Reservoirs are heterogeneous and uncertain. Multiple realizations are an important aspect of uncertainty; however, parameter uncertainty, that is, uncertainty in the histogram is even more important for uncertainty in resources and reserves. Accounting for parameter uncertainty in geostatistical simulation is a longstanding problem. Targeting specific quantiles such as P10, P50 and P90 realizations is a related challenge. This paper presents a solution to these problems. A methodology is presented to simulate realizations of continuous variables with specified position in the range of global uncertainty. The key to the methodology is the use of the generalized linear distribution in place of the uniform distribution for simulation. The theoretical validity of this method is established, implementation details are discussed and examples are presented. This has a wide range of applicability in modern geostatistical reservoir modeling where global uncertainty is an important goal.


Creating P10, P50 and P90 geostatistical reservoir models is an important task for flow simulation, risk analysis, reservoir forecasting and management. A base case model is always required. The 80% probability interval is common in the earth sciences. Higher probability intervals are often so large that they are difficult to use in risk qualified decision making.

There are some statistical methods to establish P10 and P90 reserve figures. The conventional approaches to estimate the reserves are divided into deterministic and probabilistic methods. The deterministic approach consists of volumetric, material balance and decline curve analysis and they use a single value for each parameter for estimating the reserves, there are no P10, P50 and P90 values in this method. The probabilistic approach uses a full range of values for each parameter in the reserve calculation. For example, the volumetric method could use a distribution of values for porosity, initial water saturation, formation volume factor and so on to get a range of values for the reserve. For the purpose of reserve estimation, National Instrument 51–101 (NI 51–101) defines P10, P50 and P90 (ROBINSON et al, 2004). P90 refers to proved reserves, P50 refers to proved and probable reserves and finally P10 refers to proved, probable and possible reserves. Based on NI 51–101 definition, P90 is less than P50, and P50 is less than P10. In this paper, P10 refers to a p-value of 0.9 and P90 refers to a p-value of 0.1 (the p-values in this paper are defined base on the statistical definition of cumulative distribution function). The problem with conventional statistical methods is that there are no specific realizations. It is not possible to run a flow simulator and assess the dynamic performance of the models under different conditions. It is highly desirable to have specific realizations that approximately represent the 80% probability interval.

The traditional geostatistical approach to finding P10 and P90 models is based on ranking procedures. Multiple realizations (often 100) are generated, and then some quick-to-calculate static reservoir attribute such as connected pore volume is chosen to rank the realizations. Realizations with specific position in the distribution of the static response are selected.

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