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Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*25 (02): 291–302.

Paper Number: SPE-11495-PA

Published: 01 April 1985

...Noaman El-Khatib A mathematical model is developed for waterfloodingperformance in linear stratified systems for both cases of noncommunicating layers with no crossflow and communicating layers with complete crossflow. The model accounts for

**variation**of porosity and saturation inaddition...
Abstract

A mathematical model is developed for waterfloodingperformance in linear stratified systems for both cases of noncommunicating layers with no crossflow and communicating layers with complete crossflow. The model accounts for variation of porosity and saturation inaddition to permeability of the different layers. The modelpredicts the fractional oil recovery, the water cut, the totalvolume injected, and the change in the total pressure drop, or the change in injection rate at the water breakthroughin the successive layers. A systematic procedure forordering of layers and performing calculations is outlined. Aprocedure for combining layers to avoid instability in the case of low mobility ratio is introduced. The developed model is applied to different examplesof stratified reservoirs. The effects of mobility ratio and crossflow between layers are discussed. The effects of variable porosity and fluid saturation are discussed also. It was found that crossflow between layers enhancesthe oil recovery for systems with favorable mobility ratios(lambda w/lambda o less than 1) and retards oil recovery for systems with unfavorable mobility ratios. It was found also that crossflow causes the effect of the mobility ratio on oil recovery to become more pronounced. The variation of porosity andfluid saturation with permeability is found to increase oilrecovery over that for the case of uniform porosity andsaturation for both favorable and unfavorable mobility ratios. Introduction Because of the variation in the depositional environments, oil-bearing formations usually exhibit random variationsin their petrophysical properties in both horizontal and vertical directions. Statistical as well as geological criteria usually are used to divide the pay zone betweenadjacent wells into a number of horizontal layers each with its own properties (k, phi, h, Swi, and Sor). Suchreservoirs usually are called "stratified," "layered,"or"heterogeneous" reservoirs. This variation in properties affects the performance of oil reservoirs during primary and secondary recovery processes. One of the significant factors influencingrecovery performance during waterflooding is thevariation of permeability in the vertical direction. In this case, the displacing fluid (water) tends to move faster in zones with higher permeabilities, causing earlier breakthrough of water into the producing wells and eventual by passing of some of the displaced fluid (oil). The various methods used for the prediction of waterflooding performance of stratified reservoirs differin the way the communication between the different layersis treated. Two ideal cases usually are used: completely noncommunicating layers and communicating layerswith complete crossflow. For actual stratified Systems, however, the layers are partially connected in the vertical direction, and the performance of the system lies betweenthose of the two ideal cases. For the case of noncommunicating stratified layers, the methods of Stiles and Dykstra-Parsons usually areused. Stiles' method assumes unit mobility ratio for the displacement process when computing the recovery but accounts for the mobility ratio when computing the WOR, which results in contradictory formulas for the performance. The Dykstra-Parsons method and its modified version by Johnson use semiempirical correlations based on log-normal distribution of the layers' permeability. Muskat presented analytical expressions for the performance of reservoirs having linear and exponential permeability distributions. Two methods are available in the literature forestimating the performance of communicating systems with complete crossflow the method of Warren and Cosgrove and that of Hearn. Warren and Cosgrove's method requires a log-normal permeability distribution. Furthermore, it ignores the problem of ordering of layersfor low mobility ratio, which may cause physicallymeaningless results. The method of Hearn is intended to derive pseudorelative permeability functions for the stratified system to be used in reservoir simulation. Most of these methods assume that all layers have identical properties except permeability. Also, the time is notrelated explicitly to the performance. Furthermore, noneof these methods considers the variation in injection rateand total pressure drop as the displacement process progresses. Although these points can be treated numerically for a particular case using reservoir simulation methods, the objective of this work is to developan alytical expressions for waterflooding performance inidealized linear stratified systems that will consider the previously mentioned points. Theoretical Analysis Assumption and Definitions. For both the noncommunicating and communicating systems, these assumptions are made. 1. The system is linear and horizontal, and the flow is incompressible, isothermal, and obeys Darcy's law. SPEJ P. 291^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*24 (05): 545–554.

Paper Number: SPE-10291-PA

Published: 01 October 1984

... in an actual flood, the released oil forms a bank ahead of the surfactant slug. SPEJ P. 545^ gas injection method concentration

**variation**upstream oil & gas mixture cmc injection strategy optimal injection strategy adsorbed component slug enhanced recovery chemical flooding methods constant...
Abstract

The chromatographic movement of surfactant mixtures through porous media is examined to determine possible injection strategies for minimizing the amount of surfactant required in a tertiary oil recovery chemical flood. The model used does not consider the presence of oil but does account for mixed micelle formation. Expressions are derived that represent the surfactant required to expose an entire reservoir to an "effective oil recovery mixture." This effective mixture may be either one whose overall composition is within prescribed limits of the composition of the injected surfactant solution or it may be a mixture whose overall composition varies but which contains micelles of fixed composition. Mixtures considered contain cosolvents and one, two, or three surfactant components. Initial calculations neglect dispersion, but numerical calculations including dispersion leave the conclusion unchanged; within the limitations of the model, there are optimal strategies for the propagation of surfactant mixtures through porous media. The optimal injection strategy varies, depending on the nature of the surfactant solution injected into the porous medium. Conditions for and the location of the optimum are discussed. Conclusions based on observations about these systems then are extended to cover the injection of surfactant mixtures currently available commercially. Introduction Commercial application of surfactants for EOR now appears feasible. The principle at work in such processes is the lowering of interfacial tension (IFT) between the continuous flowing water and trapped residual oil droplets to allow the oil to be mobilized. Mixtures that effectively lower oil/water IFT are often blends of various surfactant types, isomers of the same surfactant, and/or cosurfactants in an electrolyte solution. The oil recovery efficiency of the injected mixture generally is quite sensitive to changes in mixture composition. Change of composition after injection into the reservoir may occur by one or a combination of mechanisms. For example, the mixture components may partition selectively into the various phases present in the reservoir. The mechanism considered here is the chromatographic separation of the mixture into its components due to preferential adsorption of various components onto reservoir minerals-"the chromatographic problem." The recent reports of the Bell Creek Unit A micellar/polymer pilot showed 20% of the injected surfactant produced before any oil bank with negligible concomitant incremental tertiary oil production. Significantly, the surfactants produced were the lower-molecular-weight species. Though alternative mechanisms for this separation yet may be established, the hypothesis of chromatographic separation of the components in the mobile aqueous phase seems adequate. Not only did this produced surfactant not result in enhanced recovery, but since the injected solution was designed to give ultralow IFT's with the low-molecular-weight components in place, it seems likely that the oil recovery efficiency of the remaining surfactant also may have been impaired. These results emphasize the importance of understanding the mechanisms of surfactant chromatographic movement. One means of combatting the chromatographic problem is to reduce the local adsorption of the mixture components-that is, modify the adsorption isotherms of the constituents. This may be done either by changing the reservoir minerals (e.g., by a caustic flood) or by modifying the structure of the surfactant molecules. A complementary approach is to examine the dynamics of the chromatographic movement of surfactant mixtures to identify injection strategies, if they exist, that minimize the total surfactant requirement. It is this question that is considered here. The analysis considers an oil-free linear system and neglects many of the complex features that are encountered in an actual chemical flood. There are several reasons for ignoring these complicating factors. The coherence solutions apply to the systems considered here; whereas the only solutions that include the presence of oil employ numerical computations. An analytical solution is desirable; however, there is an additional more compelling argument that has been used to justify neglecting the presence of oil. The chromatographic movement of a surfactant/ cosurfactant mixture through an oil-free core should demonstrate the qualitative features of the actual oil recovery process. While multiple flowing phases do arise in an actual flood, the released oil forms a bank ahead of the surfactant slug. SPEJ P. 545^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*24 (04): 467–472.

Paper Number: SPE-10854-PA

Published: 01 August 1984

...

**variation**casing and cementing buckled rod equilibrium position lubinski helically buckled rod constraint application force-pitch relation postbuckled configuration Helical Postbuckling Configuration of a Weightless Column Under the Action of an Axial Load John B. Cheatham Jr., SPE, Rice U...
Abstract

The helical postbuckled configuration of a weightless, circular column confined within a cylinder was used by Lubinski et al. in 1962 to develop frequently applied equations describing the deformation of oilfield tubing. This paper presents an extension of their work in the form of another expression for the relation between applied force and helical pitch in the buckled state. Based on simple laboratory experiments and stability arguments it appears that the previous force-pitch equation applies during load application and the one given in this paper applies during unloading. Introduction The helical postbuckled configuration of a radially confined tubular is a phenomenon of significant import to the petroleum industry. For the case phenomenon of significant import to the petroleum industry. For the case of smaller tubulars such as most tubing, the radial clearance boundary can be large enough to both allow significant axial shortening and induce bending stresses of a magnitude sufficient to yield the tube body. For larger tubulars such as casing, large radial clearances normally are associated only with washed out or overgauge sections of the wellbore. However, even for small radial clearances, and with particular regard to intermediate casing strings, the helical configuration can present a hazard to the integrity, of the well by promoting wear during drilling operations. The majority of recent work on helical buckling can be traced to the classic paper by Lubinski et al. on helical buckling of tubing sealed in packers. Applications of this early work to a variety of downhole packers. Applications of this early work to a variety of downhole configurations have appeared in the literature; however, with the exception of some recent work on the influence of the packer on the helical configuration, the basic relation between applied compressive axial force and helical pitch has remained unaltered since its introduction. The intent of our study is to reconsider the helical postbuckled configuration by applying the principle of virtual work to deviation of the column from its straight, prebuckled configuration. The result of this effort will be two force-pitch relations (one of which is the relation derived by Lubinski et al.) that differ by a factor of two. The remainder of the discussion is then devoted to the significance of the a alternate solutions and the effect of the dual solution on tubular designs. A description of the postbuckled configuration of a tubular can be quite complex. In fact, if one includes from the outset the axial component of stress arising from the distributed weight of the tubular along its length, the resulting expressions become unwieldy. As an alternative, this discussion is restricted to an analysis of the postbuckled geometry of a weightless rod. Adjustment of the results of this analysis to include the effects of both the distributed weight of the tubular and internal and external pressure are covered in detail in Ref. 1. Geometry of the Helix At the instant a compressed, straight rod buckles, the lateral displacements will be sinusoidal in nature. However, if the lateral displacements are constrained to be less than or equal to some predetermined value (i.e., a wall constraint concentric with the undeformed predetermined value (i.e., a wall constraint concentric with the undeformed position of the rod), contact of the buckled rod with the constraint will position of the rod), contact of the buckled rod with the constraint will induce a rearrangement of the rod such that it assumes the form of a helix. Consider Fig. 1, which illustrates the geometry of a representative length of the helically buckled rod and several important variables to be used throughout the analysis. Notice in particular the definition of the pitch being the length between repetitions of the helical configuration. pitch being the length between repetitions of the helical configuration. Since the rod is assumed here to be weightless and infinite in length, the pitch will be constant throughout. pitch will be constant throughout. Now let a cartesian coordinate system be placed at some representative point along the length of the helix. point along the length of the helix. SPEJ p. 467

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*24 (03): 328–332.

Paper Number: SPE-11285-PA

Published: 01 June 1984

.... Necessary conditions for optimality are obtained through application of the calculus of

**variations**and Pontryagin's weak minimum principle. A gradient method is proposed for the computation of optimal injection policies. Introduction In recent years, EOR techniques have received much attention. This has...
Abstract

The theory of optimal control of distributed-parameter systems is presented for determining the best possible injection policies for EOR processes. The optimization criterion is to maximize the amount of oil recovered at minimum injection costs. Necessary conditions for optimality are obtained through application of the calculus of variations and Pontryagin's weak minimum principle. A gradient method is proposed for the computation of optimal injection policies. Introduction In recent years, EOR techniques have received much attention. This has been motivated by the rapid escalation in the price of crude oil, uncertainties of supply from foreign sources, and low efficiencies of current recovery technologies. There are four main EOR methods considered viable for the production of light crudes. These are polymer floods, caustic floods, CO2 floods, and micellar-surfactant floods. All methods require the injection of rather expensive fluids into oil-bearing reservoir formations. Commercial application of any EOR process relies on economic projections that show a decent return on investment. The key factors in an economic projection for emerging tertiary oil recovery processes are investment requirements such as chemical and development costs, oil recovered, time required to obtain the oil, oil price, and tax load. Because of high chemical costs, it is important to optimize EOR processes to provide the greatest recovery at the lowest cost. The goal is to determine the theoretical basis for computing the best way of injecting an EOR fluid into a reservoir formation. It is important to note that this is only a part (although an important part) of the complete optimization of an EOR system for a given reservoir. The purpose of an optimization problem is to determine the control policy that will minimize (or maximize) a specific performance criterion, subject to the constraints imposed by the physical nature of the problem. Techniques available for the solution of dynamic-optimization problems, which contain differential or integral equality constraints, are the classical calculus of variations, the minimum principle of Pontryagin, and the dynamic programming of Bellman. Each of these techniques is equivalent to the others, but each manifests itself in a different way. The fundamental theorem of the calculus of variations is applied to problems with unconstrained states and controls. Consideration of the effect of control constraints leads to Pontryagin's minimum principle. Dynamic programming is suited to the optimization of serial structures. Although there are many books on the subject of optimal control theory, only a few cover the optimal control of distributed-parameter systems. Application of these techniques to chemical and petroleum engineering operations is beginning, and dynamic-process optimization is becoming a valuable tool in the design of modern systems. Refs. 5 and 6 present surveys of recent applications of distributed-parameter systems theory. Optimization objectives can be expressed as a cost functional or performance index to be minimized. The control on the system is the composition or physical state of the injected fluids. Thus, the optimization problem is concerned with determining the injection policy that leads to a minimum in the cost functional, subject to differential equality constraints that describe the system dynamics. Mathematical models have been formulated for each of the major EOR methods. Polymer flooding is described by Chauveteau, Hirasaki, Thakur, and Shah. Caustic flooding modeling is considered by Breit and deZabala. CO2 flooding fundamentals are presented by Henderson, Metcalfe, Pontious, Todd, and Leach. Micellar-surfactant modeling concepts are given by Ramirez et al., Pope, Fleming et al., and Hirasaki. Each of these models consists of conservation relations needed to describe the dynamic state of the process given by the chemical compositions and the fluid saturation. We denote the fluid compositions by the vector 1 and the fluid saturation in a two-phase system by either the water- or oil-phase saturation. Here, x2 denotes the water-phase saturation. The state of the system is therefore given by the vectorX 1X = ...................................(1)X 2 SPEJ p. 328

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*23 (06): 870–878.

Paper Number: SPE-10849-PA

Published: 01 December 1983

...Ian D. Palmer; H.B. Carroll, Jr. Models of three-dimensional (3D) fracture propagation are being developed to study the effect of

**variations**of stress and rock properties on fracture height and bottomhole pressure (BHP). Initially a blanket sand bounded by zones of higher minimum in-situ stress...
Abstract

Models of three-dimensional (3D) fracture propagation are being developed to study the effect of variations of stress and rock properties on fracture height and bottomhole pressure (BHP). Initially a blanket sand bounded by zones of higher minimum in-situ stress is considered, with stresses symmetrical about both the pay-zone axis and the wellbore. An elliptical fracture perimeter is assumed. Fluid flows are one-dimensional (1D) Newtonian in the direction of the pay zone. Two models, FL1 and FL2, are developed. In FL1, a discontinuous stress variation is approximated by a y2 variation in the vertical coordinate, and the fracture criterion, Ki = Kc, is satisfied at both major and minor axes. The net pressure at the tip, Lf, of the long axis required by the boundary condition Ki = Kc does not seem crucial in determining fracture height or BHP (compare with one group of published models that assumes p = 0 at Lf). Model FL2 properly represents the discontinuous stresses, and satisfies Ki = Kc at the wellbore but not at the tip of the long axis. A parametric study is made, with both models, of the comparative effects of stress contrast, Kc, pay-zone height, h, and Young's modulus, E, on fracture height and BHP. Results indicate that Kc does not have as much effect as either E or, at least for large stress contrasts. Model FL2 suggests the possibility of a rapid growth in fracture height as is reduced. Such modeling may be able to give an upper or "safe" limit on the pumping parameters ( and ) to ensure good containment. When the stress contrast is high, 700 psi [4826 kPa], an analytic derivation of BHP appears to be a good approximation for the parameters we use, if everywhere the fracture height is assumed equal to the pay zone height. Although leakoff is neglected here, subsequent modeling results show that, for leak off coefficients 0.001 ft- min [3.9 × 10 -5 m.s ], the results herein are a good approximation to the case when leak off is included. Introduction In their essence, models of hydraulic fracture propagation involve elasticity theory and fluid mechanics. The first is concerned with the fracture opening or width, w(p), as a function of net pressure on the fracture faces, while the second is concerned with the pressure drop, p(w), caused by the flow of viscous fluids in the fracture. Simultaneous solution of these equations includes a boundary condition that often takes the form Ki = Kc, where Ki is the stress-intensity factor at a point on the fracture tip, and Kc is the fracture toughness. The final solution is very complex in 3D, when a vertical fracture can expand vertically as well as horizontally along the pay zone. Thus, the first solutions were essentially two-dimensional (2D), and they assumed that the fracture height, hf, was fixed at the pay zone height, h. The 2D solutions were clustered in two groups as summarized by Nordgren, Perkins, and Geertsma and Haafkens. The first grouping, based on a model by Christianovich and Zheltov, assumed that the sides of an elongated, vertical fracture were parallel (i.e., free slippage between the pay and bounding zones, or no vertical stiffness). Other papers in this grouping included Geertsma and de Klerk, Daneshy and Settari. SPEJ P. 870^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*23 (05): 717–726.

Paper Number: SPE-11133-PA

Published: 01 October 1983

... design modeling phase artificial intelligence objective function

**variation**reservoir simulation application decision variable optimal control procedure influence function enhanced recovery procedure psig output variable iteration constraint operational interval upstream oil & gas...
Abstract

This paper addresses the optimization of field operations under a given set of technical and economic constraints and demonstrates that an optimal control procedure may be applied to any industrial oil and/or gas reservoir at reasonable cost and at acceptable accuracy level. Both primary and secondary recovery processes can be considered. The method proposed here is presented in two main sections. First, the modeling phase provides an approximately and locally linear model of the reservoir. A previously calibrated reservoir simulator model is used to perform a series of experiments. and a multiple variable regression analysis is used to fit the experimental data. The experimental design was one of the key issues in this work. Second, the optimization phase is performed with a linear programming algorithm. Nonlinear effects, such as performed with a linear programming algorithm. Nonlinear effects, such as those generated by the presence of gas, are approximated by several procedural iterations. procedural iterations. The application of this method to the case of a hypothetical reservoir demonstrates the validity of the optimal control procedure and shows convergence within an acceptable number of iterations. Introduction This investigation demonstrates the application of linear programming to a set of behavior equations derived from reservoir simulation results by use of a least-squares inversion procedure. The method is intended to optimize the production schedule of any reservoir for which the producer or injector well locations have already been fixed. The accuracy of this optimization procedure depends on many factors. the most important being the approximate procedure depends on many factors. the most important being the approximate linearization of the nonlinear system. Also important is reducing the required number of simulation runs until a satisfactory cost/accuracy compromise is obtained. It appears, that the reservoir engineer may contribute in reducing the experimental and calculation cost by properly selecting the series of simulation experiments. Simulator experiments, multivariable regression, least-squares inversion, the simplex algorithm. and validation are the major steps of this theoretical optimization procedure. The application of the process is demonstrated by working out a hypothetical practical example. The optimization of field operations has been explored by many authors, and many approaches have been suggested. Lee and Aronofsky established the first principles of this type of procedure by designing a time-discretized optimization process applied to a set of single-well reservoirs. Linear programming was used, and the objective function considered was the net profit. This first approach was improved by Aronofsky and Williams, who reduced the assumptions concerning the reservoir. Attra et al. refined the Lee and Aronofsky production model by introducing additional economic and technical factors, such as sales contract requirements or gas compressor limitations. For all these methods, the linear equations constituting the reservoir model were derived from material-balance considerations, and the reservoirs generally were assumed uniform and single-phase. SPEJ p. 717

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*23 (01): 184–190.

Paper Number: SPE-11334-PA

Published: 01 February 1983

... surface isaacson representation facet upstream oil & gas boundary condition

**variation**procedure equation Steep Wave Forces on Large Offshore Structures Michael de St. Q. Isaacson, U. of British Columbia Abstract A new numerical method for calculating the interaction of steep (nonlinear...
Abstract

A new numerical method for calculating the interaction of steep (nonlinear)ocean waves with large coastal or offshore structures of arbitrary shape is described. The evolution of the flow, and in particular the loads on the structure and the runup around it, are obtained by a time-stepping procedure in which the flow at each time step is calculated by an integral equation method based on Green's theorem. A few comparisons are made with available solutions and results are presented for a typical design wave in shallow water. The method is capable of predicting forces caused by steep waves accurately and without prohibitive computer effort. Introduction The prediction of wave forces on large offshore structures on the basis of linear diffraction theory, which is formally valid for small-amplitude sinusoidal waves, is now an established part of offshore design procedure. Reviews of the approaches generally used have been given by Hogben etal ., 1 Isaacson, 2 and Sarpkaya and Isaacson. 3 To account more realistically for the effect of large wave heights, research recently has been directed primarily toward developing a second approximation based on the Stokes expansion procedure. However, such an approach is of practical value only under restricted conditions, as in the case of anundisturbed wave train described by Stokes second-order theory. In particular, nonlinear wave effects are expected to be of greatest importance for steep shallower waves, and these are precisely the conditions in which a Stokes second-order solution becomes invalid. A numerical solution to the complete boundary value problem without any wave height perturbation procedure is clearly desirable. The approach outlined here is described in detail by Isaacson. 4 In this method, the wave diffraction is treated as a transient problem with known initial conditions corresponding to still water in the vicinity of the structure and a prescribed incident wave form approaching the structure. The development of the flow then can be obtained by a time-stepping procedure, in which the velocity potential of the flow at any one instant is obtained by an integral equation method basedon Green's theorem. Comparison with known diffraction solutions can be made only for relatively restricted situations. A few such comparisons have been carried out and arequite favorable. Results also are presented for a typical design wave in shallow water, and these are found to differ significantly from linear theory predictions.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*22 (05): 743–749.

Paper Number: SPE-9815-PA

Published: 01 October 1982

... enhanced recovery hydrophobe phenol concentration pvt measurement microemulsion maximize solubilization three-phase region

**variation**Criteria for Structuring Surfactants To Maximize Solubilization of Oil and Water: Part I-Commercial Nonionics Alain Graciaa, U. de Pau Lester N. Fortney, U...
Abstract

The phase behavior of nonionic surfactants having the same hydrophilic/lipophilic balance (HLB) but differing molecular weights has been studied. It is shown that the optimal alkane carbon number (ACN) depends on the HLB, but that increasing the hydrophobe molecular weight narrows the middle phase region, increases the solubilization parameter, and decreases the interfacial tension (IFT). We found that the width of the three-phase region is in simple inverse proportion to the solubilization parameter at optimal salinity and that the multiple of IFT times the square of the solubilization is a constant. We also found it possible to synthesize nonionics that rival anionics in the properties mentioned above. Introduction There is increasing evidence that the phase behavior of surfactant/oil/brine systems and the efficiency of oil recovery with micellar solutions are connected intimately. For instance, laboratory core floods have shown that surfactant systems exhibit maximum oil recovery at the optimal salinity. The concept of optimal salinity, introduced by Healy and Reed, is especially useful because it pen-nits screening of surfactant systems by relatively simple experiments requiring the observation of the number and the types of phases that coexist at equilibrium when surfactant/oil/brine mixtures are blended. Optimal salinity, defined as that middle-phase microemulsion system containing equal volumes of oil and water, is not difficult to determine, and, thus, conditions for the most efficient surfactant system can be established. It is now well known that many different surfactant systems have the same optimal salinity. Further, it generally has been assumed, but not definitely established by laboratory experiments that the preferred surfactant system, selected from a group of systems having the same optimal salinity, will be that which solubilizes the largest volume of oil and brine per unit mass of surfactant. We do not necessarily subscribe to this simple view. since there are many factors other than solubilization (such as surfactatant retention) that may influence oil recovery efficiency however, all other factors being equal, it is reasonable to attempt to maximize solubilization, especially because it has been found synonymous with minimal IFT's-an equally important factor governing effectiveness of oil recovery. This paper seeks to identify some surfactant structural features that will lead to increased solubilization and decreased IFT. We have addressed this important question in past publications but have met with only limited success. The difficulty has been that changing the surfactant structure dictates that a second corresponding change be made so that the resulting system would remain optimal. For instance, one can increase the length of the hydrocarbon tail of the surfactant molecule and at the same time compensate for this change either by decreasing the amount of hydrophobic alcohol added to the system or by decreasing the salinity of the system. The results obtained in this manner have remained difficult to interpret because all changes can and most often do alter the solubilization of oil and water in the middle-phase microemulsion. Therefore, it was not possible to separate that pan of the resulting solubilization change caused strictly by the modification of the surfactant structure. In the study discussed here, we made compensating changes in the surfactant structure, keeping all other variables fixed. For nonionic surfactants, compensating changes can be made in several ways. SPEJ P. 743^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*22 (04): 514–522.

Paper Number: SPE-9092-PA

Published: 01 August 1982

... drilling fluid chemistry consistency limit lower concentration upstream oil & gas drilling fluid selection and formulation guanidine

**variation**inhibitory action drilling fluid formulation chloride drilling fluid property shrinkage index clay suspension potassium chloride drilling fluids...
Abstract

This study compares the effectiveness of potassium chloride with guanidine chlorhydrate in the prevention of-clay swelling. The results given on various swelling tests on calcic montmorillonite led to the conclusions that guanidine chlorhydrate is more effective than potassium chloride, especially in low concentrations, and water immersion of samples treated by both solutions shows the permanent feature of the inhibitive action of guanidine chlorhydrate in swelling on one hand and the important increase in swelling of immersed samples treated by potassium chloride on the other. The viscosity measure of montmorillonite suspensions, before and after solution ion elimination by dialysis. confirms these observations. Introduction Among the grounds encountered in drilling, swelling clays are those that raise major problems for wall firmness. These clays, commonly called "gumbos," raise many problems related to swelling, dispersion, and a strong tendency of the cuttings to aggregation. These problems result from the interaction of the drilling mud with the terrain traversed. An analysis of the effectiveness of various materials used in drilling muds to stabilize clayed zones led us to focus on the action of two solutions capable of inhibiting swelling: potassium chloride. which is normally used in drilling wells, and guanidine hydrochloride, which displays a strong fixation tendency on montmorillonite. The purpose of this study is to compare the inhibitory action of these two salts and the influence of their concentration in the solutions by means of relatively simple tests. This research work is limited to treatment of a montmorillonite clay. The Material Investigated To examine the behavior of swelling clays in the presence of the two solutions selected, it would be ideal to carry out tests on samples representative of the horizons that raised problems during drilling operations. However, it is difficult to extract enough clay of stable composition from the cuttings; moreover, the clay is polluted by the drilling mud. Composition analyses of clays that are difficult to drill because of swelling showed that montmorillonite was present in all the samples investigated. Hence this study is limited to an examination of the action of KCl and guanidine hydrochloride on a previously investigated montmorillonite. The montmorillonite used was an Italian natural calcium montmorillonite, supplied untreated and finely ground. The composition analysis carried out in the Compagnie Francaise des Petroles (CFP) laboratory at Bordeaux yielded the results given in Table 1. Table 2 gives the geotechnical properties of the clay investigated. Swelling Tests of Compacted Samples Unidirectional Swelling This test was performed in the measurement cell shown in Fig. 1. The cylindrical sample was hooped laterally and its swelling measured in a single direction. The procedure was as follows. Five grams of clay dried in an oven at 100 deg. C were placed in a measurement cell. The clay was saturated by it KCl or guanidine hydrochloride solution for 24 hours (by filtration) (the degree of saturation measured on some samples after compacting was found to exceed 95%). SPEJ P. 514^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*22 (01): 117–131.

Paper Number: SPE-9270-PA

Published: 01 February 1982

... later. SPEJ P. 117^ 1982. Society of Petroleum Engineers upstream oil & gas interpretation resistivity trans correlation well logging artificial intelligence identification cloud symposium thickness electrofacies axis log analysis information log response porosity

**variation**...
Abstract

Until recently, the lithology and sedimentology of formations penetrated by a well could be studied only through the analysis of cores. Now a technique has been developed using multivariate analysis of logging data to give a high-resolution sedimentological description of any sequence of formations. The number of different logs and their range allow determination of many of the physical characteristics of the rock. The definition of a lithofacies has been extended by introducing the concept of an electrofacies, constructed on the basis of all the logging data at any depth interval. Thus, each logging datum is considered a descriptor for purposes of establishing electrofacies in a logged interval. Once established, electrofacies then can be correlated with actual geologic facies, if the logged interval has been cored. Introduction The evaluation of the potential and performance of hydrocarbon reservoirs includes the study of the sedimentary series in which they are found. Essential tasks to be performed during such a study include description of the rock facies and the relationships between facies, determining the geometry of the sedimentary bodies, reconstruction of the vertical column and analysis of sequences and cycles of facies types, and an accurate estimation of the common petrophysical parameters (porosity, permeability, etc.). The sedimentary environment in which deposition took place and diagenetic changes that may have occurred since are inferred from the data. There are several possible sources of information for the sedimentologist, including surface outcrops, surface geophysical measurements. cuttings obtained while drilling cores, wireline core samples and wireline log data. While the study of surface outcrops always has been useful, it has become less important as the depth of drilling has increased and more importance is placed on the analysis of deep-seated structures such as faults and traps. Cuttings obtained during drilling are a valuable source of information. However, there is always some depth uncertainty, and, of course, the samples obtained are not large enough for accurate measurement of porosity, permeability, and so on. Additionally, some samples are lost because of their size (silts) or by dissolving in the drilling mud (salts), and there is often a problem of cuttings falling in from above the bit (shales or sands). Cores, on the other hand, provide an excellent source of information since their depth can be located accurately and they are large enough for analysis. However, coring is an expensive process and is sometimes impossible because of safety reasons or formation conditions. For these reasons, extensive coring usually is not done, even in exploration wells. With these constraints on sample data, the trend in recent years has been toward the use of wireline log data, not only to predict general petrophysical parameters but also as a tool for sedimentologists and reservoir engineers. Toward this end, a number of wireline tools (e.g., LDT TM, NGS TM and GST TM) and computer programs (e.g., GEODIP TM) have been developed, and more recently, as outlined in this paper, detailed information on sediments has been extracted from log data. An advantage of logs is that they usually cover the entire interval of interest and at a sampling rate providing exceptional detail. Logs often can be obtained in conditions where coring is impossible, and they are cheaper, all costs considered. A combination of (short) cored sections and log data gives the sedimentologist an excellent information base. We consider the referencing of log data against cores later. SPEJ P. 117^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*21 (04): 435–443.

Paper Number: SPE-8942-PA

Published: 01 August 1981

.... Theoretical models are applied to analyze some aspects of the dynamics of fracturing near material interfaces. The results of these calculations indicate that

**variation**of material properties across a well-bonded interface can cause dynamic material response resulting from the fracturing, which could enhance...
Abstract

We are conducting a theoretical and experimental program on the hydraulic fracturing process. One primary objective of the program is to determine those reservoir properties or characteristics that can control the created fracture geometry. Theoretical models are applied to analyze some aspects of the dynamics of fracturing near material interfaces. The results of these calculations indicate that variation of material properties across a well-bonded interface can cause dynamic material response resulting from the fracturing, which could enhance propagation across the interface. Effects of friction also are analyzed theoretically; however, in the frictional calculations, the wave mechanics are ignored. These calculations show that frictional slip along the interface tends to draw a pressurized fracture toward the interface; this motion tends to reduce the chances of penetrating the material across the frictional interface.Small-scale laboratory experiments are performed to study the effects of frictional characteristics on hydraulic fracture growth across unbonded interfaces in rocks. Various lubricants and mechanical preparations of the interface surfaces are used to vary the coefficients of friction on the interface surfaces. It is found that the frictional shear stress that the interface surface can support determines whether a hydraulically driven crack will cross the interface. Experiments also are being performed to study the effects of pre-existing cracks, which perpendicularly intersect the unbonded interface, on hydraulic crack growth across the interface. It also is found that the presence of these pre-existing cracks impedes the propagation of the hydraulic fracture across the interface. The experimental results on the effects of friction on the interface and the effects of pre-existing cracks on hydraulic fracture penetration of interfaces are consistent with the predictions of the numerical model calculations. Introduction Massive hydraulic fracturing (MHF) is a primary candidate for stimulating production from the tight gas reservoirs in the U.S. Hydraulic fracturing has been widely used as a well completion technique for about 30 years. MHF is a more recent application that differs from hydraulic fracturing in that more fluid and proppant are pumped to create more extensive fractures in the reservoir. Application of MHF to increase production from the tight reservoirs has provided mixed and, in many cases, disappointing results - especially in lenticular reservoirs. For MHF to be successful in enhancing gas production from tight reservoirs, it is important that the fractures be created in productive reservoir rock with large drainage surfaces in the low-permeability material and conductive channels back to the wellbore. We are faced then with the problem of containing fractures in a given formation.Under the U.S. DOE's unconventional gas recovery program, Lawrence Livermore Natl. Laboratory is conducting a research program on the hydraulic fracture process. The general goal of this research is to determine if and to what extent reservoir parameters control the geometry of the created fractures. From theories implied and demonstrated, hydraulic fractures propagate perpendicular to the least principal stress. Hence, except for very shallow applications, the fractures will be primarily vertical, with the azimuthal orientation controlled by the in-situ stress. The vertical gradient in the horizontal stresses also could be a factor in the control of the shape or vertical extent of fractures. SPEJ P. 435^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*21 (01): 51–62.

Paper Number: SPE-8372-PA

Published: 01 February 1981

... & gas multicomponent trajectory distance time diagram composition

**variation**grid tie line fluid dynamics fractional flow concentration chemical flooding methods displacement equation equivelocity curve construction extension composition space two-phase region Theory...
Abstract

The basis of a general theory of multicomponent, multiphase displacement in porous media is presented. The theory is applicable to an arbitrary number of phases, an arbitrary number of components partitioning between the phases, and variable initial and injection conditions. Only the effects of propagation are considered; phase equilibria and dependence of fractional flows on phase compositions and saturations are required as input, but any type of equilibrium and flow behavior can be accommodated. The principal simplifying assumptions are the restriction to one dimension, local phase equilibria, volume additivity on partitioning, idealized fluid dynamic behavior, and absence of temperature and pressure effects. The theory is an extension of that of multicomponent chromatography and has taken from it the concept of "coherence" and, for practical application, the tools of composition routes and distance/time diagrams. The application of the theory to a surfactant flood is illustrated in a companion paper. 1 Introduction A key problem in modern methods of enhanced oil recovery is that of multicomponent, multiphase displacement in porous media. This term means the induced flow of any number of simultaneous, not fully miscible fluid phases consisting of any number of components. The components may partition between the phases; moreover, the physical properties of the phases (densities, viscosities, interfacial tensions, etc.) depend on composition and, therefore, on partitioning of the components. Multicomponent, multiphase displacement may be viewed as a generalization and combination of two different and independent approaches. The first of these is the highly developed theory of multicomponent chromatrography, 2 which allows for any number of components affecting one canother's distribution behavior but admits only one mobile and one stationary phase. This theory has to be extended to more than one mobile phase. The second is the fluid dynamic theory of immiscible displacement in porous media, allowing for more than one mobile phase but not for partitioning of components. This theory was developed in the 1940's for two mobile phases 3 and so far has not been stated in general form for more than two phases. It has to be extended to include partitioning of the components between the phases and its effects on phase properties. A summary of the start of the art, including recent work on systems with up to three components and two phases, has been given by Pope. 4 This paper describes the extension of the theory to multicomponent, multiphase displacement with partitioning and for arbitrary initial and boundary conditions. The theory concerns itself only with transport behavior. Phase equilibrium and flow properties of the phases (relative permeabilities) as a function of composition are considered as given. Application of the theory, therefore, requires as input either empirical correlations of experimental data on phase equilibria and properties or theories predicting these. Morever, the theory concentrates exclusively on multicomponent, multiphase effects and does not attempt to account for the complex fluid dynamic situation in real, three-dimensional, and nonuniform reservoirs.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*20 (02): 105–112.

Paper Number: SPE-8020-PA

Published: 01 April 1980

... extension), and perforation, recasing, or bottomhole extension), and (6)

**variations**in baseline chemistry (to distinguish between average operating values and unique well conditions or to specify unusual brine flow patterns).These six points are essential for meaningful comparisons of the brine compositions...
Abstract

A brief report is given of studies of brine chemistry on both high- and low-salinity geothermal fields in support of a field corrosion testing program being conducted by the USBM in the Imperial Valley, CA. Specific results are reported for four geothermal wells: Mesa 6–1, Mesa 6–2, Magmamax No. 1, and Woolsey No. 1. These results demonstrate the necessity for careful reporting of the specific well operating conditions and brine sampling techniques under which the brine analyses were obtained. In particular, information related to recent well shut-in particular, information related to recent well shut-in periods, total stabilization time, recent production periods, total stabilization time, recent production engineering, brine flow rate from the well, and identification of nonturbulent-structure brine-flow configurations must be documented carefully with any reported analyses. Introduction For the past several years, the USBM has been involved in the nation's geothermal program, with primary responsibility for developing technology for primary responsibility for developing technology for recovering important metals and minerals from geothermal brines. Because the most accessible U.S. geothermal mineral resources occur in extremely corrosive hydrothermal fluids, the bureau also has conducted research to identify construction materials for process plants designed to recover these resources.The largest identified geothermal resource area in the U.S. containing substantial quantities of potentially recoverable metals and minerals is in the potentially recoverable metals and minerals is in the Imperial Valley. Of six known geothermal resource areas (KGRA's) there, the Salton Sea KGRA contains brines with the highest mineral content - 25 to 32% total dissolved solids (TDS). The brines from the Salton Sea KGRA, however, are among the most singularly corrosive natural fluids to be found, and during any type of brine processing a wide range of scaling phenomena occurs that can create havoc within a geothermal resource recovery plant. Early attempts to recover these geothermal resources were abandoned, partly due to the failure to overcome these corrosion and scaling problems.This paper presents on-site brine chemical analyses for the early stages of production for four geothermal wells and discusses how these analyses can be influenced by operational conditions. In addition to specifying the analytical and sampling procedures used for geothermal brine analyses, a procedures used for geothermal brine analyses, a number of important conditions concerning the geothermal well in question must be specified for meaningful interpretation of the analytical data. Much of the data reported in the literature does not include this type of information, thus limiting its value. These conditions, defined here as the "reportable conditions for geothermal brine chemistry data," are (1) sampling procedure (to include temperature, pressure, date, type of sampling port, and either suspected or known phase of the port, and either suspected or known phase of the preextracted sample - i.e., brine, steam, or mixed preextracted sample - i.e., brine, steam, or mixed phases), (2) total flow rate from the well (in volume phases), (2) total flow rate from the well (in volume per time interval), (3) shut-in time (if well is being per time interval), (3) shut-in time (if well is being restarted after a period of nonflow), (4) total operating time (of actual brine-flowing operations), (5) production engineering (including any recent perforation, recasing, or bottomhole extension), and perforation, recasing, or bottomhole extension), and (6) variations in baseline chemistry (to distinguish between average operating values and unique well conditions or to specify unusual brine flow patterns).These six points are essential for meaningful comparisons of the brine compositions of different wells, the variations in brine chemistry with time for a single well, and the sampling and analytical results for brines from the same well obtained by different organizations. SPEJ P. 105

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*19 (05): 301–312.

Paper Number: SPE-6734-PA

Published: 01 October 1979

...P.J. Closmann; W.B. Bradley The analysis of underground oil-shale recovery processes requires knowledge of the mechanical properties of oil shale at various temperatures. The tensile strength, compressive strength, and Young's modulus are of special importance. The

**variation**of these properties...
Abstract

The analysis of underground oil-shale recovery processes requires knowledge of the mechanical properties of oil shale at various temperatures. The tensile strength, compressive strength, and Young's modulus are of special importance. The variation of these properties with temperature is important when assessing the strength of underground columns and confining walls for process cavities. This paper presents the results of an experimental study to quantify this temperature dependence. We found that both tensile and compressive strengths of oil shale show a marked decrease in strength as temperature increased, for a given richness. For example, for 15.6 gal/ton oil shale, the tensile strength at 400 deg. F is only 28% of its room temperature value. For 19.2 gal/ton shale, the compressive strength at 400 deg. F with 500-psi confining pressure is 43% of its value at room temperature. At a given temperature, both the tensile and compressive strengths decrease as richness increases, although the rate of decrease diminishes at richnesses of about 42 gal/ton and higher. Equations are developed to permit estimates of the various parameters involved. The compressive Young's moduli show a considerable decrease with temperature. At 400 deg. F the modulus is reduced to 51% of its room temperature value. Introduction In-situ processes for recovery of oil from nahcolite-bearing oil shale usually involve some heating or pyrolysis of the shale. Wet processes (steam, hot water) also involve dissolution of nahcolite to generate pore space and to create permeability. If the leaching of nahcolite is conducted at a sufficiently high temperature, some stress will develop in the rock beyond the heated cavity boundary because of CO2 generation and possibly water vapor, as follows. 2NaHCO3 goes to Na2CO3 + H2O + CO2. When the decomposition pressure of nahcolite is high enough, the rock tends to fracture ("popcorn effect"). Rubbling of the formation then can occur. To predict conditions suitable for fracturing and rubbling, we need to know how the rock tensile strength varies with temperature. McLamore measured the oil-shale tensile strength as a function of orientation of stress. So far as we know, no measurements of tensile strength as a function of temperature have been reported for oil shale. We also need to know the variation of nahcolite decomposition pressure with temperature. This pressure variation was measured by Templeton. The variation of Young's modulus, compressive strength, and Poisson's ratio also have been reported for various richnesses. Logan and Heard studied the compressive Young's modulus and thermal expansion as functions of richness. Compressive strength of oil shale has been studied extensively. This parameter was measured as a function of oil-shale richness for various confining pressures in triaxial tests at temperatures up to 300 deg. C (572 deg. F). The effect of temperature on rocks other than oil shale has also been studied. Knowledge of the compressive strength is important when assessing the possibility of failure of underground supporting walls in mines or with process cavities. Since the reacted oil shale probably will support the walls or the roofs of the process cavities very little, the strength of the supporting walls and roof under process conditions will determine the tendency for subsidence or intercavity communication. SPEJ P. 301^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*18 (06): 418–434.

Paper Number: SPE-6771-PA

Published: 01 December 1978

... with fluids ahead and behind or by loss to, or gain from, the rock over which it travels, but otherwise conserving its integrity. Multicomponent theory shows this idea is too naive. Rather, an injection sets off a set of "waves" (composition

**variations**) that advance at different speeds and between which new...
Abstract

The interfacial activity and mobility control of a chemical flooding process are affected by the concentrations of the cationic and anionic species that travel with the surfactant and polymer. In this paper we use equations from the literature to paper we use equations from the literature to describe the environmental composition changes resulting from cation exchange that occurs as a chemical flood traverses a reservoir. This paper presents examples of two or three exchanging cations (calcium, sodium, and magnesium) with and without mobilized oil present and with up to four fluids in a typical chemical flooding sequence (connate water, preflood, slug, and polymer drive). The results indicate how cation polymer drive). The results indicate how cation exchange and adsorption may be incorporated into a chemical flood design. The general theory from which the results are developed is based on the concept of "coherence." This theory allows any number of exchanging cations to be present and allows adsorption of surfactant, polymer, or other species and their interaction with cation exchange to be included. Introduction A key requirement for a successful chemical flood is to provide an adequate ionic environment for the surfactant, to ensure that the desired interfacial activity, phase behavior, and mobility control are maintained. Aside from the inplace and injected ionic compositions and mixing through dispersion, crossflow, etc., this environment may be affected deeply by cation exchange with clays, solubility of minerals, and adsorption on rock. The importance of cation exchange effects in chemical flooding recently has been stressed and need not be reiterated here. We describe a fundamental theoretical analysis of cation exchange and adsorption phenomena in reservoir flooding. The treatment is applicable to multicomponent systems with any kind of equilibrium relations, specifically including interactions between components, but presumes idealized behavior with respect to fluid dynamics, absence of dispersion, immiscibility of aqueous and oleic phases, and conservation of local equilibrium. The treatment is an adaptation of multicomponent chromatographic theory to practical problems of chemical and related floods. The bask problem is that the components involved in a chemical flood--water, cations, surfactant, polymer, and oil--are coupled with respect to their transport properties, and only a theory of coupled, multicomponent systems can adequately describe their dynamic behavior. At first glance, one may be inclined to assume that a mixture injected as a slug might traverse the reservoir as such, changing its composition a little by mixing with fluids ahead and behind or by loss to, or gain from, the rock over which it travels, but otherwise conserving its integrity. Multicomponent theory shows this idea is too naive. Rather, an injection sets off a set of "waves" (composition variations) that advance at different speeds and between which new compositions arise that bear little resemblance to the injected and previously present compositions, or any that could be formed from these by mixing. Moreover, the wave patterns generated by successive injections of different fluids may overlap and interfere and, thereby, modify injected compositions. Injected components thus generate their own environment through dynamic interactions. To be sure, it is not impossible, in principle, to operate under conditions ensuring that an injected active surfactant slug retains its favorable environment and thus its activity through most or all of the reservoir, but this often may prove impracticable. The task then is to design the flood so that a favorable environment is generated in-situ. This paper tries to present a theoretical basis that will paper tries to present a theoretical basis that will facilitate such design. SPEJ P. 418

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*18 (06): 399–408.

Paper Number: SPE-6309-PA

Published: 01 December 1978

... in presented at the Eighth Annual Offshore Technical Conference, held in Houston, May 3-6, 1976. Abstract Excessive riser-force

**variation**on the upper joint in a riser string can lead to buckling and excessive fatigue. This**variation**is caused by two components of the riser support system - the riser...
Abstract

Original manuscript received in Society of Petroleum Engineers office Feb. 24, 1976. Paper accepted for publication June 14, 1977. Revised manuscript received Aug. 21, 1978. Paper (SPE 6309, OTC 2648) first presented at the Eighth Annual Offshore Technical Conference, held in presented at the Eighth Annual Offshore Technical Conference, held in Houston, May 3-6, 1976. Abstract Excessive riser-force variation on the upper joint in a riser string can lead to buckling and excessive fatigue. This variation is caused by two components of the riser support system - the riser tensioning system and the telescopic, or slip, joint. Using specific examples, two conclusions are reached. First, the force variation at the top of the riser string may be much greater than that indicated by monitoring the tensioner system's air-tank pressure. Second, a major contribution to this pressure. Second, a major contribution to this variation can be pressure drop in the air valves. Introduction The riser tensioners and slip joint (Fig. 1) form the support system for the riser string used in floating drilling operations. Although tensioners are the primary support mechanism, their forces are transmitted through the slip joint to the upper joint in the riser string. In many deep-water drilling operations, the riser string is isolated in bending by an upper ball joint from the more massive telescopic joint. This upper ball joint interacts directly with the riser string; therefore, the forces seen at that joint become riser-string forces because of the tensioner support system. Ideally, the tensioner support-system forces at the upper ball joint should provide a net axial load on the riser string and should be constant in magnitude as well as direction. However, the nonideal behavior of the riser tensioners - as well as the inertia and geometrical effects associated with vessel, slip joint, and riser-string motions - result in load variations. Generally, the upper ball-joint force vector depends on time. No limits as yet have been determined for allowable variations of the riser-string forces resulting from the riser support mechanism. Nevertheless, measuring these variations analytically and qualitatively is important when assessing the effectiveness of the support mechanism or when providing important information about the boundary providing important information about the boundary conditions necessary to analyze the riser string. Our paper has two purposes. First, to emphasize by numerical examples the strong dependence of riser-tensioner force variations on the character of the assumed losses (pressure chop) in the tensioner-system air valves. Second, to present an analytical expression and numerical results for the tensioner-system force variations at the upper ball joint, thereby emphasizing the strong effects of vessel motion on riser-string force. TENSIONER ANALYSIS The typical drilling riser tensioner is a hydropneumatic mechanical system (Fig. 2) that provides tension in the cable attached to and supporting the outer barrel of the slip joint. Kozik studied the cable tensioner variation (r) resulting from cable motion. A convenient form for his equation is .....................(1) SPEJ P. 399

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*18 (02): 117–122.

Paper Number: SPE-5892-PA

Published: 01 April 1978

... (developed in the Appendix) includes the following assumed mechanisms and their corresponding mathematical expressions. SPEJ P. 117 1 4 1978 1 4 1978 1978. Society of Petroleum Engineers Upstream Oil & Gas

**variation**boundary condition time necessary flow in porous media...
Abstract

This paper presents a new theory of the incompressible flow of two fluids (water displacing oil) in a fractured porous material composed of two distinct media - matrix blocks of low transmissibility embedded in a highly transmissible medium. This general description includes heterogeneous porous media not necessarily of the fractured type. The theory accounts for an important fact not considered in framer analytical model found in the literature. The blocks downstream in a reservoir subject to waterflood are exposed to a varying water saturation resulting from the water imbibition of the upstream blocks. Expressions for the water-oil ratio and the cumulative-oil production are derived, allowing a complete economic evaluation of a fractured-reservoir waterflood project. Comparison of experimental curves reported in the literature with curves obtained using this theory show a good fit. Introduction Imbibition is a most important mechanism of oil production in the waterflooding of fractured production in the waterflooding of fractured reservoirs. Using the action of capillary forces, it allows the recovery of oil from the interior of blocks that cannot be reached by the externally applied gradients of the waterflood. Previous papers assume a function to describe the time rate of exchange of oil and water for a single matrix block. In a lineal reservoir, a water table advances as water is injected with the matrix blocks progressively exposed to water, depending on their position. The oil released by the matrix blocks is assumed transferred instantly to the water-oil interphase,. In this way, the oil production is an added function of individual block contributions. An analytical approach to this problem, and a numerical model, use the problem, and a numerical model, use the simplifying assumption of a water front. This may be a sound description in the presence of vertical high-transmissivity fractures where oil may segregate readily, but in fractures with a discrete transmissivity, it is expected that water imbibition and the simultaneous release of oil by these blocks will give rise to a varying saturation in the fractures that will affect the imbibition rates of the downstream blocks. Braester's analytical approach assumes relative permeabilities of both wetting and nonwetting permeabilities of both wetting and nonwetting phases, intermediate between the fracture's and the phases, intermediate between the fracture's and the matrix's relative permeabilities; these intermediate permeabilities are impossible to measure. The permeabilities are impossible to measure. The model also uses an approximation of the fluid interchange between fractures and blocks. The model may be used for predictions after finding parameters to match observed oil and water parameters to match observed oil and water productions. productions. Kleppe and Morse conducted laboratory experiments on matrix blocks surrounded by fractures and numerical simulations (with rather coarse numerical grids) of Braester's laboratory system. Their numerical simulation computations agree well with the experimental results. This numerical formulation is exact or causalistic; capillary pressures and relative permeabilities are computed pressures and relative permeabilities are computed at every grid block. Their experimental and numerical results are used to test the theory presented here. presented here. Another numerical formulation assumes an approximation for the fluid interchange between fractures and matrix blocks. This approximate formulation did not try to reproduce the exact formulation results of Kleppe and Morse, nor their laboratory experiments. The theory presented here analitically accounts for varying saturations in the fractures by introducing a convolution. A somewhat similar approach -was used successfully to describe the transient one-phase flow in a fractured reservoir. THEORY An outline of the subject theory (developed in the Appendix) includes the following assumed mechanisms and their corresponding mathematical expressions. SPEJ P. 117

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*18 (01): 33–41.

Paper Number: SPE-6088-PA

Published: 01 February 1978

... of Petroleum Engineers relative fracturability Fracturability experimental result fracture Carthage Limestone hydraulic fracturing

**variation**fluid pressure hydraulic fracture interface propagation hydraulic fracture propagation experiment fracture propagation limestone layered formation...
Abstract

This paper reports theoretical and experimental developments involving propagation of hydraulic fractures in layered formations. Unobstructed fractures are shown experimentally to propagate with a decreasing fracturing fluid pressure. This general trend is in agreement with pressure. This general trend is in agreement with theoretical predictions. Restrictions in fracture propagation result in an increase in fluid pressure. propagation result in an increase in fluid pressure. The relative fracturability of rocks can be determined by a direct experiment, the results of which are clear, easy to interpret, and include all pertinent parameters, such as physical and pertinent parameters, such as physical and mechanical properties of rocks, as well as the reactions between formation and fracturing fluid (for example, leak-off). Fracturing experiments with layered samples show that with strong bonding between rocks it is difficult to contain a fracture in a formation totally. The strength of the interface between adjacent formations is shown theoretically to be an important factor in fracture containment. With a weak bonding, fracture containment is possible and is associated with slippage at the interface. The pattern of propagation then will depend on the relative propagation then will depend on the relative mechanical properties of fractured formations. Introduction Most industrial hydraulic fractures are created in layered formations. During propagation, these fractures encounter various formations with different physical and mechanical properties. This paper physical and mechanical properties. This paper discusses the effect of those properties on propagation of the fracture. propagation of the fracture.Most of the theoretical studies on fracture propagation have been extensions of Griffith's propagation have been extensions of Griffith's work. Based on an energy criterion, Griffith developed a relationship among fracture shape, material properties, and the external force needed for fracture propagation. The energy source in hydraulic fracturing is the fluid pressure inside the fracture. The relationship between this pressure and material properties is (1) (2) in which L = fracture extent (length of a two-dimensionalfracture or radius of a penny-shapedfracture) E = Young's modulus of material mu = Poisson's ratio of material gamma = effective fracture surface energy of material sigma = least in-situ principal stress A similar equation for a three-dimensional fracture is derived in Appendix A in the form of (3) in which hf = fracture height E(k) = complete elliptic integral of the secondkind K(k) = complete elliptic integral of the first kind k = parameter of the elliptic integrals Eqs. 1 through 3 show p to decrease with increasing L (Fig. 1) As the fracture becomes larger, it needs less pressure for propagation. In deriving these equations, no allowance has been made for fluid leak-off into the formation. SPEJ P. 33

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*18 (01): 27–32.

Paper Number: SPE-6089-PA

Published: 01 February 1978

... Reservoir Characterization stress-intensity factor

**variation**in-situ stress hydraulic fracture reservoir geomechanics containment pay zone loading representation tectonic stress gradient interface propagation stiffness permeability fracture barrier layer Containment of Massive Hydraulic...
Abstract

Hydraulic fracture containment is discussed in relationship to linear elastic fracture mechanics. Three cases are analyzed, the effect of different material properties for the pay zone and the barrier formation, the characteristics of fracture propagation into regions of varying in-situ stress, propagation into regions of varying in-situ stress, and the effect of hydrostatic pressure gradients on fracture propagation into overlying or underlying barrier formations. Analysis shows the importance of the elastic properties, the in-situ stresses, and the pressure gradients on fracture containment. Introduction Application of massive hydraulic fracture (MHF) techniques to the Rocky Mountain gas fields has been uneven, with some successes and some failures. The primary thrust of rock mechanics research in this area is to understand those factors that contribute to the success of MHF techniques and those conditions that lead to failures. There are many possible reasons why MHF techniques fail, including migration of the fracture into overlying or underlying barrier formations, degradation of permeability caused by application of hydraulic permeability caused by application of hydraulic fracturing fluid, loss of fracturing fluid into preexisting cracks or fissures, or extreme errors in preexisting cracks or fissures, or extreme errors in estimating the quantity of in-place gas. Also, a poor estimate of the in-situ permeability can result in failures that may "appear" to be caused by the hydraulic fracture process. Previous research showed that in-situ permeabilities can be one order of magnitude or more lower than permeabilities measured at near atmospheric conditions. Moreover, studies have investigated the degradation in both fracture permeability and formation permeability caused by the application of hydraulic fracture fluids. Further discussion of this subject is beyond the scope of this paper. This study will deal mainly with the containment of hydraulic fractures to the pay zone. In general, the lithology of the Rocky Mountain region is composed of oil- and gas-bearing sandstone layers interspaced with shales (Fig. 1). However, some sandstone layers may be water aquifers and penetration of the hydraulic fracture into these penetration of the hydraulic fracture into these aquifer layers is undesirable. Also, the shale layers can separate producible oil- and gas-bearing zones from nonproducible ones. Shale layers between the pay zone and other zones can be vital in increasing successful stimulation. If the shale layers act as barrier layers, the hydraulic fracture can be contained within the pay zone. The in-situ stresses and the stiffness, as characterized by the shear modulus of the zones, play significant roles in the containment of a play significant roles in the containment of a hydraulic fracture. The in-situ stresses result from forces in the earth's crust and constitute the compressive far-field stresses that act to close the hydraulic fracture. Fig. 2 shows a schematic representation of in-situ stresses acting on a vertical hydraulic fracture. Horizontal components of in-situ stresses may vary from layer to layer (Fig. 2). For example, direct measurements of in-situ stresses in shales has shown the minimum horizontal principal stress is nearly equal to the overburden principal stress is nearly equal to the overburden stress. SPEJ P. 27

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*16 (02): 97–109.

Paper Number: SPE-5489-PA

Published: 01 April 1976

... variability coefficient loading strength

**variation**bearing capacity Soil property foundation resistance cohesion correction factor London Clay Probabilistic Investigation of Foundation Design for Offshore Gravity Structures L. M. KRAFT, JR. J.D. MURFF ABSTRACT This paper lays the groundwork...
Abstract

This paper lays the groundwork for establishing foundation safety criteria for offshore gravity structures. The concepts are explained in terms of first- and second-order uncertainty analyses. Various uncertainties associated with foundation analyses are identified and applications are illustrated with examples. Introduction Gravity structures play a prominent role today in North Sea oil development. These structures are not supported by piles, as are most ocean structures, but rather sit directly on the ocean bottom and depend on their foundation geometries and large weights m resist severe environmental loadings. A number of structural and foundation configurations have been proposed; however, attention is restricted here to a general configuration typical of the most prominent structures presently being constructed. prominent structures presently being constructed. An example of a gravity structure is illustrated in Fig. 1. The structure foundation consists of a large caisson placed directly on the unprepared sea-bed surface. The deck is supported by large columns extending from the caisson. Various combinations of steel and reinforced concrete have been proposed, but most structures are being constructed almost entirely of reinforced and prestressed concrete. prestressed concrete. One of the primary engineering concerns with these structures is foundation design. Because of the variability associated with the environmental forces, as well as the basic soil properties, this problem lends itself well to modem probabilistic problem lends itself well to modem probabilistic procedures. Such procedures provide a rational, procedures. Such procedures provide a rational, quantitative means for evaluating uncertainties affecting appropriate design, even though a degree of subjectivity will always remain in any such evaluation. The probabilistic method requires the engineer to formally and consistently recognize die variability of many of the important design parameters. The method gives management and parameters. The method gives management and others responsible for setting design criteria an opportunity to appraise cost/benefits of design levels required for given reliability levels. It also quantifies reliability to permit direct comparison with other options. This paper presents a method for analyzing the reliability of gravity-structure foundations in terms of simple loading and resistance models. The sources of variability in estimating resistance to loads are discussed, with particular emphasis on the nature of soil-property variability and uncertainty. These concepts are illustrated through an analysis of a typical gravity-structure foundation. SPEJ P. 97