CLOSED-FORM VAN DER WAALS CRITICAL POINT FOR PETROLEUM RESERVOIR FLUIDS by TALAL HUSSEIN HASSOUN, B.S., M.S. A DISSERTATION IN PETROLEUM ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved Akanni S. Lawal Chairperson of the Committee Lloyd R. Heinze James F. Lea Accepted John Borrelli Dean of the Graduate School May, 2005 ACKNOWLEDGEMENTS I wish to express my sincere thanks for the advice, guidance, and encouragement given by my supervisor, Dr. Akanni S. Lawal. I would also like to thank the members of my committee, Dr. Lloyd R. Heinze, and Dr. James F. Lea for their time and efforts. A special thanks is extended to Dr. U. Mann for his assistance and willingness to help. Finally, I thank Mr. S. Andreas, Mr. N. Kumar, and Mr. Tarek Hassoun for their help and discussion contributed to this dissertation. I would like to acknowledge the Petroleum Engineering Department for providing the financial support during the course of my doctoral studies. This Dissertation is dedicated to my father, my mother, my wife, Majida, my daughter Amani, and to my sons, Ashraf, Heitham, and Tarek for providing me with inspiration and confidence. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS ii ABSTRACT v LIST OF TABLES vii LIST OF FIGURES ix NOMENCLATURE xi CHAPTER I. INTRODUCTION 1 1.1 Importance of Critical State 7 1.2 Approaches to Critical State Prediction 12 1.3 Retrograde Reservoir Fluids 14 1.4 Objectives of Work 16 II. CRITICAL PROPERTY CORRELATION METHODS 18 2.1 Criteria of the Critical State 19 2.2 Empirical Models 22 2.3 Corresponding States 25 2.4 Convergence Pressure 31 2.5 Equation of State Models 45 III. CLOSED-FORM VAN DER WAALS EXPRESSIONS 54 3.1 Van der Waals Equations of State Theory 54 3.2 Closed-Form Equations for Fluid Critical Point 61 iii 3.3 Closed-Form Critical Property Computation Methods 69 IV. CRITICAL PROPERTIES FOR RESERVOIR FLUIDS 74 4.1 Critical Pressure Data for Complex Hydrocarbon Mixtures 74 4.2 Calculation of Critical Properties 4.3 Results and Discussion 75 4.4 Comparison Between Calculated and Experimental Data 80 83 V. CONCLUSIONS AND RECOMMENDATIONS 86 5.1 Conclusions 86 5.2 Recommendations 88 REFERENCES 89 APPENDICES A. ANALYTICAL SOLUTION FOR CUBIC EQUATIONS 99 B. VAN DER WAALS EXPRESSIONS FOR FLUID CRITICAL POINT 112 C. PREDICTION RESULTS OF CRITICAL PRESSURE, CRITICAL 130 TEMPERATURE, AND HEPTANE PLUS PROPERTIES iv ABSTRACT The prediction of critical points is of great practical importance because the classification of petroleum reservoir fluids as a dry gas, gas condensate, volatile oil, and crude oil depends largely on the knowledge of the critical properties of the reservoir fluid. Also, the critical pressure and critical temperature of reservoir fluids are important properties for describing the reservoir fluid phase behavior, predicting volumetric properties of reservoir fluids and designing supercritical fluid processes. Previous work for determining critical pressure, and critical temperature for reservoir fluids include, empirical correlations, corresponding states method, and pseudo- critical property methods. The generality of these previous correlations is limited to the range of conditions and parameters used in the establishment of the correlations. Methods based on the Gibbs criteria have also been used with Redlich-Kwong and Peng-Robinson equations for prediction of critical properties. However, the Gibbs criteria have not been applied to predicting critical properties of reservoir fluids. A closed-form equation is developed for predicting the critical properties (Tc, Pc) of complex reservoir fluids by using the Lawal-Lake-Silberberg (LLS) equation of state with the criticality criteria established by Nobel Laureates van der Waals (VDW) in 1873. By inverting the parameters of the LLS EOS in terms of the mixing parameters that are based on the constituent substances and composition of the reservoir fluids, experimental critical pressures and temperatures are predicted with interaction parameters expressed in terms of molecular weight ratios of the binary constituent of reservoir fluids. v The prediction results of critical pressures and temperatures based on the VDW criticality criteria show that experimental data consisting of 85 reservoir fluid mixtures are within average absolute percent deviation of 3% to 5% of the measured critical pressures and temperatures. In contrast to the previous work, this research project provides an accurate method for computing the critical properties of reservoir fluids and it is easy to use because the parameters of the criticality equation are readily available. This project is useful for unifying near-critical flash routine with phase equilibria of the compositional reservoir models. The project is also very attractive for establishing reservoir models that are based on the critical composition convergence pressure concept. vi LIST OF TABLES 2.1 Modification to the Attractive Term of van der Waals Equation of State................52 2.2 Modification to the Repulsive Term of van der Waals Equation of State................53 3.1 Parameter of Selected Equations of State ..................................................................62 3.2 Relationship of EOS Constants to Critical Parameters..............................................64 4.1 Sample of Experimental Data Used in Calculations of Mixture 1.............................76 4.2 Calculated Critical Data of Heptane-plus Fraction Correlation.................................77 4.3 Calculated Critical Data of Heptanes-Plus Fraction for Data Set 1...........................78 4.4 Calculated Results for Pure Component parameters .................................................78 4.5 Calculated Results for Mixtures Parameters..............................................................79 4.6 Predicted Critical Pressure, Pc, Critical Temperature, Tc for Mixtures .....................81 C.1 Critical Pressure Prediction for Complex Mixtures………………………………130 C.2 Critical Pressure Prediction for Complex Mixtures ……………………………...131 C.3 Critical Pressure Prediction for Complex Mixtures...….…………………………132 C.4 Critical Pressure Prediction for Complex Mixtures...… …………………………133 C.5 Critical Pressure Prediction for Complex Mixtures……………………………....134 C.6 Critical Pressure Prediction for ComplexMixture………………………………...135 C.7 Critical Pressure Prediction for Complex Mixtures ……………………………...136 C.8 Critical Pressure Prediction for Complex Mixtures ……………………………137 C.9 Critical Pressure Prediction for Complex Mixtures ……………………………138 C.10 Critical Temperature Prediction for Complex Mixtures ……………………….139 vii C.11 Critical Temperature Prediction for Mixtures ………………………………….140 C.12 Critical Temperature Prediction for Complex Mixtures………………………..141 C.13 Critical Temperature Prediction for Complex Mixtures…………………….….142 C.14 Critical Temperature Prediction for Complex Mixtures ……………………….143 C.15 Critical Temperature Prediction for Complex Mixtures ……………………….144 C.16 Critical Temperature Prediction for Complex Mixtures ……………………….145 C.17 Critical Temperature Prediction for Complex Mixtures ……………………….146 C.18 Critical Temperature Prediction for Complex Mixtures ……………………….147 C.19 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus....….….….….….….….….….….….….….….….….….….….148 C.20 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus……………………………………………………………. ….149 C.21 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus…………………………………………………………...……150 C.22 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus…………………………………………………...……….. ….151 C.23 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus …………………………………………………………….….152 C.24 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus …………………………………………………………….….153 C.25 Critical Pressure, Temperature, and properties Prediction for Heptane Plus……………………………………………………………. ….154 C.26 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus……………………………………………………………. ….155 C.27 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus……………………………………………………………. ….156 viii LIST OF FIGURES 1.1 Pressure-Temperature Phase Diagram of Petroleum Reservoir Fluids……………5 1.2 Pressure-Volume Diagram of Pure Components………………………………... 8 1.3 Pressure-Volume Diagram of Mixtures………………………………………….10 1.4 Specific Weight of Liquid and Gas for Propane in the Critical Region………....12 1.5 Pressure-Temperature Diagram of Retrograde Gas Condensate………………...14 2.1 Pressure-Volume Plot for a Single-Component System…………………………19 2.2 Critical Point Representation in a Multi-Component System……………………22 2.3 Compressibility Factors of Methane, Ethane, and Propane as a Function of Reduced Pressure………………………………………………………………...27 2.4 A Deviation Chart for Hydrocarbon Gases…………………………………… 28 2.5 Approximate Temperature of the Reduced Vapor Pressure…………………… 30 2.6 Equilibrium Ration for a Low-Shrinkage Oil……………………………………33 2.7 Equilibrium Ratio for a Condensate Fluid……………………………………….34 2.8 Illustration of Quasi-Convergence Pressure Concept……………………………38 2.9 Comparison of Equilibrium Ratios at 100°F for 1000 and 5000 psi Convergence Pressure……………………………………………………………40 2.10 Equilibrium Ratios of Heptanes-plus Fraction…………………………………. 42 2.11 K vs Pressure with C10+ Curve Required to Match Check Point Data…………...43 2.12 K vs Pressure with Curve Showing Effect of Choosing a Convergence