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By H. Kazemi, G. W. Thomas, M. S. Seth

The double-porosity model of Warren and Root7 for examining pressure drawdown and buildup phenomena in naturally fractured reservoirs has been extended to intetpret interference test results. Both analytical and numerical solutions are presented. The practicality of some of the assumptions in the Warren-Root model was also investigated. The standard procedure for interpreting interference tests in naturally fractured reservoirs beretofore employed an equivalent homogeneous reservoir model. A comparison of our results with the equivalent homogeneous model indicates the latter is inadequate for early time responses. For late time responses, however, the simpler equivalent homogeneous model yields favorable results. Finally, the effects of well spacing and the magnitude of the interporosity flow parameter are briefly cited. INTRODUCTION Interference tests can be used to provide reservoir information not obtainable from conventional pressure drawdown and buildup tests. The technique involves measuring the response in one or more wells arising from a pressure perturbation introduced into another well. It has been. successfully applied to single-layer, homogeneous, unfractured, isotropic and anisotropic formations to arrive at values of well storage capacity and a measure of directional permeability.l The techniques of interference testing usually can be extended to heterogeneous systems, provided the permeability contrast between any existing rock layers is not excessive.9 In such cases, the practical interference effects in an observation well of an interconnecting heterogeneous reservoir can be predicted using an equivalent homogeneous model. The calculated properties are then average values. Russell and Prats2 have demonstrated the utility of this approach for a single-phase, two-layer system. For naturally fractured reservoirs, some different approaches usually are necessary. The major departure from homogeneous or nearly homogeneous reservoir treatment is to characterize the fractured system as an entity having "double-porosity"; namely, we suppose the existence of both primary and secondary porosity, where the former is contained within a systematic array of identical rectangular matrix blocks and the latter is contained within an orthogonal system of continuous, uniform fractures (see Fig. 1). It is sufficient to assume that the permeability of the rock matrix is so small that movement of reservoir fluids to the producing wells is entirely through the fracture network. This assumption has been validated by Barenblatt3 and .employed by others?-7 Warren and Root7 used the double-porosity model to develop an interpretation of pressure drawdown and pressure buildup tests in naturally fractured

Jan 1, 1970

By James E. Berryman

A method is presented for calculating the pressure depletion performance of a gas-condensate reservoir from the original reservoir fluid composition using the laboratory determined pressure-production history and retrograde liquid accumulation as guides. Equilibrium vaporization ratios for this calculation were modified from published data until the calculated dew point pressure and retrograde liquid accumulation agreed with the values observed in the laboratory. The calculated results are compared with the produced gas composition determined by low temperature analysis during constant volume laboratory experiments of pressure depletion behavior. This comparison shows that the calculation procedure is adequate for determination of gas phase composition. Data from field separation tests in low temperature separators are presented which verify the applicability of published equilibrium vaporization ratios for a wide range of separator conditions. INTRODUCTION A reliable prediction of the pressure depletion performance of a gas-condensate reservoir is necessary in determining reserves and evaluating field separation methods. The predicted performance is also used in planning future operations and studying the economics of projects for increasing liquid recovery. Because the flow of fluid in a volumetric gas-condensate reservoir is usually single phase flow, the pressure-depletion per-formake is primarily a function of the phase behavior of the reservoir fluid. The properties of a gas-condensate system of major importance are: (1) compressibility factor at reservoir temperature and pressure; (2) dew point pressure; (3) composition changes of the vapor phase with pressure depletion; (4) recovery of original in-place hydrocarbons at any pressure; and (5) retrograde condensate accumulation. These properties are most generally determined by a pressure depletion-type experiment on recombined separator gas and liquid samples in the laboratory. ldeally the pressure-depletion performance can be calculated from the composition of the original reservoir fluid if the reservoir temperature and initial pressure are known.1,2,3 It has been reported "... comparison of such computations with depletion-type ex-periments on recombined samples in the laboratory have yielded results that appeared to be entirely adequate.". The success of such calculations is determined by the equilibrium vaporization ratios chosen, especially those for the higher boiling point hydrocarbons. It is the purpose of this paper to present a method for computing the change in composition of the vapor phase during pressure depletion by production using laboratory determined properties of the system to guide the calculation and to illustrate the use of the method with data from the Cockfield "D" sand of Washington field, St. Landry Parish, La. Results of field tests and laboratory analysis are included. As a result of this reservoir fluid study, a cycling program has been evaluated, the fidd has been unitized, and the cycling started in July, 1956.

Jan 1, 1958

By William Hurst

Muskat's depletion performance equation is here derived considering the expansion behavior of the reservoir hydrocarbon system and a simple fractional-flow equation. This nietkod of derivation leads logically to the two extensions that follow. The first of these is concerned with gravity segregation in a depletion-drive reservoir. The second is concerned with including an empirically determined tern? for water influx in the performance equation. The more general equation for gravity segregation when there is a primary gas cap and empirically-determined water influx is stated for completeness. These equations have been found useful in reservoir performance calculations in Eastern Venezuela. A discussion on the methods of solving these equations follows, and considers firstly the effect of taking finite intervals in the numerical integration, and sec-ondly, methods of incorporating the time functions involved in segregation in with the expansion behavior. The paper concludes with a brief general discussion on further extensions to the depletion performance equation. INTRODUCTION The two fundamental sources of energy by which oil is produced from a reservoir result from pressure depletion inside the boundaries of the reservoir and fluid encroachment across the boundaries of the reservoir. The wells in either case form low pressure outlets through which oil and gas may be produced by the expansive force of the reservoir fluids and associated encroaching fluids. When the reservoir pressure is higher than the bubble-point pressure of the oil, so that there is no free gas in the reservoir, these expansive forces are the only ones available for the production of oil. However, when the reservoir pressure is less than the bubble-point pressure of the oil, free gas is vaporized as the pressure falls. With both oil and free-gas phases present, the additional forces of gravity and capillarity may operate on the gas-oil system, as they have previously operated on the oil-interstitial water system. Gravity tends to segregate the free gas from the oil due to their density difference. Capillarity opposes and eventually balances gravity as the more extreme free gas and oil saturations are reached, preventing the independent move- ment of free gas until it is above a certain saturation, and the independent movement of oil when it is below a certain saturation. The type of depletion performance equation chosen for predicting the future performance of a reservoir depends on the amount of past history available. When the reservoir is somewhere past the halfway mark in depletion, some form of decline curve is often used. With less past history, material balance equations which incorporate empirical factors based on the past performance are often used. When, however, the amount of past history is small, the Muskat depletion performance equation will usually be used. The distinguishing feature of this type of equation is that empirical factors based on the over-all or macroscopic reservoir behavior are almost or entirely absent. Each parameter affecting the reservoir performance is ascribed an independent set of values based on measurements made on laboratory samples; that is, incorporating microscopic empirical factors. In establishing Muskat-type depletion performance equations, it is necessary to consider the reservoir as consisting of a number of associated blocks, in each of which the saturations and pressures may be considered uniform, and in each of which all substances have uniform pressure-volume characteristics. Thus, a primary gas cap can usually be considered as one block and an aquifer as another. Gravity segregation may be negligible for practical purposes when the rock and oil properties are adverse and/or the dip or thickness of the reservoir is too small. In this case the whole oil leg may be considered as one block, except in very large reservoirs. In very large reservoirs the fluid and rock properties may vary enough, particularly in the dip direction, for it to be necessary to divide the oil leg into a number of blocks, in each of which the relevant quantities may be considered uniform. When gravity segregation of the oil and free gas is not negligible, it is necessary to consider the space occupied by the initial oil leg as divided into two blocks, a secondary gas cap and an effective oil leg, in each of which saturations may be considered to be uniform. The total volume of these two blocks is thus constant, but the secondary gas cap grows continuously at the expense of the oil leg. Muskat1 derived depletion performance equations for the basic case of an oil reservoir with closed boundaries and without segregation, and for the case of an oil

By M. Sheffield

This paper presents a technique lor predicting the flow of oil, gas and water through a petroleum reservoir. Gravitational, viscous arid capillary lorces are considered, and all fluids are considered to be slightly compressible. Some theoretical work concerning the fluid flow in one-, two- and three-space dimensions is given along with example performance predictions in one- and two-space dimensions. INTRODUCTION Since the introduction of high speed computing equipment, one of the goals of reservoir engineering research has been to develop more accurate methods of describing fluid movement through underground reservoirs. Various mathematical methods have been developed or used by reservoir engineers to predict reservoir performance, 193-5,7,8 The work reported in this paper extends previously published work on three-phase fluid flow (1) by including a rigorous treatment of capillary forces and (2) by showing certain theoretical mathematical results proving that these equations can be approximated by certain numerical techniques and that a unique solution exists. DISCUSSION The method of predicting three-phase compressible fluid flow. in a reservoir can be summarized briefly by the following steps. 1. The reservoir, or a section of a reservoir, is characterized by a series of mesh points with varying rock and fluid properties simulated at each mesh point. 2. Three partial differential equations are written to describe the movement at any point in the reservoir of each of the three compressible fluids. All forces influencing movement are considered in the equations. 3. At each mesh point, the partial differential equations are replaced by a system of analogous difference equations. 4. A numerical technique is used to solve the resulting system of difference equations. Capillary forces have been included in two-phase flow calculations.1,4,8 The literature, however, does not contain examples of prediction techniques for three-phase flow that include capillary forces. Where capillary forces are considered, each of the three partial differential equations previously discussed has a different dependent variable, namely pressure in one of the three fluid phases. Therefore, three difference equations must be solved at each point in the reservoir. Where large systems of equations must be solved simultaneously, an engineer might question whether a unique solution to this system of equations actually exists and, if so, what numerical techniques may be used to obtain a good approximation to the solution. It is shown in the Appendix that a unique solution to the three-phase flow problem, as formulated, always exists. It is also shown that several methods may be used to obtain a good approximation to the solution. The partial differential equations and differenceequations used are shown in the Appendix. Matrix notation has been used in developing the mathematical results. Two sample problems were solved on a CDC 1604. They illustrate the type of problems that can be solved using a three-phase prediction technique. SAMPLE ONE-DIMENSIONAL RESERVOIR PERFORMANCE PREDICTION A hypothetical reservoir was studied to provide an example of a one-dimensional problem that can be solved. Of the several techniques available, the direct method of solution as shown in the Appendix was used. The reservoir section studied was a truncated, wedge-shaped section, 2,400 ft long, with a 6' dip. (A schematic is shown in Fig. 1.) This section was represented by 49 mesh points, uniformly spaced at 50-ft intervals. The upper end of the wedge was 2 ft wide, and the lower end was 6 ft wide. The section

Jan 1, 1970

By V. J. Kniazeff, S. A. Naville

The problem of unsteady-state condensate-gas flow through porous media leads to a set of second-order non-linear partial differential equations. Such a set of equations is numerically solved in the case of radial two-phase flow around a well, taking into consideration both the thermodynamical properties of the fluid and the mechanical properties of the reservoir. The fluid properties, reflecting the PVT relationship of the gaseous and liquid phases, are expressed by using the partial specific masses of the two main separator products in these phases. The flow properties of the reservoir rock are expressed by the generalized Darcy's law for the liquid phase and by a quadratic relationship between the rate of flow and the pressure gradient for the gaseous phase. The numerical solution of the equations for pressure and saturation US radius and time is worked out through programs written for a computer. The evolution of bottom-hole pressures, well productivities or- deliverabilities and effluent compositions with the depletion of the reservoir is easily derived. The application to the Saharian gas-condensate field Hassi R'Mel led to a better understanding of the drainage mechanism. A zone of fairly high liquid saturation develops around the wells, reducing the effective permeability, and represents a loss of condensible products in addition to the PVT-like retrograde condensation. lnside this zone, near the well, the deviation from Darcy's law for the flow of the gaseous phase governs the well deliver-ability. A back-pressure test has been computed and correlates with the field results. INTRODUCTION Two-phase flow of volatile hydrocarbons, like condensate gas or light crude oil, may be treated as the flow of a binary mixture by an arbitrary division of the chemical components into two groups. This is translated into two equations of mass continuity, which constitute a set of relation- ships for the pressure and the saturations vs the space coordinates and the time. The equations contain the laws governing the composition and the motion of the phases. The problem so defined is solved with the assumption that the compositions of the phases at any pressure are respectively the same as those observed in a PVT measuring cell under differential liberation. In a first series of computations, it was assumed that the flow obeys the generalized Darcy's law. A satisfactory representation of the retrograde condensation around the well was thus obtained. In addition, the trend toward decreasing effective permeabilities was obtained, and the computed composition of the effluent checked the laboratory values. However, it has not been possible within this basic assumption to represent the non-linear relationship between the production rate and the bottom-hole pressure drawdown as observed for gas wells in the field. Following the advice of A. Houpeurt it was decided to consider the relative permeability to gas as a function of the velocity of the gas Phase.1 The necessary physical determinations were made by E. COstaséque using a method devised by A. Houpeurt and R. Iffly. As numerical processing of the equations progressed, several difficulties were encountered which were overcome through collaboration with the computer manufacturer. This mathematical model of two-phase flow in porous media had been primarily intended for and extensively applied to the case of the Hassi R'Mel gas-condensate field, operated by SN Repal for SEHR, a joint subsidiary of SN Repal and CFP (A). The programs have also found their applications to forecast the behavior of several fields in the Sahara area containing light and volatile hydrocarbons. BASIC EQUATIONS We will consider a zone in the porous reservoir where the flow properties and the in situ composition of the fluids can be assumed to be uniform. A part of the equations of transient flow can be written with the specific gravity of the fluids being taken equal to the sum of the contributions due to the

Jan 1, 1966

By L. J. Health

Synthetic rock with predictable porosity and permeability has been Prepared from mixtures of sand, cement and water. Three series of mixes were investigated primarily for the relation between porosity and permeability for certain grain sizes and proportions. Synthetic rock prepared of 65 per cent large grains, 27 per cent small grains and 8 per cent Portland cement, gave measurable results ranging in porosity from 22.5 to 40 per cent and in permeability from 0.1 darcies to 6 darcies. This variation in porosity and permeability was caused by varying the amount of blending water. Drainage-cycle relative permeability characteristics of the synthetic rock were similar to those of natural reservoir rock. INTRODUCTION The fundamental behavior characteristics of fluids flowing through porous media have been described in the literature. Practical application of these flow characteristics to field conditions is too complicated except where assumptions are overly simplified. The use of dimensionally scaled models to simulate oil reservoirs has been described in the literature. 1-3 These and other papers have presented the theoretical and experimental justification for model design. Others have presented elements of model construction and their operation. In most investigations the porous media have consisted of either u ncon soli da ted sand, glass beads, broken glass or plastic-impregnated granular substances—materials in which the flow behavior is not identical to that in natural reservoir rock. The relative permeability curves for unconsoli-dated sands differ from those for consolidated sandstone. The effect of saturation history on relative permeability measurements is discussed by Geffen, et a1.4 Wyga15 has shown quite conclusively that a process of artificial cementation can be used to render un consolidated packs into synthetic sandstones having properties similar to those of natural rock. Many theoretical and experimental studies have been made in attempts to determine the structure and properties of unconsolidated sand, the most notable being by Naar and Wygal.6 Others have theorized and experimented with the fundamental characteristics of reservoir rocks. This study was conducted to determine if some general relationship could be established between the size of sand grains and the porosity and permeability in consolidated binary packs. This paper presents the results obtained by changing some of the factors which affect the porosity and permeability of synthetically prepared sandstone. In addition, drainage relative permeability curves are presented. EXPERIMENTAL PROCEDURE Mixtures of Portland cement with water and aggregate generally are designed to have certain characteristics, but essentially all are planned to be impervious to water or other liquids. Synthetic sandstone simulating oil reservoir rock, however, must be designed to have a given permeability (sometimes several darcies), a porosity which is primarily the effective porosity but quantitatively similar to natural rock, and other characteristics comparable to reservoir rock, such as wettability, pore geometry, tortuosity, etc. Unconsolidated ternary mixtures of spheres gave both a theoretically computed and an experimentally observed minimum porosity of about 25 per cent.7 By using a particle-distribution system, one-size particle packs had reproducible porosities in the reproducible range of 35 to 37 per cent.6 For model reservoir studies of the prototype system, a synthetic rock having a porosity of 25 per cent or less and a permeability of 2 darcies was required. The rock had to be uniform and competent enough to handle. Synthetic sandstone cores were prepared utilizing the technique developed by Wygal.5 Some slight variations in the procedure were incorporated. The sand was sieved through U. S. Standard sieves. A 18-20 sand denotes that the sand passed the Number

Jan 1, 1966

By J. C. Melrose

The contact angle is one of the boundary conditions for the differential equation specilying the configuration of fluid-fluid interfaces. Hence, applying knowledge concerning the wettability of a solid surface to problems of fluid distribution in porous solids, it is important to consider the complexity of the geometrical shapes of the individual, interconnected pores. As an approach to this problem, the ideal soil model introduced by soil physicists is discussed in detail. This model predicts that the pore structure of typical porous solids will lead to hysteresis effects in capillary pressure, even if a zero value of the contact angle is maintained. The model is generalized to situations in which the contact angle takes on values between zero and 40 degrees. For the imbibition branch of the capillary- pressure function, the model predicts a considerable departme ftom the usually assumed cos 0 relationship. In fact, according to the model, it is possible that a displaced wetting phase will not be able to reimbibe, even when the contact angle does not exhibit hysteresis. INTRODUCTION The subject of capillarity in porous media has long been of interest in many branches of engineering and applied science. The earlier investigators were those concerned with the physics of soils. 1-5 More recently, petroleum engineers and others dealing with the problems of petroleum production from reservoir rock have given much attention to the subject.6-10 Also, important additions to the literature of capillarity have been contributed from the field of chemical engineering.11-12 These attest to the wide range of industrial applications in which capillary phenomena play a role. The present paper is concerned with the role which wettability plays in capillary action in porous media. As is well known, capillary-rise (or capillary pressure) phenomena have frequently been interpreted with the aid of a model employing the concept of a cylindrical capillary tube. This approach has enjoyed a certain degree of success in correlating experimental results.13 The generalization of this model, however, to situations which involve varying wettability, has not been established and, in fact, is likely to be unsuccessful. In this paper another approach to this problem will be discussed. A considerable literature relating to this approach exists in the field of soil science, where it is referred to as the ideal soil model. Certain features of this model have also been discussed by Purcel114 in relation to variable wettability. The application of this model, however, to studying the role of wettability in capillary phenomena has not previously been attempted in detail. In the present paper, additional features of the model are introduced. These features are critical in determining the quantitative behavior of the model. GENERAL FEATURES OF CAPILLARY HYDROSTATICS BASIC PRINCIPLES When the interstices of a typical porous solid are occupied by two or more immiscible fluid phases, the fluids are microscopically commingled. Hence, fluid-fluid interfaces are found within a certain fraction of the pore openings. The fundamental equation of capillarity specifies the configuration of these fluid-fluid interfaces. This is known as the Laplace equation, when derived from mechanics, and as the Gibbs-Kelvin equation when derived thermodynamically.15 Given two fluid phases, a and ß, in hydrostatic equilibrium, the Laplace equation states that the respective fluid pressures in regions close to the interface are related by Here oaß is the surface or interfacial tension and rl and r2 are the principal radii of curvature of the surface or interface. The pressure difference, Pa-Pß, is the capillary pressure, PC. As Buff15 has shown, Eq. 1 states the condition for hydrostatic equilibrium within the two-phase confluent region, which is referred to as the interface. It thus can

Jan 1, 1966

By T. G. Richardson, F. M. Perkins

In the recent paper of Richardson and Perkins entitled "A Laboratory Investigation of the Effect of Rate on Recovery of Oil by Water Flooding,"' the authors found very little appzrent effect on oil recovery of having free gas saturation present prior to water flooding in unconsolidated sands. We agree that the effect in this particular system was small but do not feel that the data presented should be regarded as disproving work of others which has shown an appreciable effect. The authors do not make such a claim, but neither do they advance an explanation of the difference. In our experience the following factors must be carefully considered when studying the relation between residual oil saturation after water flooding and initial free gas saturation: (1) the magnitude of the inherent residual oil saturation, i.e., the residual oil saturation with no free gas; (2) the increase in oil-water viscosity ratio with pressure lowering if the gas saturation is obtained by solution gas drive; and (3) the oil shrinkage which occurs with pressure lowering and gas evolution. The system studied by Richardson and Perkins gave residual oil saturations as low as 20 per cent without free gas present, and one should perhaps not expect much further lowering. As an illustration of the magnitude of the effect when the residual oil saturation is inherently higher we submit the accompanying figure. These data were obtained on an unconsolidated sand under elevated pressure in a system similar to the one described by Richardson and Perkins. Gas saturation was established by gas drive before water flooding. The viscosity ratio was 600:1 under the conditions of the displacement. Here the gas was extremely effective in reducing residual oil saturation. This fact was also indicated by Kyte, et al.2 Their plots of residual oil saturation vs initial gas saturation show trends of greater free gas effect with higher oil viscosities. The pressure-dependent effects were not discussed by Richardson and Perkins. They should be separated from the free gas effect if the objective of the study is a correlation of residual oil saturation vs initial free gas saturation. The increase in oil viscosity when gas is released from solution tends to increase residual oil saturation. This effect is rather small in the viscosity range of kerosene. A Buckley-Leverett calculation, assuming an increase in viscosity ratio from 1.2 to 1.8, indicates an increase in residual oil saturation of about 1 per cent of pore volume. Thus, one would expect this system, if operated at constant viscosity ratio, to show a slightly greater free gas effect. As Dyes- as shown, oil shrinkage can offset the beneficial effect of free gas on waterflood oil recovery, if the gas saturation is established by depletion. Since

Jan 1, 1958

By G. M. Webster, G. E. Dawsongrove

The use of percussion-type side-wall cores as an aid in detection and evaluation of hydrocarbon shows and in examination of rock properties has become increasingly popular in recent years. Sidewall coring has an advantage over normal bottom-hole coring methods in that the zones may be surveyed with the electric log prior to selection of sample depths. This allows an efficient sampling of all zones of interest, a procedure not possible with bottom-hole coring techniques unless the costly procedure of continuous coring is utilized. Percussion sidewall coring presently provides the most rapid and lowest cost method of obtaining sidewall cores. It has been recognized for some time that certain discrepancies exist between core analysis data obtained from percussion cores and data obtained from other types of core Samples. This paper presents existing field and laboratory data that were available to the authors. Several examples of comparative core analysis data obtained from various fields are presented along with photomicrographs of thin sections of conventional and percussion cores. The results of this investigation are presented to throw more light on the possible disturbed condition of percussion sidewall cores. SAMPLE DISTURBANCE AND FRACTURING The principal cause of discrepancy between conventional core analysis data and percussion sidewall core analysis data is believed to be the shattering and re-adjustment of the grains due to the large impact shearing forces produced by mechanics of percussion coring. Figs. 1 through 5 are photomicrographs of thin section slides of formation samples obtained by means of diamond cores, conventional cores, and percussion sidewall cores. Figs. 1 and 2 show samples of the

By N. Mungan, J. L. Thompson

Laboratory displacement tests were performed to study oil recovery elficiency by gravity drainage in fractured systems under miscible conditions. The porous media used were cylindrical Berea and Blue Jacket sandstone cores containing a number of well-defined, artificially formed, vertical and subvertical fractures. Butane and Soltrol 130 were the two miscible fluids used. The purpose of the study was to examine the influences of the displacement rate, fracture density, fracture orientation, fracture permeability, matrix permeability, crossflow, core length and connate water on the oil recovery. It was found that displacement rate, matrix permeability and the subvertical fractures affected oil recovery most. The critical flow rate, based on the matrix permeability, was found to be a significant factor in the process. For displacement rates below the critical flow rate, the oil recovery efficiency appeared to be unaffected by the density of the subvertical fractures. At the high displacement rates, the fracture density becomes important, with the recovery being the most efficient in the core having the greatest number of the subvertical fractures. The magnitude of the fracture permeability, the fracture orientation, the core length and the connate water have little effect on the oil recovery efficiency. INTRODUCTION Gravity drainage under miscible conditions from relatively thick reservoirs can be a very efficient recovery process, especially at low flow rates where the gravity forces are dominant and, consequently, the adverse viscous fingers associated with the unfavorable viscosity ratio are minimized or eliminated. Such a process might involve injection of an LPG or some solvent bank at the crest and then driving the bank with dry gas. For some favorable combination of reservoir temperature, pressure and oil composition, there may be no need to inject any solvent bank, since enrichment of the dry gas by the light ends of the crude creates an in-situ solvent bank. Slobod and Howlett2 have studied the effects of gravity segregation in vertical unconsolidated porous media under miscible conditions in the laboratory. The main variables in their study were the viscosity ratio, the density differences and the rate of flow. In this study, the objective was to find the influence of fractures on gravity drainage under miscible conditions. A greater number of related variables were also studied. LABORATORY STUDY CORE DESCRIPTION Displacement tests were performed mostly in cylindrical Berea sandstone cores. In a few cases in which a much lower matrix permeability was desired, Blue Jacket sandstone cores were used. The vertical fractures were formed by cutting the cores with a thin circular blade or by parting them along a bedding plane. The vertical fracture plane always contained the axis of the cylindrical core. The subvertical fractures, cut with a saw, were inclined 45' from the horizontal. The line of intersection between the planes of the vertical and subvertical fractures was parallel to the circular faces of the cores in all cases except one, as depicted by VF3HF-2 in Fig. 1. Fracture geometry of the other cores is also given in Fig. 1. Tables 1 and 2 give the physical properties of the solid and fractured cores, respectively. The suffixes and prefixes shown for each core have been used throughout the paper so that the reader may discern the fracture geometry from the core numbers.

Jan 1, 1970

By Charles R. Stewart, William W. Owens

Reservoir performance predictions based on laboratory core test data assume that fluid flow is laminar for the laboratory test. A study has been made to determine the validity of this assumption for laboratory tests on various types of porosity found in producing limestone formation. Data are presented which show that turbulence and slippage can occur during laboratory tests on hetero-geneous-type porosity limestones, thus causing serious errors in measured single-phase permeabilities and two-phase relative permeability characteristics. In single-phase flow tests it is possible to eliminate turbulence and correct for slippage or to eliminate both factors by controlling test conditions. It is not always possible to control test conditions and thereby eliminate turbulence and slippage in two-phase .flow tests. A correction method is presented which can be used to calculate the true two-phase laminar flow relative permeability characteristic even though furbulence and slippage exist. .INTRODUCTION It is customary to make use of Darcy's law and modifications of this law, together with laboratory data on formation core samples to predict the performance of producing reservoirs. Such predictions are based on an assumption that fluid flow is in the laminar or streamline region for the laboratory test. It was the purpose of this inves- tigation to determine the extent to which turbulent flow may occur in laboratory fluid flow tests on hetero-geneous porosity limestones. Considering that turbulent flow conditions might exist in some laboratory fluid flow tests, additional emphasis was placed on the development of a method to correct for turbulence when laminar flow conditions could not be attained. FLUID FLOW CONCEPTS FOR POROUS MEDIA The Influence of Pore Geometry on Fluid Flow One of the more important factors influencing fluid flow in porous media is the geometry of the pore space which includes such characteristics of the pores as size, shape, distribution, roughness, uniformity, etc. In general, oil- and gas-producing formations can be divided into two broad types on the basis of pore geometry. One has been called sandstone-type porosity media, which is characterized by a small range in pore size, uniformity in shape of the pores, smooth pore surfaces and a regular and uniform distribution of pores. The other type has been called heterogeneous porosity media and is usually limited to the dolomites and limestones. This type is characterized by a wide variation in the size, shape, and distribution of the pores and rough, irregular pore surfaces. It is therefore apparent that conditions are much more favorable for turbulent flow* in heterogeneous-type porosity media than in sandstone-type porosity media. Interrelationship Between Turbulence and Gas Slippage In studying the problem of turbulent flow in laboratory tests on porous media, it is necessary to be aware of the interrelationship between slippage and turbulence for gas flow. As a result of slippage or the Klinken-berg effecta, apparent perrneabilities to gas are greater than the true value because there is no stationary layer of gas in contact with the walls of the flow channels. Gas slippage decreases as the mean free path of the gas molecules decreases. Since the mean free path of any gas decreases with increasing density, increases in static pressure result in lower apparent gas permeabilities. However, a reduction in gas permeability can also be due to turbulence. Therefore, in studying only turbulent flow in porous media, it is necessary to hold gas density, and slippage, constant or to reduce slippage to a negligible value by operating at high static pressures. Presentation of Laminar and Turbulent Flow Data A graphical relationship between permeability and a pseudo-Reynolds number, N,, will be used to show the two types of fluid flow, i.e., laminar and turbulent. The usual graphical method for such a description has been the use of friction factor-Reynolds number charts4. On such a logarithmic diagram, the laminar region appears as a straight line having a slope of 45 degrees. As the friction factor decreases and the Reynolds number increases, the turbulent region is reached and appears as a deviation from the 45-degree slope line. However, in petroleum engineering literature resistance of por-

By L. L. Handy

The most common method of identifying hydrocarbon-bearing strata in a well that penetrates many different formations involves measurement and interpretation of the electrical properties of the formations as determined by electrical logs. Even though this method is used extensively, and even though in a great many instances it is capable of indicating presence of oil or gas, situations arise for which it is extremely difficult, if not impossible, to deduce the presence of hydrocarbons. These situations may involve the following. 1. A thin formation, bounded by highly resistive formations, in which it is impossible to obtain the actual resistivity of the uninvaded zone with existing logging devices. 2. A formation in which invasion has been so extensive that a value for the uninvaded zone resistivity cannot be obtained. 3. A very shaly formation in which the resistivity index, I, is lower than that usually associated with productive formations. 4. Laminated formations comprised of thin productive sands separated by thin shale streaks in which the individual sand and shale streaks are too thin to permit measurement of uninvaded-zone resistivity with existing logging devices. 5. Productive formations in which the water saturation is high. To extend the utility of electric log interpretation to identification of hydrocarbons in all types of formations, there is strong incentive to find a method not subject to these limitations. Some time ago, in connection with research on the wettability of reservoir rock, an investigation was conducted in which the resistivities of cores were measured shortly after they were removed from a core barrel, and again after they had been extracted and restored to their original oil and brine saturation.' The resistivities after extraction were generally lower. Other tests made on the cores indicated that they were more nearly water wet after they were extracted; thus, it was assumed that the observed changes in resistivities were due to a change in wettability of the cores. Other experiments'," have shown that resistivities of rock samples are sharply dependent on wettability. These experiments have shown that oil-wet samples are more resistive than water-wet samples. To obtain an understanding of how the wetting properties of the surfaces of core material affect electrical resistivity, a series of experiments was conducted. Two groups of core samples were prepared for testing. One group contained brine, but no residual oil. The other group was saturated with brine, flooded with oil to a low water saturation, then flooded with brine to a final residual oil saturation. Resistivity measurements were made on each group. Both groups were then flooded with the original brine to which a chemical had been added that renders sand and clay surfaces preferentially oil wet, a so-called reverse-wetting agent. very little change in resistivity was observed in cores containing only water. The group containing residual oil, however, showed resistivity increases of 100 to 200 per cent. These experiments showed that the resistivity of a core containing oil could be altered by changing wet-tability of the core. Moreover, the possibility was introduced that reverse-wetting agents might be employed as the basis for a logging method for identification of oil-bearing strata. Since behavior of a porous rock containing gas and water might be expected to be similar to that of a rock containing oil and water, such a method should also be applicable to identification of gas-bearing zones. In principle the wettability of the invaded zone could be reversed without altering conductivity of the interstitial water or the hydrocarbon saturation therein. Those strata showing significantly increased invaded-zone resistivities would, therefore, contain hydrocarbons; those with no significant change would be filled only with water. Addition of a reverse-wetting agent to a hydro carbon -bearing zone which is, by nature, already preferentially oil wet would not result in an enhancement of its resistivity. It is generally believed, however, that most hydrocarbon-bearing strata are preferentially water wet.

By R. C. Hassinger, D. V. von Rosenberg

Transverse dispersion has received considerably less treatment in the literature than has longitudinal dispersion. Different methods for determining transverse dispersion coefficients have been used in different investigations, and the results obtained have not been consistent enough to permit accurate generalizations as to the effect of various physical parameters on the magnitude of these coefficients. A numerical solution to the differential equation describing transverse dispersion in the absence of longitudinal dispersion was obtained to enable one to calculate the dispersion coefficient from experimental results. The more general dispersion equation including longitudinal dispersion also was solved numerically to give quantitative limits of a dimensionless group within which the assumption of negligible longitudinal dispersion is justified. Possible experimental procedures were examined, and one utilizing a cylindrical packed column was chosen for the determination of transverse dispersion coefficients. Values of these coefficients were determined for a system of two miscible organic fluids of equal density and viscosity, for two sizes of packing material over a wide range of flow rates in the laminar regime. The dispersion coefficient was found to decrease, for a constant value of the product of packing size and interstitial velocity, as the size of the packing material particles increased. INTRODUCTION Longitudinal dispersion has received extensive treatment in the literature, and consequently is better understood than its orthogonal counterpart, transverse dispersion. Many mathematical models of dispersion processes assume that transverse dispersion is rapid enough to damp out any radial** concentration gradients and therefore may be neglected. Laboratory and production results, however, indicate that this is a poor assumption.l,2 Various experimental procedures for determining transverse dispersion coefficients have been used in previous investigations, but the results have generally been expressed by similar correlations. The transverse dispersion coefficients obtained, however, have often varied considerably for given values of the correlation parameters. We feel that further experimental determinations of transverse dispersion coefficients will help alleviate some of the inconsistencies in these empirical correlations. One assumption implicit in all previous investigations is that of negligible longitudinal dispersion in the experimental system. An attempt to justify this assumption often is made using intuitive reasoning, but it is apparent that this reasoning must break down as the condition of zero flow rate is approached. A mathematical examination of the equations describing the system yields physical limits outside of which the assumption of negligible longitudinal dispersion is invalid. BACKGROUND In a porous medium, the "effective molecular diffusivity" De is less than the molecular diffusivity D measured in the absence of a porous medium, due to the tortuous path which a diffusing molecule must travel. Various authors3-5 have reported values of the ratio De/D in the range of 0.6 to 0.7. When there is fluid flow within the porous medium, mass transfer occurs by convective dispersion as well as by molecular diffusion. These are separate phenomena and can be treated as such on a microscopic scale. However, the mathematical complexity is such that only extremely simple geometries could be considered, and the results hardly would be applicable to the complex geometries existent in actual porous media. A more

Jan 1, 1969

By L. H. Robinson

Triaxial compression tests have been performed to determine the strength characteristics of limestone, sand-stone and shale rocks subjected to controlled stress conditions. This control tuns exercised by varying the liquid pressures within and around a plastic-encased rock specimen. The pressure in the pores of the rock was varied throughout the range from atmospheric to 15,000 psig; the external pressure was changed over the same range with various positive pressure differences between is and the intertnal pressure. The data show that the rock strength increased and the mode of failure changed as the pressuire. surrounding the rock became greater than the pressure in the pores of the rock. These observations and the resu1ts of microbit drilling experiments indicate that the increased rock strength under pressure may bc an important effect in reducing drilling rare, but that other factors are probably of even greater importance. INTRODUCTION A large part of the research conductcd to reducc thc cost of drilling is directed toward improving present drilling technology and dcveloping new drilling methods. In spite of the fact that drilling is essentiaIly the act of making rocks fail, many details of the mechanism of failure are not known. It seems probable, therefore, that continuing studies of the fundamentals of rock failure can lay a foundation for future improvements in drilling technology. One of the factors which affects rock failure is the stress applied to the rock. This effect has been known for many years.'.'.; As an example, stress studics have been published by engineers' concerned with the strength characteristics of concrete in large dams. More recently, a number of investigators4-12 have explored the effects of stress on the strength of geologic formations and the geologic implications of their findings. During the last few years some research'" on rock strengths has also been conductcd in an attempt to gain additional insight into the factors which affeet rock drillability. In detcrmining the strength characteristics of rocks under different stress conditions, investigators have conventionally used triaxial compression equipment. wherein a jacketed rock cylinder is uniformly loaded from all directions and then compressed longitudinally. The results of this work have demonstrated that the mode of failure (brittle or malleable) is dependent upon the loading pressure, and that the strength of the rock increases as the loading pressure increases. These results clearly manifest that the stress, which is determined by the loading pressure, affects rock strength, and this implies that rocks under stress should be stronger and hence harder to drill. During the drilling of a porous rock the stress conditions are determined not only by the pressure surrounding the portion of the rock undergoing failure (confining pressure) but also by the pressure of the interstitial fluids within the rock (pore pressure). The importancc of both these pressures on the drilling rate of rocks has already been demonstrated by a number of investigators. Murray and Cunningham" found in microbit drilling expcriments that at constant pore pressures the drilling ratc decreased as the hydrostatic pressure surrounding an unjacketed, impermeable rock increased. Later Eckel" found that for jacketed Iimestone specimens the differcnce between hydrostatic (wellbore) pressure and formation (pore) pressure had an important effect on drilling rate. These observations of the effect of pressure on drilling rate emphasize that for drilling studies triaxial compression tests should be conducted under conditions of confining and pore pressure which simulate conditions underground. In the past, triaxial tests have been made under conditions which do not reproduce those underground. In particular, most of the tests on porous materials have keen made at varying confining pressures but with atmospheric pore pressures. In only one group of tests' was any data on the effect of pore pressurc reported, and these few data were on materials not representiltive of those encountered in drilling earth formations. The research described herein was initiated to deter-mine the importance of both internal pore pressure and confining pressure on failure characteristics of limestone. sandstone and shales. Triaxial loading equipment was used to obtain strength data at pressures up to 15,000 psig. An additional objective of this research was to apply these results to drilling rate studies by comparing them with the previously published drilling data of Murray and Cunningham" and of Eckel12. The work reported here is but one step in determining and evaluating the fundamental mechanisms which control drilling rate. Additional research is needed to clarify other aspects of the drilling process. The better

By J. C. Melrose, W. R. Foster, J. G. Savins, E. R. Parish

Many differences can be imagined between gas-oil flow in which the gas is supplied at the face of the core and gas-oil flow in which the flowing gas was originally dissolved in the oil. If capillary pressure characteristics and flow requirements control gas saturation distribution, the gas would be expected to be located at preferred sites within the porous medium as determined by pore sizes. On the other hand, during solution gas drive the gas first appears as bubbles through a nuclea-tion process. Nothing in self-nucleation theory specifies at which sites the first bubbles should be formed. In all probability they will be randomly distributed throughout the porous medium. Furthermore, it is not at all certain that even at low rates of production the gas will redistribute itself after nucleation to the channels normally occupied by gas in simple gas flow. Stewart, et at, have shown that at least for some limestone samples, oil recoveries could not be predicted for all rates of production using any one set of relative gas and oil permeabilities. An important factor in controlling recoveries during solution gas drive was the rate of bubble formation, higher rates giving higher recoveries. Stewart, et al, attributed the increase in recovery to a better distribution of the gas phase in heterogeneous limestone samples than is obtained by simple external gas drive. Differences in recovery from these causes were not reported for sandstone cores. In the experiments to be reported here, oil recovery, pressure and producing GOR history were measured during solution gas drive for a 5-ft sandstone core. The results were compared with predictions from the Muskat method for computing solution gas-drive behavior using external gas-drive relative permeability. The effects of changing the rate of production and oil viscosity were studied. At high laboratory rates of average pressure decline, two observations were made which would not have been predicted by Muskat's depletion theory: (1) oil recovery increased with increasing rate of production for a given viscosity oil, and (2) oil recovery increased with increasing oil viscosity for a given high rate of production. Both of these observations are explained as consequences of diffusion control of gas saturations superimposed on the normal gas-oil flow requirements, Fur- thermore, discontinuous gas phase flow appears to be significant during solution gas drive. The laboratory tests were performed at rates of average pressure decline many times greater than the maximum possible rate of average pressure decline in an actual oil field. It is, therefore, not possible to draw any direct conclusions regarding the effect of rate on recovery for the solution gas-drive mechanism under actual field conditions. However, at the lower laboratory rates, recoveries were nearly independent of rate and could be predicted by the Muskat method, using external gas-drive relative permeability data. These results suggest that at normal oilfield rates the effect of rate on recovery for the solution gas-drive mechanism is negligible. EXPERIMENTAL PROCEDURES The core material for the pressure depletion studies was Bandera sandstone from an outcrop in the Mid-Continent. This sandstone was selected because of its low permeability (about 10 md), which would permit the development of substantial pressure gradients in the corn at moderate flow rates. The core was 5-ft long and 2-in. in diameter. Its properties are listed in Table 1. Relative gas-to-oil permeability ratios were measured by an external gas-drive method2. The results are shown for a short 2-in. core and for the 5-ft core in Fig. 8. Oils used in the pressure depletion experiments were kerosene and a highly refined white oil (standard white oil No. 3) with gas-free viscosities of 1.8 and 25 cp, respectively. The gas was a naturally occurring methane from Gough field, Inglewood, Calif. The oil viscosities, gas solubilities and formation volume factors are plotted as functions of pressure at 75°F in Figs. 1 and 2. Methane viscosities and compressibilities were obtained from the literature"'. A core mounting was required which could withstand up to 2,500 psi internal pressure. This was obtained by first encasing the core completely in a plastic resin (Scotch Cast, manufactured by Minnesota Mining & Manufacturing Co.) The plastic covered core was then inserted into a steel pipe equipped with screw caps so that the plastic coating could be pressured from

By P. van Meurs, C. van der Poel

From observations of the linear displacement of oil by water from a porous medium as visualized in transparent models, new insight into the mechanism of vircous fingering, as occurring in the case of unfavorable oil-water viscosity ratios, has been obtained. A mathematical description of the observed phenomena leads to formulas. expressing both oil production and pressure drop across the formation as a func-tion of cumulative water injection with the oil-water viscosity ratio as a parameter. Theoretical results are compared with those obtained from scaled laboratory experiments in sand-filled rubes in which water was injected at the one end and production was obtained from the other end. In the second part of the paper the same principles are used to calculate a more realistic production scheme in which production of oil and water is obtained from wells regularly distributed over the field. Results show satisfactory agreement with production data of an actual oil field. Application of the theory is extremely simple as only a limited number of readily available parameters have to be known and results are presented in an analytical form. INTRODUCTION When oil is displaced from a porous medium by water having a viscosity lower than the oil. the oil-water interface is essentially unstable and has the tendency to break up into what are called "fingers" or "streamers". This Viscous fingering has been noted before by Engelberts and Klinkenberg' and has recently been demonstrated in transparent models by van Meurs-. It is highly questionable whether it is at all possible to perform relative permeability measurements In the laboratory reproducing the same distribution of water and oil in the cores as Present in practice. Therefore, we based our descriptive theory on an idealized model which was "tailored" to the fingering mechanism as observed in transparent models under scaled conditions. This approach was the more attractive, as a highly simplified model had already yielded an adequate description of the process. Though we are aware our theoretical approach can be extended in several ways, it is given here in its simplest form to explain the fundamental principles and to avoid complicated formulas. SIMPLEST CASE OF A LINEAR DRIVE Observations and Theoretical Model In Fig. 1 a series of photographs shows how water penetrates into a formation saturated with an oil which is considerably more viscous than the water. The technique for obtaining these photographs, using transparent models, has already been described?. The fingers can be clearly observed and are seen to be mainly parallel to the flow direction. Note that the contours of these fingers are not smooth but show protrusions of water enclosing oil pockets.

By D. R. Parrish, W. E. Lamoreaux, W. W. Owens

Several methods are now being used by the industry for determining the gas-oil flow characteristics of reservoir rock samples. Most of the laboratory experirnerltal rlzethods can be classified either as steady-state, unsteady-state, or Hassler-type tests. Although the validity of the results obtained by any specific method is not questiorzed, in general, the complexity of these methods leaves much to be desired. This investigation dealt primarily with the development of a simplified unsteady-state (gas drive) laboratory test method for measuring kg/ko re1ationships of small core samples. A procedure for selecting uniform and representative sandstone rock samples was developetl since it was observed that the reliability of the test resu1ts was dependent upon having a sample which was uniform in porosity and permeability. The importance of factors known to influence the results of this type test, such as oil viscosity, gas expansion, and pressure differential were eva/uated. The method of calculating the test results was found to give valid data over a wide range of gas expansion. The limits of pressure differential and oil viscosity which would insure reliable results were defined. In cornparisorz with other presently used testing methods, the gar drive method requires considerably less apparatus, is less time consuming, and is more readily adaptable to a routine type core test. INTRODUCTION The detcrmination of the gas-oil flow characteristics of reservoir rocks has been given concentrated study by the oil industry for the past decade. Both analytical and experimental methods have been considered. The analytical methods have generally relied on a simplified characterization of porous materials in order that the fundamental laws of fluid flow and capillarity could be utilized. More direct experimental methods have been proposed by numerous production research laboratories with the primary goal being a simple and inexpensive, yet reliable method for measuring the gas-oil flow characteristics of reservoir rock samples. The experimental methods which have been reported in the literature generally can be grouped into one of the following classifications: 1. Steady state1,2,3,4 in which a fixed ratio of flowing fluids is forced through the test sample until equilibrium conditions of saturation and pressure gradient are established. 2. Hassler or stationary liquid method5,6,7 in which a uniform liquid saturation is attained, usually through capillary desaturation, then gas or oil is flowed through the sample at a pressure drop insufficient to disturb the pre-established saturation. :. Non-steady state method1,8,9,10 In which only the gas phase is injected into the sample, causing displacement of the oil phase and resulting in a system of continually changing average saturation and saturation gradient. The steady-state and Hassler methods have been most popular because, no doubt, of the simplicity of the actual relative permeability calculations. The problems involved in these methods were limited to apparatus design and measurement techniques. In the steady-state method, for instance, a constant flowing fluid ratio is established which, at equilibrium, results in an essentially uniform saturation over the entire length of the test sample. Therefore, the basic flow characteristics could be obtained by correlating the average saturation of the test sample with the kg/ko determined at various flowing gas-oil ratios.

Jan 1, 1957

By J. G. Richardson, J. W. Graham

The purpose of this paper is to present the results of theoretical and experimental studies of water imbibition. The imbibition processes are involved in recovery of oil from stratified and fractured-matrix formations in natural water drives and water flooding. An understanding of the role of inhibition in implementing the recovery of oil from such formations is deemed essential to proper control of these reservoirs to achieve maximum recovery. The theoretical studies involved development of the differential equations which describe the spontaneous imbibition of water by an oil-saturated rock. The dependence of the rate of water intake by the rock on the permeability, interfacial tension, contact angles, fluid viscosities and fluid saturatiorls is discussed. A few experiments were performed using core samples to determine the effects of core length and presence of a free gas suturation. The role of water imbibition in recovery of oil from a fractured-matrix reservoir by water flooding was investigated by use of a laboratory model. This model was scaled to represent one element of a frac-tured-matrix formation. Water floods were made at various rates with several fracture widths. Interpretations were made of the behavior expected in a system containing many matrix blocks. The presence of a free gas sntu.ration was found to reduce the rate of water imbibition. In the reservoir prototype of the fractured-matrix model, water imbibition rather than direct displacement by water was the dominant mechanism in the recovery of oil at low rates. INTRODUCTION Imbibition may be defined as the spontaneous taking up of a liquid by a porous solid. The spontaneous process of imbibition occurs when the fuid-filled solid is immersed or brought in contact with another fluid which preferentially wets the solid. In the process of wetting and flowing into the solid, the imbibing fluid displaces the non-wetting resident fluid. Common examples of this phenomenon are dry bricks soaking up water and expelling air, a blotter soaking up ink and expelling air and reservoir rock soaking up water and expelling oil. As increasingly better lithological descriptions have been made of the characteristics of petroleum-bearing formations, it has become obvious that imbibition phenomena which were once considered laboratory curiosities are of practical importance. For instance, in reservoirs composed of water-wet sand strata of different permeability in intimate contact, the tendency of water to channel through the more permeable stratum is offset by the tendency for water to imbibe into the tight sand and expel oil into the coarse sand. Also, in fractured-matrix formations the tendency of water to channel through the fractures is offset by water-wet matrix blocks. As some imbibition of the water into the of the largest fields in the world are fractured-matrix reservoirs, it has become increasingly important to understand all the factors involved in the imbibition process. Examples of fractured-matrix reservoirs are the Spraberry field in West Texas which produces from a fractured sandstone', the giant Kirkuk field in Iran', the Dukhan field in Qatar, Persian Gulf2, and the Masjid-I-Sula-main and the Haft-Kel fields in Southwestern Iran, which produce from fissured limestone3. Research into recovery of oil from fractured-matrix formations was stimulated by the rapid decline of oil productivity of wells in the Spraberry formation. One result of this research was the water imbibition process developed by the Atlantic Refining Co.4 Another idea was that much of the Spraberry oil could be recovered by conventional water-flooding procedures5. Subsequently, pilot floods were conducted in this field to test the feasibility of these ideas. It was felt that an understanding of the role played by imbibition processes in displacement of oil from a fractured-matrix reservoir could not be obtained from field data alone because of the many complicating factors and uncertainties involved. Therefore, theoretical and laboratory studies were undertaken to provide this understanding. Study of the equations which describe the linear, countercurrent imbibition process provided an insight into the role of various factors in the process, such as the permeability of rock and inter-facial tension. In addition to the theoretical studies, imbibition experiments were conducted with core samples to determine the effect on the rate of imbibition of such variables as core length and free gas saturation. The principal experimental studies were conducted by water flooding a scaled model of an clement of a frac-tu red-matrix reservoir to evaluate

By C. S. Brooks

Several of the commonly used methods for deter-mination of effective grain densities of minerals have been evaluated for accuracy and precision, and their applicability to various types of petroleum reservoir minerals tested. Consideration was given to the effect of consolidation, swelling clay, and wettability on these grain-density determinations. Procedures with helium displacement, a modified low-pressure Boyle's law porosimeter, and liquid displacement (water, n-decane, and n-octanol) were tested. The helium-displacement procedure was used primarily as an independent reference for which the effects of fluid-solid interaction, e.g., adsorption and wettability, were negligible. The procedures with the Boyle's law porosimeter and with liquid displacement are the methods in common use. Minerals investigated included calcite, West Texas limestone core samples, California sandstone core samples, and Wyoming bentonite. The number of samples tested, while not large, was adequate to permit application of statistical methods. Grain-density measurements on calcite (Iceland spar) provided a basis of comparison for the accuracy and precision of the procedures under investigation for a material with a high degree of chemical and physical homogeneity and for which fluid-solid interactions were at a minimum. Liquid-displacement procedures (water or n-decane) provided the greatest reliability and precision for measurement of the grain densities of the limestone tested. Gas-displacement procedures (helium or Boyle's law porosimeter) provided the greatest reliability and experimental convenience, if not the greatest precision, for the measurement of the grain densities of the swell- ing clay, Wyoming bentonite, and the shah sandstones which -contained appreciable amounts of swelling, in-terstitial clays. These experiments were designed to evaluate the applicability of the grain-density procedures in common we to some of the less tractable minerals—tight limestones, shaly sandstones, and swelling clays. The results should provide a useful basis for a selection of the method most suitable for application to the various petroleum reservoir minerals encountered and especially for use in the estimation of formation porosity. INTRODUCTION These experiments were undertaken to evaluate the accuracy and precision of several commonly used methods for determination of effective grain densities of minerals, and to test their applicability to various types of petroleum reservoir minerals. Consideration was also given to the effect of consolidation, clay, and wettability on these grain-density determinations. Procedures used included measurements with helium displacement, with a modified low-pressure Boyle's law porosimeter, and with liquid displacement (water, n-decane, and n-octanol). The minerals investigated included calcite, West Texas limestone core samples, California sandstone core samples, and Wyoming bentonite. The accurate determination of the porosity of petroleum reservoir formations, in particular those of low porosity, has received much attention. Both bulk-volume and grain-volume measurements necessary in the evaluation of porosity have been examined by many investigators with the objectives of improving both the accuracy and the precision of these determinations. In this investigation statistical criteria for selection of the most reliable and suitable procedure for use with a given type of mineral are demonstrated. The grain-volume measurement procedures appear to be the most susceptible to improvement.

Jan 1, 1958

By P. E. Baker

An experimental study of heat flow in steam flooding was carried out with steam displacing water in a plane-radial fluid-flow model. Temperature distributions in the model reservoir, overburden, and substratum were measured at frequent intervals during flooding by means of a fixed array of thermocouples. Heat losses to overburden and substratum and the division of heat between the steam and hot-water zones in the reservoir were calculated from the temperature data by performing numerical integrations over the appropriate volumes. Temperature profiles and heat distribution results are presented. Some results were (1) the fraction of injected heat that is lost to overburden and substratum, when expressed as a function of time, did not depend on injection rate; (2) a significant portion of the injected heat was contained in the hot-water zone ahead of and underlying the steam front in the reservoir. The division of heat between the steam and hot-water zones did depend on injection rate; and (3) considerable gravity override was observed. The experimental results are compared with those of several published theoretical studies of heat transfer in hot-fluid injection; approximate agreement on heat losses is found with some of the most idealized theories. A valid comparison cannot be made in regard to steam us hot-water zone; basic interpretation of the hot-water zone, in terms of a vertical steam front, is clouded by gravity override in the experiments. INTRODUCTION The experimental study reported here primarily concerns two problems of heat transfer in steam flooding that have economic and engineering importance. These are (1) the fraction of injected heat lost to the overburden and substratum, and (2) the division of the remaining heat between a zone containing steam and a zone of hot water at lower-than-boiling temperature. The volume of the steamed region, which has a direct effect on oil recovery, can be calculated readily from these results. Several valuable theoretical studies have been published on this problem, but no experimental tests of the theories have previously appeared. Tests were needed because the effects of the necessary simplifications in the theories could not be determined otherwise. The basic, untested theoretical problem is the effect of "decoupling" heat and mass transfer, and the resulting inability to include the effects of heat transfer and condensation on fluid flow. The theoretical problem solved so far has been a heat transfer problem in which fluid is present as a heat carrier and flows in a stipulated manner, either linear or plane-radial. The simplifications that have sometimes been made in regard to thermal conductivity by, for example, Marx and Langenheim,l Lauwerier2 and Thomas3 apparently do not have an important effect on heat loss calculations, although the results of the different methods disagree to a certain extent, as shown by Spillette4 and by Ramey.5 A recent paper by the author6 compared the results of experiments in hot-water flooding with several theoretical results and showed a rather poor agreement. It will be seen that the experimental heat loss results for steam flooding, however, show acceptable agreement. These comparisons are discussed in a later section. EXPERIMENTAL APPARATUS AND PROCEDURE The system studied was one in which a sand pack, initially water saturated and at a uniform temperature, was flooded with steam at a constant rate and temperature. The temperature distribution in the "reservoir" and adjacent "strata" was measured with an array of thermocouples, and heat losses were determined from these measurements. The sand pack contained no oil because the only concern of the experiments was heat flow, and it was important to have the flow pattern as simple

Jan 1, 1970