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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 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 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 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. 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

By J. J. Arps

This paper reviews the methods currently in use for estimating primary oil reserves and discusses the principles on which these methods are based. Particular emphasis is placed on how these methods change with the type of information available during the life cycle of an oil property. This paper contains various novel estimating methods and shortcuts heretofore unpublished. INTRODUCTION Estimating oil reserves is one of the most important phases of the work of a petroleum engineer since the solutions to the problems he deals with usually depend on a comparison of the estimated cost in terms of dollars, with the anticipated result in terms of barrels of oil. His recommendations to management regarding the best course of action are therefore normally based on the most favorable balance between these two. Specific engineering problems which require such a knowledge of recoverable oil reserves and a projection of future rates are: [a.] the exploitation and development of an oil reservoir; [b.] the construction of gasoline plants, pipelines and refineries; [c.] the division of ownership in unitized projects; [d.] the price to be paid in case of a sale or purchase of an oil property, and the magnitude of the loan which it will support; [e.] the proper depreciation rate for the investment in oil properties; and [f.] evaluation of the results of an exploration program. This discussion will be confined to the various methods and tools which are currently in use for estimating oil reserves to be obtained during the primary phase of an oil-producing reservoir and for a projection of the future production rates. Reserves which may be obtained by secondary recovery methods or fluid injection programs and gas and gas condensate reserves will not be discussed in this paper. Unfortunately, reliable oil reserve figures are most urgently needed during the early stages when only a minimum amount of information is available. Management's interest in the oil recovery from a property—aside from its use for accounting purposes—usually declines when the property approaches its economic limit, just at the time when the reliability of the estimates is at its best. To illustrate this general idea a chart is presented (Fig. 1) showing the three periods in the life of an imaginary oil property. Time is shown on the horizontal axis, while the cumulative production and estimated ultimate recovery are plotted vertically. No particular units are used and this schematical chart is not to scale. During the first period, before any wells are drilled on a property, any estimates will of necessity be of a very general nature, based on experience from similar pools or wells in the same area, and usually expressed in barrels per acre. This will

Jan 1, 1957

By R. D. West

Miscible displacenzent recovers all oil in the area contacted by the injected .fluid, whereas water or immiscible gas drives usually leave substantial amounts of oil as residual. However, the Door mobility ratios associated with a gas-driven miscible displacement cause the sweep pattern efficiency to be much lower than that obtained with water flooding. One way in which the sweep eficiency in a miscible displacement process can be increased is by decreasing the mobility behind the flooding front. This can be achieved by injecting water along with the gas which drives the miscible slug. This water reduces the relative permeability to gas in this area and thus lowers the total mobility. The main operating conditions for the simultaneous injection -vrocess are that a zone of gas exists between the miscible slug and the leading edge of the water and that a su,@cient amount of gas be injected with the water to form the pas volume which is being left in the water zone. Laboratory model studies have shown that the ultimate sweep pattern efficiency can be as high as 90 per cent for a five-spot flooding system. If gar alone is used as the driving medium an ultimate sweep-out efficiency of about 60 per cent would be obtained in the same system. I INTRODUCTION The miscible displacement processes are a step towards total oil recovery. Conventional gas or water drives usually leave 25 to 5.0 per cent of the oil as residual in the swept portion of the reservoir. This residual can be eliminated if the oil is driven by a fluid with which it is miscible. At some reservoir conditions natural gas will become miscible with the oil. This is the "high pressure gas process".' More often, the oil does not contain enough light hydrocarbons to cause the gas to become miscible with the oil at reasonable pressures. In these cases a small band of fluid which is miscible both with the oil and gas must be kept between them2. Less than 2 per cent of the reservoir volume of the slug material is needed to keep the displacement miscible. Both processes work in the same manner, recovering all of the oil in the portion of the reservoir contacted by the injected fluids. The only difference is the manner in which the miscibility between the oil and the injected gas is obtained. Previous publications have contained detailed descriptions of these processes.1,2,3,4 However, total displacement of the oil in the swept region does not guarantee an efficient recovery process. The amount of oil to be recovered is also determined by the fraction of the reservoir contacted by the flood. This fraction is largely determined by the mobilities of the fluids. (The fluid mobility is the permeability of the rock to that fluid divided by the fluid's viscosity, k/p). This. dependence of the fraction swept on the mobility ratio has been shown in previous studies. Fig. 1 shows the ultimate fraction swept in a five-spot system as a function of the mobility ratio. The small drawings show the location of the areas left unswept for two different mobility ratios. The ultimate fraction of the reservoir swept is here considered to be attained when the producing stream contains less than 5 per cent oil at reservoir conditions. THE GAS-DRIVEN MISCIBLE DISPLACEMENT Since there is no oil left in the swept region after miscible displacement the mobility in this region is very high. It is often 50 times the mobility in the unswept regions. This means that the fraction of the reservoir contacted by the injected fluid will be less for a gas-driven miscible displacement than for a conventional water or gas drive. For a five-spot injection system, water would contact the entire reservoir volume, and the low pressure gas would contact about 90 per cent of this volume, while a gas-driven miscible displacement would only contact about 65 per cent of the reservoir. This poor sweep efficiency often offsets the benefits obtained through miscible displacement. Fig. 2 shows what the recovery curves for the three processes might look like for a five-spot system. The curves show the fraction of the in-place oil recovered as a function of reser-

By B. H. Caudle, A. B. Dyes

In a recent publication' it was shown that wells with a free surface in a homogeneous gravity-drainage reservoir have a hyperbolic decline with index n '. This paper reports efforts to extend this theory to practical field cases. As a first step in making the extension, a large number of lease production curves from gravity-drainage fields in the stripper stage were fitted with the general hyperbolic decline equation, q/q, = 1/(1 + y) y and n in each case being determined by statistical methods. It was found that this equation is reliable for decline curve extrapolation for periods up to at least 25 years. The resrilts obtained were disturbing in the sense that values for the index, n, were often smaller than one, which fact leads to the prediction that a well would produce all infinite amount of oil in infinite time. A theoretical investigation based on the results of the previous paper provides a suitable explanation for these low values of n. It suggests that these are obtained if production occurs from two or more layers of diflerent permeabilitv and thicktress or two lnyers having different skin effects. It is predicted that as the more permeable layers are exhausted, the index of decline, n, will rise from a value smaller than 2, to n = 2. A practical method is given for fitting the general hyperbolic decline Eq. 6 to production decline data. When this method is used, the index of decline observed in gravity-drainage reservoirs in the stripper stage when wing only data over early years 01 decline usually agrees quite closely with that determined using data over 25 years or more. This indicates that the hyperbolic decline equation is reliable for predicting behavior of such reservoirs. A method is also suggested for taking into account any changes in the value of the index INTRODUCTION As a result of both experimental work with models and theoretical deductions, it was recently concluded' that wells having a free surface in a homogeneous reservoir producing by pure gravity drainage should have a hyperbolic decline with n = 2. The present report details efforts to extend this work on ideal homogeneous reservoirs to actual field cases. It should be remembered that both the previous work and the present extension are based on the assumption that the reservoir is at very low pressure, and hence, nearly all the potential available for driving fluid to the well is the gravity head. Attempts to apply conclusions of this study to reservoirs which have considerable pressures supplementing the gravity drive may lead to difficulty. WELLS WITH NO FREE SURFACE In a dipping gravity-drainage reservoir such as that shown in Fig. 1, some of the down-dip wells have no free surface. The previous study' showed that the up-dip wells, which have a free surface, should follow Eq. 1. The following approximate analysis, of a type which was justified experimentally for up-dip wells in the work just mentioned, leads to results for the down-dip wells.

By L. H. Simons, H. H. Spain

I NT RODUCTION Theory indicates that linear water-floods should exhibit scaling and stabilization properties in both oil-wet and water-wet porous media'. Experimental verification of these proper-ties in oil-wet media was obtained some time ago', and more recently, Perkins', employing a high permeability unconsoli dated sand and a low oil-to-water viscosity ratio, has obtained flooding data in a water-wet medium which follow the pattern of scaling and stabilization. Also, Root and Calhoun3 have found similar trends in the displacement of gas by liquids from unconsolidated sands. In general, however, the experimen-tal evidence appears still somewhat conflicting since scaling and stabilization of water floods in water-wet media has not been observed in most of the studies to date'.".'. The purpose of this paper is to provide a more comprehensive picture of waterflood behavior in water-wet media and to corroborate the validity of the scaling and stabiliza-tion concepts. The need for differentiating between the intrinsic nature of water-oil displacements inside a porous medium and perturbating secondary end effects is discussed, pertinent experimental procedures are outlined and the results of flooding tests conducted at low and at high oil-to-water viscosity ratios on con- solidated, water-wet Alundum cores are analyzed. END EFFECTS IN WATER-WET MEDIA Outlet End Effects The peculiar feature of the outlet end effect in water-wet cores is that it not only results in an excessive wetting phase saturation at the outflow face,'.' but that it also retards the moment of water breakthrough: i.e., causes an undue delay in the appearance of water in the produced effluent. Conditions existing in a water-wet core when water first arrives at the outflow face are schematically shown in Fig. 1. At this flooding stage the saturation distribution, Fig. la, has not yet been distorted by the outlet end effect. A water-invaded pore at the outlet core face, opening into the oil-filled void space between the end of the core and the end plate of the flooding cell, is illustrated in Fig. lb. The curvature of the water-oil interface is concave toward the outlet, and the pressure on the water side of the interface is lower than on the oil side by an amount equal to a certain capillary pressure (Fig. lc). Before the water (of this particular pore) can be produced from the core, it is apparent that the pressure in the water phase must exceed that in the oil outside the core. Accordingly. when water first reaches the outlet face, it will not leave the core but will accumulate inside, and only oil will be produced. Eventually, as the water injection is continued. sufficient pressure is built up in the water to reverse the curvature of the interfaces in the outlet pores, and only then will water be produced from the core. This stage, illustrated in Fig. 2. corresponds to the conventional "water breakthrough" observed in flooding experiments. For purposes of discussion, it is convenient to distinguish between the flooding stage at which water first arrives at the outlet face of a core (Fig. 1) and the stage at which water is first produced from the core ( Fig. 2). The former will be referred to as "water arrival" and the latter water breakthrough. In oil-wet media

By Homer N. Mead

This report develops equations for decline curve analysis based upon the premise that the rate of change of the reciprocal of decline for succeeding time intervals is constant when the reservoir is produced under a fixed set of conditions. A method is shown for predicting maximum production rate against cumulative production for any reservoir. A method is also presented for predicting future production rate by an analysis of past production performance after decline has been esrahlished. INTRODUCTION No significant contribution to the analysis of decline curves by the loss-ratio method has been made since J. J. Arps paper1. The concept of loss-ratio that was developed by Arps and his contemporaries has been redefined in another way and the concept of instantaneous loss-ratio at time zero has been added. In the mathematical developmerit by Arps, production rate was related as a continuous function with time and his equatlons were developed on that basis. In this paper, production is considered to be a series of segments for equal time intervals and equations have been developed based upon these finite differences. It is realized that decline curve analysis is not the answer to all predictions of reservoir behavior. However, as production must decline from an initial maximum rate to zero in any reservoir, if such decline can be expressed as an infinite series, this series should accurately predict production. Decline curve analysis should be considered a valuable tool that may be used in conjunction with predictions of future recoveries by other methods. Various uses of the decline curve method will be discussed in this paper. Equations predicting production rate at any time and cumulative production by exponential and hyperbolic decline will be developed. Characteristic values of b related to various types of drive will be discussed. After that a hypothetical reservoir will be studied and ultimate recovery by natural depletion and pressure maintenance compared, then actual field examples will be shown and discussed. The rate-cumulative curve introduced by H. N. Marshs in 1928 is an important one for use in estimating future production rate and ultimate recovery. This graphical method is the one that is used in conjunction with the equations to be developed for predicting production behavior. EXPONENTIAL DECLINE In exponential decline each succeeding production rate per unit of time is a constant percentage of the production rate just before it. Since (1 — r) is the common ratio, where r is the constant decline rate, this condition may be expressed as a geometrical progression; thus Pn = P (1 - r)n-1 . ......(1) Where r is less than unity, r Since P1 (1 - r)n = Pn+1 P1 - Pn+1 C =P1-Pn+1/r........(2) LOSS-RATIO In this paper the calculation for loss-ratio has been changed from the method presented by Johnson and Bollens2. The first loss-ratio to be calculated is a,; it is the initial production per unit of time after decline has set in divided by the difference between the initial and the second production rate. This loss-ratio is for the first time interval. After b (which is the constant change

Jan 1, 1957

By A. M. Rowe, I. H. Silberberg

A computer program was written to predict the phase behavior generated by the enriched-gas-drive process. This program is based, in part, on a new concept of convergence pressme, which is then used to select vapor - liquid equilibrium ratios (K-factors) for performing a series of flash calculations. The results of these calculations are the equilibrium vapor and liquid phase compositions which define the phase envelopes. The program was used to predict phase envelopes for 11 different hydrocarbon systems on which published experimental data were available. The predicted and experimental results compare favorably. INTRODUCTION The reservoir engineer is frequently faced with the problem of predicting what will happen if gas is injected into a reservoir. One aspect of this, general problem is predicting the phase changes that will occur when a non-equilibrium gas displaces a reservoir fluid. In particular, a "dry" gas, upon displacing a volatile oil, will pick up intermediate components from the oil. On the other hand, a "wet" gas, containing a high concentration of intermediates, will lose some of these components to a relatively low-gravity, non-eouilibrium1- crude. It is this latter Drocess. occurring in the enriched-gas displacement, which is treated in this paper. In the past, these phase changes have been determined experimentally and the results incorporated into various modifications of the Buckley-Leverett analysis.112 Such experimental work is time consuming, and the results are sensitive to numerous experimental errors. With the wide availability of high-speed digital computing equipment and numerous correlations pertaining to the vapor-liquid equilibria of hydrocarbon systems, it is now practical to calculate such phase behavior. This paper describes a computer program for performing these calculations. THE ENRICHED GAS DISPLACEMENT PROCESS Experimental results have shown that oil recovery can be significantly increased by enriching the displacing gas with intermediate hydrocarbon c0m~onents.3 The essential features of the phase behavior generated by this enriched-gas-drive process are commonly illustrated with ternary diagrams such as Fig. 1.4 In this figure, Gas D, which contains a high concentration of intermediate hydrocarbons with respect to the undersaturated Crude A, is injected into the reservoir. When D contacts A, gas goes into solution until the oil becomes saturated (Point. B). Further contacting of Gas D and saturated Oil B results in a Mixture C which separates into Vapor Y(c) and Liquid X(c). Liquid X(c) is contacted by additional Gas O, resulting in Mixture E which separates into Vapor Y(e) and Liquid X(e). Repeated contacts of the liquid by the injected gas will eventually result in Liquid X(4 of maximum enrichment existing in equilibrium with Gas Y(d). The equilibrium tie-line X(4 Y(4, when extended, passes through the Point D representing the enriched injection gas. For systems of more than three components, the predicted equilibrium states are dependent upon not only reservoir temperature and pressure, but also the compositions of the crude oil and injected gas. If the gas is sufficiently enriched, a miscible displacement is generated. Line If is tangent to the phas,e envelope at the critical point (Point Z) and represents the limiting slope of the tie-lines as the critical state is approached. Point 1 therefore represents the minimum enrichment of injection gas required to generate a miscible displacement. Point G represents the minimum enrichment required for initial miscibility of the injection gas with Crude A. Accra has presented a method to be used for prediction of oil recovery by the enriched gas displacement process.l To develop the phase behavior data needed, he designed the experimental procedure described in the following quotation from his paper: The original liquid was contacted by a volume of displacing gas and allowed to come to equi-

Jan 1, 1966

By A. M. Sarem, J. M. Campbell

Molecular refraction is introduced as a new and improved third parameter for prediction of the PVT behavior of hydrocarbon systems. This parameter, characterizing the complex as well as the pure hydrocarbon systems (1) is a direct measure of London dispersion forces which affect the PVT behavior; (2) can be measured directly, accurately and easily for complex as well as pure hydrocarbon systems; (3) varies significantly between the reference substances thus allowing accurate interpolation of PVT properties; and (4) has a fundamentally sound basis. Molal average molecular refraction was used with the pseudocritical mixing rule proposed by Leland and Mueller to predict the molal volume of liquid and vapor for 84 published data points on nC4-nCI0, Cl-nC5, Cl-nC4-nC10 systems in the region of coexisting vapor-liquid and at the critical point. The same technique was applied to predict the densities of liquid hydrocarbon mixtures in the two-phase and single-phase regions near the critical point for seven different systems each having different properties of the heptanes-plus fraction, but all simulating the fluids of either gas-condensate or volatile oil reservoirs. A modification of the Leland-Mueller mixing rule for the determination of pseudocritical properties is also presented which yields more accurate liquid density predictions than the original form when used with the proposed third parameter. INTRODUCTION The PVT behavior of multicomponent systems has received widespread attention in the literature in recent years. A large portion of the reported work has stemmed from attempts to overcome the inherent difficulties encountered when using the standard compressibility-factor chart which shows 2 as a function of reduced temperature T, and and reduced pressure p,. This chart assumes that z, is constant for all systems. The error in this assumption is not pronounced for pipeline-type gases high in methane and low in heavy ends, particularly when contained at moderate pressures. For such gases the error in the calculated z is usually less than 2 to 3 per cent. However, the reliability of this approach becomes somewhat nebulous for gas condensate and volatile oil systems at high pressure. One logical approach has been to recognize the variation of 2,. This idea was first suggested by Meissner and Seferianl and used by Lyderson, Greenkorn and Hougen2 to prepare extensive tables of thermodynamic properties. Other third parameters have been proposed by Stockmeyer, Kihara, Riedel, Lightfoot, Bloomer and Pitzer. These were tested by Satter and Campbel13 for many gas mixtures and it was shown that they were inter-related. All of these third parameters have a common fault — they cannot be directly evaluated for the heptanes-plus fraction. Consequently, one necessary requisite of an improved method for predicting the density of multicomponent hydrocarbon systems, was development of a third parameter that could be evaluated for that fraction. PREVIOUS MULTICOMPONENT DENSITY PREDICTION METHODS Sage and Lacey4 suggested a method of using partial molal volumes for computing the density of hydrocarbon gas mixture for pressures to 3,000 psi. In this technique, they approximate the complex multicomponent gaseous mixtures by a quarternary system of methane, ethane, propane and butane. Sage, Hicks and Lacey5 suggest a method of using partial molal volumes for computing the density of hydrocarbon liquids. The results agree within 3 per cent of the experimental values. However, the composition range is limited to about 10 per cent by weight of methane. Consequently, this correlation does not cover the low-molecular-weight liquid similar to natural gasoline and high-pressure gas condensates.

Jan 1, 1966

By P. B. Baxendell

The performance of a water-drive reservoir having a gas cap depends primarily on the movement of the gar-oil and oil-water contacts. The movement of the contacts during production depends in turn on fluid withdrawals and how the reservoir pressure changes as fluids are produced from the reservoir; that is, on how effectively the aquifer maintains pressure by replacing withdrawals. inasmuch as pressure changes, fluid withdrawals, contact positions, and produced gas-oil and oil-water ratios are interdependent, the analysis and prediction of the performance of a reservoir produced in a given way must take into account this interdependence throughout depletion. This paper presents an analysis which, within the limitations of the assumptions made, yields an engineering approach to predicting future performance based on reservoir pressure and production history. The most significant assumption is the method of extrapolation of future gas-oil and water-oil ratios. The extrapolation procedure smoothly increases both ratios to preselected values as the remaining oil column undergoes a specified decrease in thickness. This preselection is made on the busis of previous field experience in depletion of similar reservoirs under similar conditions. For computing future performance a vdumebic balance L combined with the differential equation defining pressure distribution in the aquifer to obtain positions of water-oil and gas-oil contacts. From these positions are extrapolated produced water-oil and gas-oil ratios. Reservoir performance can be investigated when oil production rates are dependent upon various factors including the performance of the reservoir itself. Examples of practical application of the procedure are included. INTRODUCTION To predict performance of water-drive reservoirs with gas caps and thin oil columns, it is necessary to describe the motions of the fluids within the reservoir during the entire production period to depletion. These motions depend on the pressure changes and on the withdrawal of oil, gas and water. Production of oil from the oil zone primarily causes water to move in to take its place; production of gas from the cap tends to cause oil to migrate into its place. In addition, the volumes of oil and gas remaining in the reservoir depend on changes in the pressure, since a decrease in pressure causes fluid expansion, gas liberation and oil shrinkage. A method of relating the future pressure to total withdrawals is used to describe the motions of fluids within the reservoir under conditions arising in possible modes of production. The analysis is based on two relationships and will reduce the problem to one amenable to digital computation. The first of these concerns the dependence of the pressure distribution in the entire aquifer furnishing the water drive upon the total reservoir withdrawals. This dependence is dictated by the permeability distribution and the extent of the aquifer; at present, such information is most readily obtained from the performance history by means of the resistancecapacitance reservoir analyzer. The second relation involves withdrawals, pressure in the reservoir, and movement of oil, gas, and water within the reservoir. The analysis is subject to certain simplifying assumptions that are necessary to permit solution of the problem. A comparison of the methods of this paper with those presently practiced is pertinent. One method of analysis is to use the reservoir analyzer to predict reservoir behavior based on aquifer characteristics determined from production history. Inasmuch as the total withdrawal rate depends upon the gas-oil and water-oil ratios, which depend in turn upon, among other things, the positions of the gas-oil and water-oil contacts, these ratios must be assumed in advance. In this paper these ratios are, instead, related to the computed positions of gas-oil and water-oil contacts. Thus, our method can be applied to problems in which the oil production rate is limited by produced gas-oil ratio, or to problems in which it is desired to determine the variable gas injection rate that will maintain the gas-oil or water-oil contact stationary. These problems cannot be worked satisfactorily on the analyzer. Another method presently used is based on a paper by Hurst.' Since his procedure is dependent upon using the solution of the heat flow equation, which requires constant permeability within the aquifer, and our procedure recognizes variations of permeability in

By J. J. Arps

n approach to the study of the electrical properlies of drilling muds, their cakes and their filtrates was made by treating the drilling fluid as a porous medium. Lluring this study it was found that the weight pararneter was omitted from previous investigations on electrical properties. Patnode in 1949 made a similar approach to ,his investigation. His work showed that there was a variation of resistivity relationships with per cent Puid present in a given suspension. This present study ex-tends Patnode's investigation to one in terms of the solids fraction of the drilling fluids. Previous investigations were made on low weight muds where the effect of the rolids fraction was obscured. The following equationr were fitted to data taken from 94 field muds and 47 laboratory muds (see Tables I and 2). The parameter, C, is a complex function of the solidr content, or mud weight, of a drilling fluid. It was also found that for all drilling muds examined. A nomograph is presented combining Eqs. I and 2. INTRODUCTION It is generally recognized that the materials present in drilling muds affect the relationship between mud filtrate and mud resistivities. However, most electric log interpretations proceed with the assumption that the mud resistivity and temperature are the only important parameters needed to obtain a filtrate resistivity or a mud cake resistivity. Lamont1 suggests that only average relationships exist between mud cake, mud filtrate and mud resistivities. He indicates that there is limited accuracy when these intercorrelated factors are used since no other measure-able variable is apparent. A previous paper by Patnode indicates that drilling fluids may be considered to be a system of fluids in solid particles which may be correlated electrically in a manner similar to rock solids. 1 he equation for the resistivity factor. F = (per cent fluid) is satisfied for a given clay type in water, but the value of m varies as different materials are used in the water. The resistivity factor is not constant for a given amount of clay. As is suspected it varies as the solids fraction conducts electrical current. Since both weighting material and clay solids are present in commonly used drilling fluids, it is expected that both materials contribute to the variation of the m factor in Eq. 3. An extension of Patnode's work is presented in this paper to show some of the interrelations which exist between the resistivities of mud cake, mud filtrate and mud. Use of these data enable a more accurate calculation of mud filtrate and mud cake resistivities if only the mud resistivity and weight are measured. EXPERIMENTAL The measurements of mud weight, fluid and solids resistivity, alkalinity, and chloride content of mud samples were made in this research. Mud samples representing drilling fluids from 94 Louisiana and Texas wells (Table 1) and 47 laboratory-constructed systems (Table 2) were used in the experimentation. The measurements of the mud properties were made both in the field and in the laboratory. The tests on the properties of weight, alkalinity and chloride ion were standard tests as described by API specifications. The resistivity measuring devices were constructed in the laboratory for this study. For details on the resistivity equipment see Fig. 1. The fluid cell