## 1 Introduction  
The first edition of The Guide to the Expression of Uncertainty in Measurement (GUM) was published in 1993 by the Comite International des Poids et Mesures (CIPM) and the International Bureau of Weights and Measures (BIPM) as ISO/IEC Guide 98 [1] and since then it has been used as an industry standard for estimating measurement uncertainty. This method, also known as GUF (GUM Uncertainty Framework) is based on the law of propagation of uncertainty (LPU). Nevertheless, it has certain limitations when the model function is not linear and when the Probability Distribution Function (_PDF_) is not Gaussian. It is common to use GUM to estimate the uncertainties of custody transfer meters (turbines and positive displacement meters) involved in large economic and fiscal transactions [2; 3; 4; 5]. A reduction or increase in the uncertainty in this type of operation implies economic consequences, so the uncertainty evaluation should be performed as accurately as possible. Therefore, these inherent limitations of GUM do not allow a correct optimization of the calibration uncertainties of these meters, which has considerable importance from an economic as well as a legal point of view [6]. Hence, this work proposes to replace the classical GUM method of variance propagation by the propagation of _PDF_ through the Monte Carlo method (MCM) [7; 8]. The main strength of the MCM is that, to estimate the expanded uncertainty, it avoids using the central limit theorem and the application of Student's t-distribution with certain effective degrees of freedom, whenever that theorem is not applicable. The purpose of this paper is to address a key challenge for the petroleum industry: the implementation of the GUM-S1 scheme to improve the calibration of the measurement systems used to perform custody transfer. Currently, the uncertainties related to the API-MPMS regulation [9]--internationally accepted by the entire oil industry--are calculated by applying the classical GUM method. There is still a research gap regarding the application of the GUM-S1 scheme to overcome the limitations of the classical GUM method. There are no references in the literature to the application of the GUM-S1 scheme in relation to the measurement systems regulated by API-MPMS [9]. Therefore, a first application is presented for one of the most important cases: the calibration of a master meter against a standard proving tank. This case can be used as a reference for the calibration of a master meter against a bidirectional prover, and even for the calibration of the prover itself. In this way, the calibration of standard meters ensures the metrological traceability of all the logistics operator's dynamic volume measurement systems. There are different examples in the literature where the GUM method has been validated by MCM [10; 11; 12; 13; 14]. Examples of hydrocarbons metrology can also be found, though not many [2; 3; 4]. For this purpose, the GUM-S1 (Monte Carlo method (MCM)) scheme of Supplement 1 of the GUM document has been used as a starting point, and methodological innovations are proposed to optimize the application of the MCM, which can be useful beyond custody transfer meters. It must be kept in mind that the number of samples \(M\) in MCM is critical since different results could be obtained when repeating the simulation [15; 16]. In fact, according to GUM-S1, there is no guarantee that a specific pre-assigned number will suffice. Specifically, inside the GUM-S1 scheme, the adaptive implementation of the MCM (AMCM) is chosen to optimize \(M\). Examples of application of this method can be found in the literature [17; 18; 19; 20]. In this work, the following innovations are proposed when applying AMCM in the calibration of a flow meter against a proving tank:  
1. Regulation of AMCM convergence by applying the Student's t-distribution coverage factors (with the possibility to select the confidence level). 2.