Fast and Robust Algorithm for Compositional Modeling: Part I - Stability Analysis Testing
- Abbas Firoozabadi (Reservoir Engineering Research Inst.) | Huanquan Pan (Reservoir Engineering Research Inst.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2002
- Document Type
- Journal Paper
- 78 - 89
- 2002. Society of Petroleum Engineers
- 5.2 Reservoir Fluid Dynamics, 5.2.2 Fluid Modeling, Equations of State, 5.2.1 Phase Behavior and PVT Measurements, 5.1.5 Geologic Modeling, 5.8.8 Gas-condensate reservoirs, 4.6 Natural Gas, 5.5 Reservoir Simulation
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Given pressure, temperature, and composition of a fluid, one desires to determine whether the single phase state is stable. This problem is, in principle, much simpler than phase-behavior calculations. For certain applications, such as compositional-reservoir modeling, stability testing can be the most important item for efficient phase-behavior calculations.
In this paper, we use the tangent-plane-distance (TPD) in the reduced space to perform stability analysis testing. The results reveal that there are major advantages in the reduced space. One interesting feature of the transformation is that the TPD surface becomes smooth and has one minimum. The combination of a single minimum and the surface smoothness contributes to a remarkable robustness in calculations.
Phase-behavior computations are an important element of thermodynamics of phase equilibria in general, and of compositional modeling of hydrocarbon reservoirs and production facilities in particular. Compositional modeling in hydrocarbon reservoirs demands robustness for hundreds of millions of phase partitioning computations with varying conditions of pressure and composition, including in the critical region.
There are five basic requirements and desirable features for phase behavior-related computations in a compositional-reservoir simulator. Given the temperature, T, the pressure, p, and the overall composition vector, z = (z1,z2, . . . , zc,) one desires to:
Determine the phase stability.
Recognize the state of the phase (that is, gas or liquid).
Perform flash computations.
Calculate phase derivatives with respect to p and zi.
Use an adequate number of components/pseudocomponents for certain accuracy.
Item 1 above determines the need for flash computations. When it is determined that the fluid system is in single state from stability testing, there will be no need for flash computations; flash computations are more costly and complicated than stability testing when one takes advantage of proper stability analysis testing. The TPD from stability can also be used to perform flash computations efficiently, as will be discussed in the second part of this work.
In compositional-reservoir simulators, it is necessary to recognize the phase state; one needs to know whether the single phase is a gas or a liquid phase. In Item 4 above, the total derivatives for the flow expressions are estimated. These derivatives are (??L/ ?P)z,T, (??L/?ni)T,P,ni, , and other similar derivatives; ?L = the liquid phase density, ni = the number of moles of component i in the feed, ni=(ni, . . . , ni-1,ni+1, . . . , nc), and c=the number of components. Depending on the process, sufficient accuracy may require the use of from 4 to 12 (or more) components. When the nonlinear equations in flash are performed using the Newton algorithm, the computational time is proportional to c3. Therefore, there will be an increase of about 27 in computational time when c is increased from 4 to 12. The most important element of phase-behavior calculations in compositional modeling is, however, robustness of computations rather than speed. Robustness is the prime goal of our work.
Despite remarkable progress in the period from 1980 to 1990 in phase-behavior computations,1,2 a comprehensive and unified theoretical framework continues to be the primary goal in performing the computations both robustly and efficiently for the five items outlined above. There may be a direct link between all five, though such a link has not yet been recognized.
One major difficulty in phase-behavior computations results from the ragged shape of the Gibbs free energy surface. To be more specific, both the raggedness and the number of minima of the TPD affects the complexity of phase-behavior calculations very pronouncedly. Those issues will be discussed in detail in this paper. Another difficulty arises from the nonlinear Rachford-Rice3 equation, especially near the critical region. A lesser complication is the number of nonlinear equations. In this work, we will demonstrate how we can alleviate all the above complexities. This paper presents Part I of the work covering the stability analysis, which is the key item for efficient flash computations in our work. Part II will address flash computations.4
Our algorithm is based on the reduction of the dependent variables (that is, the composition at constant T and p) of the Gibbs free energy through transformation. In the following, after a brief literature review on the reduction method, we first present a new formulation of the reduction method based on the Spectral theory of linear algebra.5 Then we provide stability analysis formulation in terms of the reduction variables for the TPD,6 followed by minimization of the TPD using the simple method of Lagrange multipliers. Next, we examine the TPD surface for threecomponent systems for both the conventional and the reduction variables. The results for stability testing of four-multicomponent mixtures of various degrees of complexity are discussed, followed by concluding remarks.
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