Modern cubic Equations-of-State (EOS) are used to describe reservoir fluid phase-behavior and for volumetric prediction under varying pressure, temperature, and fluid composition. These equations require calibration to the measured laboratory data for reliable prediction. Typical techniques use linear regression or gradient descent methods to calibrate an EOS to measured data. This results in a single solution, whereas such calibration is an inverse problem with a non-unique solution. In addition, these calibration techniques are limited to cases where the initial fluid composition is known. Bayesian inference accelerated by response surface modeling, also termed proxy modeling, is a technique commonly used to calibrate subsurface models to historical production data. This paper extends the application of a proxy modeling approach to regressing an EOS while simultaneously determining the initial fluid composition of a multi-component hydrocarbon mixture.
The proposed technique is demonstrated through its application to a PVT model based on a black-oil fluid sample obtained from an oil field in the Gulf of Mexico. The initial fluid composition of the fluid sample was unknown, but the sample was characterized using two PVT experiments including CCE (Constant Composition Experiment) and DLE (Differential Liberation Experiment). The PVT model was initially parametrized by uncertain input parameters with prior distributions. The fluid composition of a typical black-oil fluid sample was used as an initial guess in the PVT model. An initial proxy model was created using the parametrized PVT model with the objective of reducing the mismatch between simulated and user-selected measured PVT data.
The proxy model was continuously improved using a sequential design algorithm which involves Latin Hypercube (LHC) sampling, genetic algorithm followed by the gradient optimization. This sequential design ensures that multiple calibrated PVT models with an acceptable degree of accuracy are found while exploring the entire solution space of possible PVT models. In addition, the proposed technique helps determine the initial fluid composition which traditional regression approaches lack. Results show that the mismatch between the simulated and the measured PVT data is significantly less than using traditional approaches. Comparison of prior versus posterior ensembles of PVT models generated using the proxy model reveals that the mole fractions of various components gradually converge to a single value and the uncertainty in the phase envelope is significantly reduced.
The proxy model used in our proposed technique provides a robust minimization method which chooses and works with most significant EOS parameters, alleviating the tedious and time-consuming process of regression parameters selection. New regression parameters can be introduced midway during regression and the tuned parameters are always within reasonable physical limits since they are sampled from the user-defined prior distribution. Unlike traditional approaches for PVT regression, the proposed approach does not place a limit on the number of uncertain parameters that can be changed during regression.