Summary This paper presents two kinetic models for representing the thermal cracking of crude oils, which incorporate the cracking rate parameters and stoichiometric coefficients to correlate experimental data. The models presented show that the first-order kinetics generally accepted for pure components are unsatisfactory for multicomponent systems characterized by pseudocomponents. We conclude that three corrections to the existing first-order model are needed for modeling thermal cracking of mixtures. First, the apparent reaction order is always greater than one. Second, the reaction order is a decreasing function of temperature. Third, coke may also be formed from intermediate products. These corrections are incorporated into the models. In the first model, crude oil is split into two pseudocomponents, while in the second model, crude oil is represented by three pseudocomponents. The models can be easily extended to any number of pseudocomponents. The models successfully correlated experimental data of four systems available in the literature. Furthermore, it was confirmed that coke is not always the sole source of the fuel burned in an in-situ combustion process.

Abstract This paper presents a set of curvilinear coordinate transformations that lead to no mixed derivative terms in transformed flow equations. The transformations are created by examining the transformed flow equations and by showing that the mixed derivative terms are zero if the transformation satisfies a system of differential equations that depend on the geometry and rock property distribution within the reservoir. Numerical examples with a black-oil simulator are presented to show the increased accuracy resulting from the use of the curvilinear coordinate system and the importance of eliminating the mixed derivative terms.

Abstract A new mathematical model is developed for analyzing single-well pulse test data from a reservoir consisting of two permeable zones separated by a less permeable zone. The model considers well-bore storage and skin effects and allows the three zones to have different rock properties. The line-source assumption used in developing the analytic solution is validated via comparison of results with another model also developed in this work. This latter model assumes identical rock properties for the two permeable zones but allows the wellbore radius to be finite. The numerical Laplace inversion methods available in the literature are evaluated with respect to some simple functions and the analytic Laplace space solution of the problem of interest. It is found that the Fourier-series and Zakian's methods can serve as alternatives to the well-known Stehfest algorithm. The effects of wellbore storage and skin on single-well pulse testing of layered reservoirs are investigated. Consistent with the findings in the literature for multiwell pulse testing, it is shown that the influence of wellbore storage and skin is to reduce the response amplitude and to increase the time lag of the pulse response. Among the wellbore storage and skin factors involved in a single well pulse testing, the skin associated with the observation interval is least influential on the pressure response. Using field data, it is shown that the mathematical model presented can be used, in conjunction with an optimization subroutine, to estimated the parameters of interests. It is further shown numerically that neglecting the effects of wellbore storage and skin could result in an underestimation of the formation permeabilities of interest. The problem of non-uniqueness of solution for parameter estimation using optimization techniques is also discussed for the subject problem.

Abstract This paper presents the results of numerical simulation of dry, forward combustion tube experiments. The kinetic aspects of in-situ combustion processes also are discussed. The goals of the study are to investigate the fuel deposition mechanism and to identify the key parameters affecting the performance of in-situ combustion processes. The thermal simulator developed at Gulf R&D Co. was used in the study. It was modified to include the capillary outlet effects for a more realistic description of the oil and water productions. The following experimental data were matched: cumulative water and oil productions, position of the combustion front as a function of time, fuel consumption, temperature as a function of time and position, and the pressure drop across the tube. History matches were performed for two crude oils with distinctly different physical properties (gravities of 26.5 and 13°API [0.896 and 0.979 g/cm3]). The agreements between experimental data and simulation results were excellent. Results indicate that the component equilibrium K-values and the kinetics of cracking reactions are the most important parameters affecting the fuel deposition, and that the fuel deposition mechanism, the fuel composition, and the locations and sizes of the transient zones depend on the crude oil and reservoir rock properties. Simulation results are always sensitive to the K-values of the light oil component but insensitive to the K-values of the heavy oil component. Results are sensitive to the kinetics of cracking reaction only if the cracking reaction is catalytic or the peak temperature and the fuel consumption are sufficiently high. Furthermore, the fuel available may or may not be solely in the form of coke. Our study suggests that further investigations of the catalytic effect of reservoir rocks and reaction kinetics of the cracking reaction are needed. Also, more than two crude oil components may be required to simulate the evaporation effect of crude oil accurately.

Abstract Experimental phase equilibrium data are presented for three reservoir oils at conditions approximating those encountered in in-situ thermal recovery processes. The fluid systems involved consist of three major groups of components: flue gas, water, and crude oil. Data were measured at temperatures from 204.4 to 371.1°C (400 to 700°F) and pressures from 6996.0 to 20 785.6 kPa (1,000 to 3,000 psia). Experimental phase equilibrium data were used to develop a correlation of binary interaction coefficients of crude-oil fractions required for the Peng-Robinson equation of state. Phase equilibrium data predicted using the Peng-Robinson equation of state, using our interaction coefficients, are compared with experimental data. Generally, the Peng-Robinson equation of state predictions were in close agreement with the experimental data. Effect of feed gas/oil ratio and water/oil ratio on the equilibrium coefficients was examined through the Peng-Robinson equation of state. A study on the feasibility of representing the crude oil by only two fractions was made also. This study includes a procedure for lumping the crude-oil fractions and examples showing the importance of mixing rules in determining the pseudocritical properties of lumped fractions.

Abstract This paper describes a model for numerically simulating thermal recovery processes. The primary locus is on the simulation of in-situ combustion, but the formulation also represents fire-and-water flooding, steamflooding, hot water flooding, steam stimulation, and spontaneous ignition as well. The simulator describes the flow of water, oil, and gas, and includes gravity and capillary effects. Heat transfer by conduction, convection, and vaporization-condensation of both water and hydrocarbons are included. The rigorous but general nature of the simulator is obtained by employing conservation balance equations for oxygen, inert gases, a light hydrocarbon pseudocomponent, a heavy hydrocarbon pseudocomponent, water, coke, and energy. pseudocomponent, water, coke, and energy. Vaporization-condensation is governed by vaporliquid equilibrium using temperature and pressure-dependent equilibrium coefficients. Four pressure-dependent equilibrium coefficients. Four chemical reactions are accounted for: formation of coke from the heavy hydrocarbon component and the oxidation of coke and both heavy and light hydrocarbon components. Formulation details, numerical solution procedures, and computational results are presented. procedures, and computational results are presented. The computational results include both one- and two-dimensional cross-sectional studies. The simulator represents a major improvement in the ability to simulate thermal recovery processes under complex conditions. Introduction Considerable progress has been made in numerically simulating thermally enhanced oil-recovery processes during the last few years. This is particularly true for-processes involving steam, where we have seen a continual improvement of our ability to treat the problem. The most recent contributions provide an analysis capability for steam displacement and steam stimulation recovery methods, accounting for all the important physical mechanisms of these processes. Progress in simulating the performance of in-situ combustion processes is not so advanced. Initial simulation attempts were concerned primarily with the heat-transfer aspects of combustion. The most sophisticated heat-transfer model was developed by Chu. His numerical model considers the energy effects of vaporization and condensation on the temperature distribution, but neglects the accompanying phase changes by assuming constant fluid saturations. More recent heat transfer or heat-wave models for the in-situ combustion process were proposed by Kuo in 1969 and by Smith and Farouq-Ali in 1971. Kuo's model allows two temperature fronts-one at the combustion zone and one at a heat front. The heat-front position is predicted by gas flow that is allowed to have a velocity different from the velocity of the combustion front. The simulator proposed by Smith and Farouq-Ali is designed for proposed by Smith and Farouq-Ali is designed for predicting sweep efficiencies in confined well predicting sweep efficiencies in confined well patterns. Their numerical model accounts for heat patterns. Their numerical model accounts for heat generation by a combustion zone (assuming fixed fuel content all through the reservoir), heat transfer by conduction and convection (single-phase gas flow) in the reservoir, heat losses by conduction to adjacent formations, and different permeability-to-gas (air) flow on either side of the combustion zone. Special cases of the in-situ combustion process were studied by Gottfried and Khelil. These authors examine the heat transfer and oxygen use in reservoirs composed of an oil-bearing layer and an overlying "clean" porous zone containing only gas. These models were designed primarily to investigate the various transport mechanisms present when combustion is initiated in a reservoir present when combustion is initiated in a reservoir containing a gas cap. Because of the many assumptions invoked and the specialized geometry to which they apply, they do not satisfy the need for a general purpose simulator. SPEJ P. 37

Abstract Pressure buildup and flow tests conducted in wells that do not completely penetrate the producing formation or that produce from only a small portion of the total productive interval can generate noncylindrical flow regimes and require special interpretation procedures. Frequently a spherical flow regime is representative, and a new equation based on the continuous point-source solution to the diffusivity equation in spherical coordinates is presented for analyzing tests of this nature. The practical utility of the equation is demonstrated by practical utility of the equation is demonstrated by analyzing tests involving restricted producing intervals that cannot be treated with existing analytic methods.Practical guidelines for applying the proposed equation are developed by analyzing pressure data generated by a numerical simulator and more complex analytic solutions for variety of special completion situations. Equations for determining static reservoir pressure, formation permeability, and skin factors pressure, formation permeability, and skin factors are derived and their validity verified under theoretical test conditions. The equations presented should have a variety of applications, but are particularly suited for analyzing pressure data from particularly suited for analyzing pressure data from drillstem tests with short flow periods. Introduction The fundamental equation for analyzing pressure buildup tests of oil wells was presented by Horner in 1951. This equation is based on the "line source" solution to the boundary value problem describing the pressure distribution resulting from the cylindrical flow of a slightly compressible fluid in an infinite reservoir. To achieve cylindrical flow the wellbore of a well must completely penetrate the producing formation. Although this restriction is often satisfied, in many tests it is not; for example, oil wells producing through perforated casing may have only a small portion of the total production interval perforated, or in the case of production interval perforated, or in the case of drillstem tests only a small interval (often 10 to 15 ft) of a thick (hundreds of feet) homogeneous formation may be selected for testing. Tests involving restricted producing intervals of this type have a characteristic buildup curve as described by Nisle and by Brons. These authors demonstrated that Horner's conventional equation could also be used for restricted producing interval problems, provided the correct portion of the buildup problems, provided the correct portion of the buildup curve is used. They showed that during a short period after starting production (or equivalently period after starting production (or equivalently after shut-in) the well behaves as if the total sand thickness were equal to the interval open to flow. That is, Horner's equations apply if the total sand thickness, h, is replaced by the producing interval thickness, h. They also showed that after a transition period the late part of the buildup curve could be used in the conventional manner to calculate formation permeability and static reservoir pressure. Kazemi and Seth extended the work of pressure. Kazemi and Seth extended the work of Nisle by including the effect of anisotropy; they also presented an equation, based on an analytic solution developed by Hantush, for estimating the shut-in time required for the development of the second straight-line portion in a conventional plot --i.e., p vs In (t + Deltat/Deltat. The first straight-line part o the buildup curve usually lasts only a few part o the buildup curve usually lasts only a few minutes and may often be obscured by afterflow, whereas the latter straight-line portion may take several hours to develop and may not even occur for practical shut-in times if the formation is thick and the producing time is relatively short. This paper demonstrates that the transition period paper demonstrates that the transition period between the two cylindrical flow periods can be analyzed with the spherical* flow equations presented here. In addition, practical guidelines presented here. In addition, practical guidelines cue developed for their application.Moran and Finklea first suggested that a pressure buildup equation based on spherical now pressure buildup equation based on spherical now was necessary to correctly analyze pressure data obtained from wireline formation testers. In many respects this study is similar; in fact, the basic pressure buildup equation (although it was derived pressure buildup equation (although it was derived from a different starting equation) presented here was used by Moran and Finklea in analyzing wireline formation test data.

Abstract This paper presents and examines in detail extensions to the Galerkin method oil solution that make it numerically superior to conventional methods used to solve a certain class of time-dependent, nonlinear boundary value problems. This class of problems includes the equation that describes the flow of a fully compressible fluid in a porous medium. The Galerkin method with several different piecewise polynomial subspaces and a non-Galerkin piecewise polynomial subspaces and a non-Galerkin method specifically employing cubic spline functions are used to approximate the solution of a nonlinear parabolic equation with one spatial variable. With a parabolic equation with one spatial variable. With a known analytic solution of the problem, the accuracies of these approximations are determined and compared with conventional finite-difference approximations. Specially, the various methods are compared on the basis of the amount of computer time necessary to achieve a given accuracy, as well as with respect to the order oil convergence and computer core storage required. These tests indicate that the higher-order Galerkin methods require the least amount of computer time for a given range of accuracy. Introduction The purpose of this paper is to outline in detail the application of the Galerkin method, employing piecewise polynomials, to solve nonlinear piecewise polynomials, to solve nonlinear boundary-value problems and compare the computational efficiency of the Galerkin method with more conventional numerical methods. Numerical methods compared with the Galerkin technique include a non-Galerkin method that utilizes cubic spline interpolation and the conventional finite-difference methods. Four conventional time approximations were also studied in conjunction with the above mentioned space discretization methods. In an earlier paper, Price and Varga showed theoretically that higher-order approximations to certain semilinear convection-diffusion equations were possible by means of Galerkin techniques, but complete numerical results for such approximations were not given. Also, in a paper that introduced the Galerkin method to the petroleum industry, Price et al. demonstrated that higher-order approximations were far superior numerically to the conventional methods used to solve certain linear convection-diffusion type problems. Jennings, Douglas and Dupont and Douglas et al. have considered the application of Galerkin methods to various nonlinear problems, but again complete numerical results, problems, but again complete numerical results, including comprehensive comparisons with existing numerical methods, were not given. Thus, in addition to presenting some new and computationally efficient Galerkin formulations for nonlinear problems and numerically demonstrating their problems and numerically demonstrating their higher order accuracies, it was also desirable to test these. methods to determine if they also exhibited the same superiority in regard to computational efficiency as was demonstrated for the Galerkin methods applied to linear problems. If so, then the Galerkin technique could prove to be an important advancement toward developing faster numerical models for field application. To test and compare each method of solution, a problem involving the nonlinear gas-flow equation problem involving the nonlinear gas-flow equation in one spatial variable with a specific volumetric source term was chosen, for which a closed-form or analytic solution was known. Using this particular problem and its analytic solution, it was possible problem and its analytic solution, it was possible to determine numerically the order of convergence of each method, to compare each method on the basis of computer time expended to obtain a given accuracy, and to compare each method with respect to computer core storage required. In addition, the experimental data were used to define "consistent quadrature" and "consistent interpolation" schemes for the Galerkin methods. Finally, it was possible to formulate conclusions regarding the computational efficiency of the four time approximations investigated. SPEJ P. 374

Pulse-testing theory as originally outlined is based Pulse-testing theory as originally outlined is based on the standard exponential integral solution to the radial form of the diffusivity equation. This paper presents a new solution to the subject diffusivity presents a new solution to the subject diffusivity equation and outlines how this type of solution can be used to clarify the "tangent construction method" of analyzing pulse data. The new solution is composed of a periodic part and a transient part. The splitting of the total solution into two parts facilitates the extension of pulse-testing to include steady-state conditions The differential equation and auxiliary conditions that define the periodic pulsing problem are .......(1) with the initial condition of ...........(2) and boundary conditions of .............(3) .............(4) where ............(5) and H(t) is the unit step function. The solution to this boundary value problem is usually represented by the summation of appropriate exponential integral terms. A solution that splits the total solution into periodic and transient portions can be obtained as periodic and transient portions can be obtained as follows. Utilizing Laplace transform theory, the solution to Eqs. 1 through 4 is ................(6) where, the real part of the complex integration limits, is taken large enough so that all singularities of the integrand lie to the left of the line ( - i, + i ). Complex variable theory was used to evaluate the integral in Eq. 6; and the final solution, in field units, can be shown to be .........(7) P. 1245

Abstract One- and two-dimensional mathematical models have been developed that simulate transient, two-phase flow of hydrocarbon mixtures in porous media in a manner that accounts for interphase mass transfer. Numerical simulations of one-dimensional depletion-drive experiments using a two-component hydrocarbon fluid were used to establish the validity of the mathematical models. In addition, the experimental and numerical data were used to demonstrate that production rate had a relatively insignificant effect on the recovery of individual hydrocarbon components from the experimental system, and that attainment of equilibrium between phases is possible for a wide range of liquid and vapor velocities in reservoirs containing light hydrocarbon fluids. Results of some two-dimensional numerical simulations are also presented. Introduction This study was undertaken to develop a mathematical model that would simulate transient, two-phase flow of hydrocarbon mixtures in porous media under conditions that result in interphase mass transfer and to test the validity of the assumptions used to set up the model. In addition, the study was designed to determine if production rate influences the recovery of individual hydrocarbon components from reservoirs producing by depletion drive. Two-phase flow in porous media, with interchange of components between the two phases, is important in many petroleum recovery processes. Studies conducted within the last 3 years have outlined methods of solving multiphase flow problems incorporating mass transfer. Some of these studies have also indicated the importance of accounting for mass transfer under various producing conditions. An earlier works first demonstrated the importance of combining relative permeability data with equilibrium ratios in compositional balance methods. The mathematical model presented in this paper is formulated so that a phase behavior package, as described in previous papers, is not required as an integral part of the routine employed to solve for the primary dependent variables. The finite difference formulation is designed so that all these variables can be solved for simultaneously. This is accomplished by utilizing one basic set of equations. These innovations, which are in contrast to other models but are similar in some respects to the approach used by Taylor, render the total problem computationally simpler than any of the previously referenced formulations. The mathematical model was developed by combining Darcy's law with a continuity equation for each hydrocarbon component. The principal assumptions invoked in the formulation were that capillary forces and diffusional effects are negligible, and that thermodynamic equilibrium exists in the reservoir at all times. No assumption as to the type of vaporization process was made in formulating this model. Experimental data were required to complete this study. These were generated by conducting several depletion drive experiments. The experimental apparatus consisted of a sandstone core enclosed in a pressurized casing. The apparatus was designed in such a manner that the core could be charged with a liquid hydrocarbon mixture and depleted at different production rates. The experimental tests were designed to determine the effect of production rate on component recovery.