracy may yield only greater complexity and difficulty without yielding significantly better results. With this in mind, we kept our model as simple as possible, yet generally representative of the actual physical process. The number of state variables is limited to three, and stochastic variations in the parameters are ignored. As a consequence, use of this model should be restricted to cases in which the underlying assumptions are not severely violated. In

Jan 1, 1970

By B. S. Gottfried

A generalized mathematical model is presented, which describes the thermal recovery of oil in linear systems with convective external heat loss. Three-phase fluid flow, conduction-convection heat transfer, chemical reaction between oxygen and oil and aqueous phase change are included, making the model applicable to a variety of thermal recovery processes. The complexity of the model requires that numerical solutions be obtained with a digital computer. The numerical approximations of the partial differential equations are presented and the algorithm is discussed in detail. The analysis is applied to the detailed simulation of a forward-combustion laboratory tube experiment. The results of the simulation exhibit with reasonable accuracy all of the principal thermal and bydrodynamic characteristics observed in the laboratory (i.e., the propagation of a combustion zone, a steam plateau, and oil and water banks). Numerical variations in the parameters appearing in the external heat-loss term, the aqueous phase-change term, and the Arrhenius combustion term are examined, and their effects on the predicted temperature profiles, oil and water production histories, and the air-oil ratio are presented. In spite of an oversimplification in the chemical reaction mechanism and the considerable computer time required to obtain solutions, the model represents a major advance in the ability to simulate the phenomena observed in thermal recovery experiments. INTRODUCTION Thermal oil recovery refers to a class of recovery processes where heat is supplied to a reservoir to provide the necessary expulsive energy. This thermal energy can be supplied externally as steam or hot water, or it can be generated in situ by forward or reverse combustion. In either case, however, thermal recovery processes are charac- terized by the simultaneous flow of two or three fluid phases in a variable - temperature field, accompanied by possible chemical reaction or phase-change effects.2-5, 7- 10, 14,16-18,20-22 Although a physical understanding of the thermal recovery processes is far from complete, it is possible to construct mathematical models which describe approximately all of the principal physical and chemical phenomena. However, attempts to solve such models, even with high-speed computers, involve formidable mathematical difficulties. Consequently, theoretical solutions have been obtained only for idealized cases in which important physical phenomena are neglected. For example, consider the process of forward in situ combustion. All such theories which have been developed consider only certain aspects of the process, such as heat transfer, 2-4,8,9,l6,l8 heat transfer with phase change,10 heat transfer with chemical reaction,7 or the hydrodynamics of three-phase flow.21 A general theory including all of the above phenomena has not been developed to date. This paper presents a unified theory of thermal recovery processes in linear systems. A mathematical model is developed which explicitly includes conduction-convection heat transfer with convective external heat loss, chemical reaction between air and oil, aqueous phase change, and the hydrodynamics of three-phase flow. A system of equations is developed which can be solved numerically on a high-speed digital computer, resulting in predicted temperature, pressure, and saturation histories in space and time. The model allows a more detailed simulation of thermal recovery tube experiments than had previously been possible. THEORETICAL DEVELOPMENT Consider the linear flow of gas, water and oil in a homogeneous porous medium. Assume that the oil will react with gaseous oxygen, and that mass is transferred between the water and gas phase by evaporation or condensation. In the absence of hydrocarbon phase change, intraphase diffusion,

Jan 1, 1966

By Kenneth F. Whinery, John M. Campbell

Basically, the Buckley-Leverett theory involves two systems which are similar in nature but are differentiated by time. These systems may be described by the fractional flow and frontal advance equations which essentially characterize the mechanics of oil movement while being expelled from the reservoir. The fractional flow equation originally developed by Leverettl may be expressed in the more usable form The development of this equation is based on Darcy's law describing fluid flow through porous media and applies to the flow at only one point (it is a point function). For simplification, the fractional flow equation is written in the above form because the capillary forces always increase the fractional flow of the displacing phase regardless of the direction of flow or the displacing phase. If, for simplicity, the effects due to gravity and capillary pressure differences are neglected, the fraction of the displacing fluid, f, at any point in the flowing stream is related to the properties of the system by 1 Thus, it is seen that in the absence of capillary and gravitational effects, fd for a given sand and fluids varies only with saturation and pressure. The magnitude of the viscosity ratio, —a has an effective range (range of about 30) in the system where gas is displacing oil and a much smaller range for water displacing oil. In order to make the fractional flow equation more versatile, it is necessary to connect the fractional flow at a given point and saturation with time. This problem was approached by Buckley and Leverett" who developed the frontal advance equation, Eq. 3 states that the rate of advance of a plane that has a fixed saturation, Sd, is proportional to the change in composition of the flow stream caused by a small change in the saturation of the displacing fluid. It is, essentially, a transformation of a material balance equation representing the net rate of accumulation of the displacing phase within a homogeneous sand block. This accumulation is proportional to the difference between the rate at which the displacing fluid enters the sand and that at which it leaves. Eq. 3 describes the velocity with which a plane of constant displacing phase saturation advances through a porous system. Buckley and Leverett," Babson,' Kern," Welge, and others have adequately discussed the basic mechanism and application of the fractional flow and frontal advance equations. APPLICATIONS OF THE FRONTAL ADVANCE THEORY TO PETROLEUM RECOVERY Two general applications of the Buckley-Leverett frontal advance theory involve the system in which the oil is being displaced by an expanding gas cap overlying the oil zone and that in which the oil is being displaced by water. Displacement by Gas in the Presence of an Immobile Water Saturation A system in which gas is the displacing phase may be thought of as having two forces effecting the displacement process. These forces are the gravitational force and that force exerted by the displacing gas. The gravitational effects control the displacing efficiency of the gas. The gravitational effect will be less at higher rates of flow, thereby reducing the effectiveness of the displacement of the oil by the gas. The more efficient displacements occur at flow rates which are less than the gravity free fall rate. Capillary forces can be neglected without materially changing the magnitude of the gas saturation. The Mile Six pool is used herein to illustrate the calculating procedures in evaluating gas drive-gravity drainage field perfomlance. These calculations represent the determination of the gas-oil contact when the distribution of the hydrocarbon pore volume is considered. Two methods

By R. Ehrlich, F. E. Crane

A consolidated porous medium is mathematically modeled by networks of irregularly shaped, interconnected pore channels. Mechanisms are described that form residual saturations during immiscible displacement both by entire pore channels being bypassed and by fluids being isolated by the movement of an interface within individual pore channels. This latter mechanism is shown to depend strongly on pore channel irregularity. Together, these mechanisms provide an explanation for the drainage-imbibition-hysteresis effect. The calculation of steady-state relative permeabilities, based on a pore-size distribution obtained from a Berea sandstone, is described. These relative permeability curves agree qualitatively with curves that are generally accepted to be typical for highly consolidated materials. In situations where interfacial effects predominate over viscous and gravitational effects, the following conclusions are reached. 1. Relative permeability at a given saturation is everywhere independent of flow rate. 2. Relative permeability is independent of viscosity ratio everywhere except at very low values of wetting-phase relative permeability. 3. Irreducible wetting-phase saturation following steady-state drainage decreases with increasing ratio of nonwetting- to wetting-phase viscosity. 4. lrreducible wetting-phase saturation following unsteady-state drainage is lower than fir steady-state drainage. 5. Irreducible nonwetting-phase saturation 101lowing imbibition is independent of viscosity ratio, whether or not the imbibition is carried out under steady- or unsteady-state conditions. Experiments qualithtiveIy verify the conclusions regarding unsteady-state residual wetting-phase saturation. Implications of these conclusions are discussed. INTRODUCTION Natural and artificial porous materials are generally composed of matrix substance brought together in a more or less random manner. This leads to the creation of a network of interconnected pore spaces of highly irregular shape. Since the geometry of such a network is impossible to describe, we can never obtain a complete description of its flow behavior. We can, however, abstract those properties of the porous medium pertinent to the type of flow under consideration, and thus obtain an adequate description of that flow. Thus, the Kozeny-Carmen equation, by considering a porous medium as a bundle of noninterconnecting capillary tubes, provides an adequate description of single-phase flow. With the addition of a saturation-dependent tortuosity parameter in two-phase flow to account for flow path elongation, the Kozeny-Carmen equation has been used to predict relative permeabilities for the displacement of a wetting liquid by a gas. 1-5 It has long been recognized that relative permeability depends not only on saturation but also on saturation history as we11.6,7 Naar and Henderson8 described a mathematical model in which differences between drainage and imbibition behavior are explained in terms of a bypassing mechanism by which oil is trapped during imbibition. Fatt9 proposed a model for a porous medium that consisted of regular networks of cylindrical tubes of randomly distributed radii. From this he calculated the drainage relative permeability curves. Moore and slobod,10 Rose and Witherspoon,11 and Rose and Cleary12 each considered flow in a pore doublet (a parallel arrangement of a small and large diameter cylindrical capillary tube). They concluded that, because of the different rates of flow in each tube, trapping would occur in one of the tubes; the extent of which would depend upon viscosity ratio and capillary pressure. They limited

Jan 1, 1970

By K. P. Fournier

This report describes work on the problem of predicting oil recovery from a reservoir into which water is injected at a temperature higher than the reservoir temperature, taking into account effects of vis cosity-ratio reduction, heat loss and thermal expansion. It includes the derivation of the equations involved, the finite difference equations used to solve the partial differential equation which models the system, and the results obtained using the IBM 1620 and 7090-1401 computers. Figures and tables show present results of this study of recovery as a function of reservoir thickness and injection rate. For a possible reservoir hot water flood in which 1,000 BWPD at 250F are injected, an additional 5 per cent recovery of oil in place in a swept 1,000-ft-radius reservoir is predicted after injection of one pore volume of water. INTRODUCTION The problem of predicting oil recovery from the injection of hot water has been discussed by several researchers. l-6,19 In no case has the problem of predicting heat losses been rigorously incorporated into the recovery and displacement calculation problem. Willman et al. describe an approximate method of such treatment. 1 The calculation of heat losses in a reservoir and the corresponding temperature distribution while injecting a hot fluid has been attempted by several authors.7'8 In this report a method is presented to numerically predict the oil displacement by hot water in a radial system, taking into account the heat losses to adjacent strata, changes in viscosity ratio with temperature and the thermal-expansion effect for both oil and water. DERIVATION OF BASIC EQUATIONS We start with the familiar Buckley-Leverettg equation for a radial system:* This is sometimes referred to as the Lagrangian form of the displacement equation. If the injection water is at some temperature Ti which is above reservoir temperature TR, then lw, the flowing fraction which is water, is a function of both water saturation and fluid temperature at any given time and position in the reservoir. The temperature dependence of lw results from the temperature dependence of the viscosity ratio µo/µm of which lw is a function. Thus:

Jan 1, 1966

By A. Schild

The effect of heating a producing well on the rate of oil recovery has been analyzed in the simple case of a well producing oil by a radial drive and in the steady state. Differences of production rates with and without borehole heating have been calculated and plotted; these measure some of the benefits of the thermal recovery process. A striking feature is that the production increase tends rapidly to a constant value for large production rates. Some concrete examples, including a rough comparison with field data. are discussed. Heating requirements have been calculated and plotted on the assumption that the heaters used are 100 per cent efficient. Heating has been thought of as applying only to low volume wells producing heavy oils. However, the calculations suggest that the thermal process considered might be profitable even for some wells producing large quantities of light oil, especially in unconsolidated formations where fracturing is not possible. In the case of wells producing other fluids with the oil or of wells producing partly or wholly by gravity drainage, the theory developed does not apply in detail. It can be expected, however, that any general qualitative be-havior will remain valid. Under normal temperature conditions, high viscosity emulsions sometimes form around the wellbore, lowering the permeability in the vicinity of the well and impeding the production of oil. Borehole heating often prevents such emulsification. This important effect is not easily amenable to mathematical analysis and has not been included in the theory presented here. It should therefore be kept in mind that the benefits due to borehole heating may in fact exceed the values given by the following calculations. INTRODUCTION The recovery of viscous, low gravity oils can often be stimulated by applying heat to the production zone. The higher temperature lowers the viscosity of the oil, increasing its mobility, and therefore its rate of production. One thermal method of recovery 1,2,3,4. consists in operating a heater inside a producing well and, at the same time, producing oil from the well. In this paper a simple theoretical model is analyzed to study the effect of such heating on the recovery of oil. The reservoir mechanics of the process is the only aspect of the problem which is considered here. The details of the heating system are ignored, and it is simply assumed that the heater raises the temperature in the immediate vicinity of the producing well to a pre-assigned temperature T w. Although heating requirements are calculated, these refer only to the heat supplied to the formation and to the oil being produced; they do not include heat losses due to the inefficiency of the heater. A short outline of the theory follows: The heat supplied to the producing well flows out into the formation by means of thermal conduction. At the same time, the flow of oil carries heat by convection from the formation towards the producing well. After sufficient time has elapsed, a steady-state is reached where the temperature, the velocity of the oil, etc., no longer vary with time but are functions of position only. In the steady state the temperature decays like an inverse power from the high value Tw at the wellbore to the original reservoir temperature T. at some distance away from the well. The effect of the high temperature and low oil viscosity near the producing well is the same as if the oil were produced at reservoir temperature To but through a producing well of a larger radius. From this point of view, the energy and money being expended to provide heat

Jan 1, 1958

By T. S. Hutchinson, C. E. Kemp

The bottom water drive analysis presented by Muskat has been extended to include fields with wider well spacings. Curves are presented from which volumetric sweepout, water-oil ratio, and rate relations may be calculated. The effects of non-unity ratios on sweepout eficiency and productivity decline have been studied by an approximate method. Also, the effect of gravity on volumetric sweepout has been approximated for a particurlar geometry. INTRODUCTION The purpose of this paper is to report work done in applying the analysis of bottom water drive reservoirs given by Muskat1 to a larger class of reservoirs, to extend this analysis to include the effects of gravity and non-unity mobility ratios in an approximate manner, and to investigate the rate of decline of productive capacity. The bottom water drive type reservoir is one where the oil is underlain by a water zone of sufficient thickness that water movement is essentially vertical. This system is defined in more detail in the appendix and a one well element is shown diagrammatically in Fig. 1. Muskat obtained solutions to the equations describing the volumetric displacement efficiency of idealized bottom water drive systems by assuming that gravity effects due to differences in density betwecn oil and water were negligible and that the mobilities of oil and water were the same. If either of these assumptions is not made, the complexity of the system makes an analytical solution well nigh impossible at the present stage of mathematical development. Coles and Stade' developed an approximation to consider these factors. The authors have refined approximations, starting with the same basic assumption, which should give a somewhat quantitative picture of bottom water drive performance when mobilities are not equal and the effects of gravity are not negligible. The results of these approximate calculations can then be used to obtain the decline in well productivity as the field is depleted.

Jan 1, 1957

By H. J. Morel-Seytoux

Methods of predicting the influence of pattern geometry and mobility ratio on water flooding recovery predictions are discussed. Two methods of calculation are used separately or concurrently. The analytical method yields exact solutions in a convenient form for a unit mobility ratio piston-like displacement. A few typical pressure distributions, sweep efficiencies and oil recoveries are presented for various patterns. For non-unit mobility ratio, one may resort to a numerical method, such as that of Sheldon and Douherty. 1,2 Because the domains of applicability of the analytical and numerical techniques overlap, the exact solutions provide estimates of the errors in the numerical procedures. The advantages of the analytical and numerical methods can be combined. To develop a numerical technique as independent of geometry as possible, the physical space is transformed into a standard rectangle. The entire effect of geometry is rendered through one term, the "scale-factor", derived from mapping relations. The scale factor can be calcu-lated from the exact unit-mobility ratio solution for the particular pattern of interest. By this means recovery performances for arbitrary mobility ratio con be obtained for many patterns. A sample of tesults obtained in this manner is presented. INTRODUCTION Pattern geometry and mobility ratio are two major factors in making a waterflood recovery prediction. Because assisted recovery has become increasingly important to the oil industry, pattern configuration and mobility ratio also assume a greater significance in the assessment of the economic value of recovery projects. The influence of pattern geometry and mobility ratio in shaping a recovery curve and on the other quantities of interest to the reservoir engineer is the main subject of this paper. Much effort has already been spent on estimating quantitatively the influence of either pattern or mobility ratio or both on oil recovery. The literature reports many investigations of this nature. 3-9 However, many results or methods of recovery prediction presented in the literature cannot be considered fully satisfactory. Even for unit mobility ratio and piston-like displacement, where analytical solutions are available, the literature shows discrepancies. For non-unit mobility ratio, the divergence in the results is extreme. For infinite mobility ratio in a repeated five-spot, depending on the investigator, the sweep efficiency ranges from 0 per cent to 60 per cent. With respect to the influence of pattern on recovery, only the repeated five-spot has received much attention. Other confined patterns and pilot configurations have received very little attention. Two calculation methods are presented in this paper, either separately or concurrently: the analytical method of potential theory and the numerical method of finite-difference approximation. The analytical method is more restricted in scope than the finite-difference method, but it has the definite advantage of providing exact solutions within its range of applicability. If a unit-mobility ratio piston-like displacement is assumed, the analytical approach is possible. A few typical results are reported in this paper; the detailed description of the general method and of a great variety of results will be the subject of other articles. Fornon-unity mobility ratio, we must resort to a numerical scheme. The numerical technique is that which was described by Sheldon and Dougherty.l,2 It is not limited to piston-like displacement. However, mainly single interface results will be presented here. Because the respective domains of applicability of the analytical and the numerical method overlap, useful comparisons of exact and numerical solutions can be made for a variety of patterns. The advantages of the analytical and numerical approaches can be combined. The reason for the success of this analytical-numerical approach can be summarized in the following two points: 1. The numerical solution for arbitrary mobility ratio can be programmed most efficiently when the physical space in which the displacement actually takes place is transformed into a standard shape; and 2. This can be done with remarkable simplicity whenever an analytical expression for the pressure

Jan 1, 1966

By C. S. Matthews, H. C. Lefkovits

Drilling progress is often delayed by sticking of the drill string. The development of preventive and remedial methods has been hampered by incomplete understanding of the sticking mechanism. A recent lahorntory investigation hns indicated that one type of sticking may be attributed to the difference in pressure between the borehole and formation. This paper shows, by means of soil mechanics, that the primary cause for differential pressure sticking is cessation of pipe movement, whereas diflerential pressre and stanrtding time determine the severity of the sticking. The analysis stresses the importance of using low-weight muds with low solids content and low water loss to alleviate diflerential pressure sticking and describes why packed hole drilling, long strings of drill collars, and a large deviation from the vertical are conducive to sticking. Finally, preventrve and remedial methods ore evaluated, and a theory is presented on the release of stuck pipe by spotting oil. INTRODUCTION Since drilling with long strings of oversize drill collars has become standard practice in many areas, the incidence and severity of the stuck pipe problem has increased. It has been noticed that in the majority of these cases the sticking could not possibly be attributed to key seating or caving of shales. It appeared that, due to the differential pressure between the mud column and the formation fluid, the collars were pressed into the wall and so became "wall stuck". Points to note about differential pressure sticking are: (1) sticking is restricted to the drill collars, (2) the collars become stuck opposite a permeable formation, (3) the sticking occurs after an interruption of pipe movement, (4) circulation, if interrupted, can be restarted after the sticking is noticed, and (5) no large amounts of cuttings are circulated out after restarting circulation. Helmick and Longleyl investigated pipe sticking by differential pressure in the laboratory and found an empirical relationship between the differential pressure, the sticking time and the required pull-out force. In this paper an explanation of the mechanism is given based on Terzaghi's theory of clay consolidation. A qualitative description is given in the following paragraphs while the derivation of fonnulas is given in Appendices. This paper is a first attempt to explain pressure differential sticking and many points will require additional theoretical and practical investigation before the problem can be fully understood. PRESSURE DIFFERENTIAL STICKING AS A CONSOLIDATION PROBLEM In any borehole, where the mud pressure is higher than that exerted by the formation fluids, a mud cake is formed opposite the permeable sections of the hole and a continuous flow of filtrate takes place from the mud, through the cake and into the formation. This radial flow pattern requires a certain distribution of the hydraulic and the effective (grain-to-grain) stresses inside the mud cake. Any quantitative or qualitative change in the external pressure conditions will produce a change in the flow pattern and, consequently, also in the internal stress distribution inside the cake. In view of the low permeability and the high compressibility of a clay mud cake, the adjustment of the internal stress distribution is slow and is accompanied by a change in volume. Time dependent stresses are thus created which gradually diminish as the new state of equilibrium between internal and external pressures is approached. Some 30 years ago, Terzaghi developed his "Theory of Consolidation" to account for the time-dependent stresses and settling of clay formations under the influence of external loads. He derived a differential equation by which the time-dependent hydraulic stress and the consolidation can be computed for any point inside the layer during the consolidation process. His theory is based on the assumption that the change in stress is solely due to a change in water content and it may only be applied to one-dimensional consolidation phenomena. Other investiga-tors5,10 have expanded his theory to include processes of more than one dimension. The difference between the external pressures on the mud cake before and after sticking is a qualitative one (isolation of part of the cake by the static contact with the drill collars after pipe movement has been stopped)', and the time-dependent stresses thus created may be investigated by means of Terzaghi's theory. By this analysis the changes in the nature of the contact surface between the drill collars and the mud cake during the sticking can be explained; and the friction force between the two may be computed as a function of the sticking time, the borehole dimensions and the mud cake characteristics.

By R. D. Taylor, Jim Douglas Jr., H. H. Rachford Jr., P. M. Dyke

A major obstacle to the use of wetting agents in .secondary recovery by water flooding is the adsorption of the agents on the sand. As a result of adsorption, the surfactant always lags behind the floodwater front. Consideration of the chromatographic theory of adsorption indicates that the detergents will not lag as much if used in very high concentrations. An investigation was made of the possibility of using high concentrations economically by flowing slugs of wetting agents followed by normal flood water. The experiments consisted of adsorption studies on Alundum powder and Berea sandstone. Flow rests on a 12-in. Alundum core and 22-in. Berea core were used to determine rate of detergent movement. The results of the flow experiments indicate that the relative rate of surfactant advance is, indeed, sensitive to the concentration of the agent. A 10 per cent slug moved with a rate that war 78 to 95 per cent as fast as the rate of advance of the flood water. By contrast, one with 25 ppm (the number of parts of commercial detergent in a million parts of water on a weight basis) concentration moved less than one-fourth as fast as the flood water, and calculations indicate that in very long porous systems the rate of movement of the lower concentrations will be a small fraction of the rate of advance of the flood front. The results. of the adsorption studies were utilized to calculate the rate of advance of the detergent when only the initial concentration was known. The calculated rates showed substantial agreement with the experimental flow tests in the high concentration ranges. The adsorption results were also used to estimate the cost of the materials for a slug-type surfactant flood in the field. In addition to the faster rates of movement, the concentrated detergent slugs removed much more oil than the dilute solutions. However, the effectiveness of the slug process depends on many variables. The quantity of oil removed is increased markedly by increasing the flooding rate. The efficiency is also influenced by the type of crude, type of reservoir rock and initial water saturation. Therefore, a careful analysis of each reservoir system is required before the economic value of the process can be determined. INTRODUCTION It is well known that the displacement of oil by invading water during water flooding is far from complete. It is generally agreed that the unrecovered oil is retained in the porous medium by the capillary forces which may be relatively large compared to the forces generated by the flowing water. Therefore, it was logical that some early workers should turn to surface-active materials to reduce the capillary forces to facilitate the release of oil. As early as 1927,' a patent was granted for the use of surface-active materials in water flooding. In 1932, when soap solutions were passed through Bradford and Venango sands, it was reported that the results were inconclusive, erratic and that "further investigation is needed to determine exactly the function of the solution and to obtain a clearer insight into the phenomena involved."' Some of the modern scientific reports conclude with a similar statement,' showing that the lack of agreement on the mechanism of oil removal by wetting agents is still very widespread even though several comprehensive studies have been reported.'." Although there is a lack of agreement as to the general effectiveness of the detergents for water flooding, most investigators do agree that all of the common detergents are strongly adsorbed onto the solid surfaces of the reservoir. In the early calculations it appeared that all additives would be lost before reaching much of the formation area which contained the additional oil to be removed. Experiments indicated that if the usual small waterflood concentrations of wetting agents were used, the rate of advance of detergent through the formation would be only a small fraction of the rate of advance of the flood front. Indeed, some investigators4 felt that the use of wetting agents would never be economically feasible because of their adsorption. For example, DunningG estimated that the wetting agent in concentrations of 25 ppm, would advance only 0.05 times as fast as the flood front. Ojeda, et al,' found that a surfactant in a concentration of 10 ppm moved less than 0.01 times as fast as the flood front. It is significant, however, that both investigators found that increased concentrations of wetting agents moved faster, relative to the flood front, than solutions at the lower concentrations. Ojeda showed that an extrapolation of his data indicated a relative rate of 0.5 at 1 per cent concentration, while Dunning6 estimated a relative rate of 0.46 for a 1 per cent concentration. It was obvious that these concentrations could not be used for continuous injection because the cost of the injected detergent would far exceed the value of additional oil produced. Traditionally, detergents are used in very low concentrations for they show good

By W. W. Yankie, A. T. Chatas

This paper is written as a discussion of the paper, "Buckling of Tubing in Pumping Wells, Its Effects and Means for Controlling It" by Arthur Lubinski and K. A. Blenkarn, which war published in the March, 1957, issue of Journal oP Petroleum Technology. An analogy is drawn between the buckling of tubing in pumping wells and a fiube, fitted with frictionless pistom, linked by an inelastic rod, and which is subject to an internal pressure. A simple derivation shows that the critical internal pressure exerts a hydrostatic force on the pisfon which is close to the Euler load for compression buckling of the same tube. While the results agree with those of Ref. I, the basis of calculations has been changed and clarified. The presence of internal upsets at the pistons is discussed. INTRODUCTION In the Appendix, Lubinski and Blenkarn state that a tube with ends closed by frietionless pistons connected by an inelastic rod (Fig. 1) will behave as a column loaded with a force equal to the tension force in the rod. They note that, while the fluid may not resist a shear force, the moment in every cross section is piston force multiplied by the elastic line deflection (c.f., the loaded column). Since the equations of the elastic lines are similar in the two cases, the solution is carried over from the loaded column. On the contrary it is suggested that while the method of solution remains unchanged from the column case, the actual stress distribution is different, a fact which is taken into account in this note. A simple method of approximating the buckling pressure is given, and their reasoning in our mathematical language is continued. GENERAL EQUATIONS The equation of an elastic line, initially straight in the x direction (Fig. 2), is given by In the case of a loaded column, this may be written: For the pressurized tube it was shown, by analogy with the loaded column, that tion has a solution that, for P, = EI/P, any deflection is stable (a criterion for buckling). An attempt will be made to describe more precisely, the forces acting on the tube, and to analyze the buckling condi,tions. DISCUSSION If the tube assumed a deflected position, the volume between the pistons would increase so that in deflecting, the energy of the fluid would decrease. When the increase in elastic energy of the tube (due to bending) is equal to the decrease of fluid pressure energy, buckling conditions exist. Analyzing the forces on the tube (Fig. 3), two transverse forces, T,, T,, exerted by the pistons, are balanced by a resultant of the pressure elements which are linearly dependent on the local curvature of the elastic line (Fig. 4). Unlike most buckling cases, the forces depend on the shape of the elastic line, a condition that makes a rigorous mathematical treatment difficult. Several approximations may be used to clarify the problem.

By H. Hooykaas

A few years ago Marsal published a method of predicting oil recovery as a function of time for an edge-water-drive reservoir with several rows of wells.' The method is based solely on oil- and water-production data obtained during a short period for the first row of wells and the gross productions (past and future) of all wells. This paper presents a streamlined version of the original publication and experimental evidence showing the reliability of this version. For this purpose scaled experirnents were performed in a large model (750 X 500 X 23 cm, 24.4 ft X 16.2 ft X 8.9 in.) in which edge-water-drive reservoirs with three rows of production wells were simulated. From the results of the experiments it is concluded that the method gives a fairly accurate prediction of the future production behavior of the reservoir and enables us to select the most profitable production plan. Results obtained with the method were always slightly conservative. INTRODUCTION The calculation method developed by Marsal aims at forecasting future reservoir behavior.' It can be applied to floods in which a number of down-dip wells are rapidly going to water. Essentially it is a method for predicting the future behavior of the up-structure wells on the basis of the observed behavior of the down-structure wells. As such, the method is especially useful for predicting behavior of edge-water drives toward several rows of wells. The requirement to be met for justifying the approach is identity of production mechanism in all regions of the reservoir. The attraction of the method then lies in the fact that it applies even when we have no complete description of the prevailing production mechanism. The original publication was in the German language and for this reason, perhaps, did not attract widespread attention. In our opinion it deserves to do so because it does offer in a simple way results which are in fair agreement with observed data. The aim of the present paper is to bring the method to the attention of a wider audience. The method in its most simplified form is reviewed and experimental data in support of it are presented. PRINCIPLES To show Marsal's principle. we will first write the continuity equation for very simple conditions, linear flow (in the positive x-direction) in an incompressible porous body [cross-section A(x), porosity +(x)], saturated with incompressible fluids (oil saturation S). When the oil in this formation is displaced by water, injected at x = 0 (inlet face), we get where q(t) is the total gross displacement rate and f the fractional oil flow. The equation can be solved if we make the assumptions: (1 ) that the fractional oil flow (f) is a univalent function of saturation (S); and (2) that the function is the same throughout the porous medium. Then: f = f(S) Solving this equation with the method of characteristics, we obtain

Jan 1, 1966

By E. J. Bonet, P. B. Crawford

It is fairly common practice to reinject water into the aquifer near the oil-water interface in water-drive reservoirs. There have been many studies of aquifer behavior without reinjection,l-5 but apparently no analytical studies have been made to indicate performance when injection wells are present in the aquifer. The diffusivity equation may be used to describe the pressure behavior in an aquifer. When sources are present it can be written as A general solution of Eq. 1 may be obtained as the sum of two solutions. One is a general solution of the homogeneous equation, obtained by dropping the source term in Eq. 1 and still satisfying the required boundary conditions. Another is a general solution of Eq. 1 satisfying homogeneous boundary conditions.1° The first solution, which corresponds to the aquifer without reinjection, has been given quantitative and qualitative treatment in the literature.l-5,8,9 The latter solution will be treated here. Solutions with variable injection rates are obtained by superimposing solutions with constant rates, so we have considered only the latter. Five cases have been treated here: 1. A linear aquifer of finite width and extent. Water is injected into the aquifer at point xo, yo at constant rate. The pressure along the line x = 0 is maintained at zero pressure. (See Fig. 1.) Equations showing the pressure distribution, rates, velocity and cumulative influx have been developed. Certain numerical results are shown. 2. A linear aquifer of finite width and extent. Water is injected into the aquifer at point xo, yo at constant rate. The rate of production along the oil-water water contact is maintained constant. Equations showing the pressure distribution inside the aquifer are presented. 3. A linear aquifer of finite width and extent. Constant pressures are maintained at opposite ends of the aquifer. Water is injected at constant rates at point xo, yo. Equations showing the pressure distribution are shown. 4. A linear aquifer of finite width and infinite extent. The pressure is maintained constant at the oil-water contact. Water is injected at a constant rate at point xo, YO. 5. A linear aquifer of finite width and finite extent. The pressure at the oil-water contact is maintained constant. Water is injected at constant rate per unit length along a line x,. The Linear Aquifer The linear closed aquifer, performing at constant terminal pressure was selected as the first case to be analyzed. The development of the equations is based on uniform injection per unit of thickness, and gravity effects are neglected. In practical application this condition is probably most closely achieved when the

Jan 1, 1970

By J. S. Lasater

Resu1ts of experinmental measurernents of heat capacities and thermal conductivities of some typical porous rocks are presented. Measured heat capacities agree closely with va1ues calculated front known chenlical . compositions of the rocks. On the basis of this agreement, heat capacitir.7 of fluid-satttrated rocks were calc~ililted. Therrnal conductivities of the rock samples were measured under varioits conditions of fluid saturation. From these data therrnal dif-fvsivities were calculated. A significant variation of thermal diflusivity with trnlperature is indicated. Knowledge of the thermal characteristics of fluid-bearing porous rocks has become increasingly important with the development of thermal oil-recovery processes. Reliable thermal data for petroleum reservoir rocks are not available and approximate values are generally assumed for reservoir calculations. 'The effects of temperature, pressure and fluid saturation on thermal properties of porous rocks have not been carefully studied. The purpose of the present work was to measure and report the thermal characteristics of some typical porous rocks. A group of eight sedimentary rock samples, including limestone, shale, sandstone and silt-stone, was selected for the study. Properties reported include heat capacities, thermal conductivities and thermal diffusivities. The effect of fluid saturation on these properties was also considered. Determinations of thermal properties require precise measurements which are time consuming and difficult to reduce to a routine basis. Methods have not been developed for measurement of thermal proper-ties under conditions of pressure, temperature, and fluid saturation, which may be encountered in thermal recovery processes. As one approach to these problems, some preliminary work on methods of estimating thermal properties from other more readily measurable characteristics of the rock-fluid system is reported. The samples selected for the studies were oilfield cores from within, or closely associated with oil-productive zones. Descriptions of individual samples are given in Table 1. Clay minerals present in the samples were determined by differential thermal analysis (DTA). Quartz content was estimated from DTA cooling curves. For all but the thermal conductivity tests, a representative 200-gm sample of each core was disaggregated, thoroughly extracted in diox-ane and oven dried at 100°C. The cleaned and dried sample was care- fully split into appropriate portions for the several tests. Chemical analyses of the samples were obtained to permit calculations of heat capacities by methods which are outlined later. Resuits of chenlical analyses are given in Table 2. Heat contents of the test samples were measured by a Runsen-type calorimeter. The heat content of the unknown sample is measured relative to the known heat content of platinum. The apparatus is referred to as a relative error type because of the comparative measurements, and has the advantage of canceling systematic errors. Experimental results are reproducible within 0.5 per cent. A temperature base of 298.16 K is taken for heat content measurements. Measurements were made at five temperatures (260, 440, 620, 800 and 980° F). It was necessary to preheal the sample above the maximum experimental temperature to eliminate volatile constituents. Assuming the weight loss on pre-heat, ing was mostly water, a correction to the observed heat content was made to account for the water originally present in the sample. This latter procedure was followed in the determination of calculated heat contents as will be described later. The apparatus and the method of measure-

By T. D. Mueller, H. Dykstra

This paper describes a method for obtaining compositions of gas and oil phases in equilibrium with each other at a given reservoir temperature and pressure as lean or enriched gas is injected into a reservoir oil not in equilibrium with the gas. The method is based on calculation of equilibrium constants, K, from the equation log Kp = A + B F, where p is pressure and F is component characterization factor. A and B are constants that vary with a composition parameter. The paper explains how A and B could be determined with a minimum amount of data. The method consists of calculating equilibrium constants as a function of liquid composition and using the K's to calculate changes in compositiort of injection gas as it successively contacts original resetvoir oil at the gas-oil front and the change in composition of reservoir oil as it is successively contacted by the injection gas in the vicinity of an injection well. An iterative process is required. Two examples of the method are given which show how oil and gas compositions and properties are calculated with the aid of a computer. INTRODUCTION The calculation of recovery from lean-gas or enriched-gas injection into a saturated or under-saturated reservoir oil requires a knowledge of how phase properties change in a reservoir between injection well and producing well. To determine the phase compositions or properties experimentally would be difficult and time consuming, especially if one wanted to look at the effect of various injection gas compositions. For some fields an engineer would want to make preliminary calculations on the economics of gas injection before he makes the decision to spend a corrsiderable sum of money for laboratory work to obtain the necessary fluid properties. A method for calculatine compositions and properties based on a minimum of laboratory data is described. The minimum amount of data required is injection gas and reservoir oil composition and injection gas solubility as a function of pressure for an undersaturated oil, or the compositions of mixtures of reservoir fluid with two or more enriched gases, which mixtures have a bubble-point pressure that is the same as the expected average displacement pressure. On the other hand, if the economics appear to be attractive, more laboratory data can be obtained in the form of equilibrium constants for three or more mixtures of the reservoir fluid and injection gases. The method is based on the calculation of K's from the equation, log Kp = A + B F, where K is equilibrium constant, p is pressure and F is component characterization factor. This method of plotting K data by means of the above equation is described by Hoffman, Crump and Hocott.1 A and B are constants that vary with a composition parameter. It is shown how experimental data are used to determine how A and B vary with composition. BASIS OF METHOD EQUILIBRIUM CONSTANTS Equilibrium constant (K) data are required for this method. The K of each component is obtained from an equation that gives the product of K and pressure as a function of component characterization factor. The characterization factor F = b(l/Tb — l/T), where b is a constant characteristic of a given component, Tb is the boiling point in OR at atmospheric pressure of the given component, and T is the temperature of the system in OR. This method of plotting K data is described by Hoffman, Crump and Hocott.l The straight-line equation of the log Kp - F plot is log hp = log Kop +BFi, where Kop is at F =0, i designates the component, and B is the slope of the line. The factor F can be

Jan 1, 1966

By J. E. Kirby, L. B. Schnitz, H. E. Stamm III

The production history of a gas-cap-drive reservoir was reproduced by calculations, and predictions were made for operations under primary depletion, pressure maintenance by gas injection, and pressure maintenance by water injection. Although the general equations used in this analysis have been available to the industry for a number of years, the procedure has not had extensive use because of the unusually large time requirements for developing numerical results. The use of digital computing equipment for processing the calculation work generally reduced the time usually required for an extensive analysis of this type. Calculation techniques were developed that were equally applicable for solving the gar-cap-drive problem, whether digital computing equipment or desk calculators are used. These techniques are outlined and the procedures are illustrated by the field problem. INTRODUCTION The analysis of a gas-cap-drive res-ervoir requires the simultaneous solution of the volumetric balance1 and displacement2 equations. Equations for the analysis of combination drive reservoirs were presented by Wooddy and Moscrip in a previous paper." These equations permit the evaluation of possible operating processes in order that the optimum method of pro- ducing the reservoir may be determined. Although these equations represent the basic technique known for the detailed analysis of gas-capdrive reservoirs, the calculation processes are quite complex and time consuming; hence, the procedure has not had extensive use. Recently, these equations were satisfactorily applied with the aid of a digital computer to investigate the completion history of a gas-cap-drive reservoir under the following operating methods: primary depletion, depletion under full pressure maintenance by gas injection, and depletion under full pressure maintenance by water injection. The use of digital computing equipment substantially reduced the time necessary for making the computations and permitted a more concise investigation of the reservoir's depletion history than was possible when the desk calculator was used. This more rigorous investigation also permitted a detailed analysis of the calculation procedures necessary in the solution of the gas-cap-drive problem. This paper outlines the procedure of analyzing a gas-cap-drive reservoir and illustrates the process with an actual field problem. CHARACTERISTICS OF THE RESERVOIR ANALYZED Production in the field under consideration is obtained from a Frio formation of Oligocene age at a depth of approximately 5,740 ft sub-sea. The accumulation, which is bounded on the flanks by strand lines, is semirectangular in shape with gentle narrowing toward the base of the structure as shown by the sand limits and completion map in Fig. 1. The reservoir development indicates an oil column of 319 ft that is defined by gas-oil and oil-water contacts. Basic data relative to the characteristics of the formation and the formation fluids are shown in Table 1. The formation exhibits considerable shaliaess and thinning near the oil-water contact indicating that little or no benefit will be received from natural water influx. This is substantiated by the fact that wells completed near the original oil-water contact have not produced water. A sizeable gas cap is present; however, considerable faulting is in evidence, and it is believed that only a limited portion of the gas cap will aid in oil displacement and pressure maintenance. Although the reservoir was in an initial stage of depletion, it was believed that as soon as the equilibrium gas saturation in the oil zone was exceeded and a number of upstructure wells were invaded by the expanding gas cap, the resulting high gas-oil ratio production would rapidly dissipate the natural drive and lead to limited oil recoveries and inefficient reservoir operation. This analysis evaluated the effectiveness of the primary displacement mechanism and

Jan 1, 1958

By N. R. Morrow, C. C. Harris

The experimental points which describe capillary pressure curves are determined at apparent equilibria which are observed after bydrodynamic flow has ceased. For most systems, the time required to obtain equalization of pressure throughout the discontinuous part of a phase is prohibitive. To permit experimental points to be described as equilibria, a model of capillary behavior is proposed where mass transfer is restricted to bulk fluid flow. Model capillary pressure curves follow if the path described by such points is independent of the rate at which the saturation was changed to attain a capillary pressure point. Aemodified suction potential technique is used to study cyclic relationships between capillary pressure and moisture content for a porous mass. The time taken to complete an experiment was greatly reduced by using small samples. INTRODUCTION Capillary retention of liquid by porous materials has been investigated in the fields of hydrology,l soil science,2 oil reservoir engineering,3 chemical engineering," soil mechanics,5 textiles: paper making: and building materials In studies of the immiscible displacement of one fluid by another within a porous bed, drainage columnsl and suction potential techniques2 have been used to obtain relationships between pressure deficiency and saturation (Fig. 1). Except where there is no hysteresis of contact angle and the solid is of simple geometry, such as a tube of uniform cross section, there is hysteresis in the relationship between capillary pressure and saturation.2 The relationship which has received most attention is displacement of fluid from an initially saturated bed (Fig. 1, Curve R2), the final condition being an irreducible minimum fluid saturation Swr. Imbibition (Fig. 1, Curve A), further desaturation (Fig. 1, Curve R), and intermediate scanning curves have been studied to a lesser but increasing extent.2 ,3,9-12 This paper first considers the nature of the experimental points tracing the capillary pressure curves with respect to the modes and rates of mass transfer which are operative during the course of measurement. There are clear indications that the experimental points which describe these curves are obtained at apparent equilibria which are observed when viscous fluid flow has ceased; and any further changes in the fluid distribution are the result of much slower mass transfer processes, such as diffusion. Unless stated otherwise, this discussion applies to a stable packing of equal, smooth, hydrophilic spheres supported by a suction plate5 with water as the wetting phase and air as the nonwetting phase. The discussion can also be applied, in many

Jan 1, 1966