Physics-Based Sampling of Lognormal Reservoir Well Production/Productivity Distributions

Abstract

A large number of physical and social phenomena generate lognormal, ‘long-tailed’, or ‘fat-tailed’ population distributions. Such distributions cause considerable problems for statistical sample analysis because the underlying interactive processes giving rise to such system populations violate the central limit theorem. With no tendency for system processes to converge to normal distributions, the relation between sample data used for making decisions and actual system behaviour can be tenuous at best and catastrophic at worst. Crustal reservoirs are subject to high degrees of lognormality in well production/productivity. For reservoir engineering in general and geothermal reservoir engineering in particular, the breakdown of statistical sample analysis strongly impacts traditional reservoir modelling and greatly increases the risk/cost of reservoir drilling. However, unlike many/most systems having long-tailed populations, physical processes underlying reservoir flow lognormality are well constrained by empirical rules interpretable in terms that clearly indicate how and why reservoir engineers can sample their reservoir for flow structures at the spatial scales relevant to effective reservoir management. The physical elements of crustal reservoir flow heterogeneity and its appropriate spatial sampling scale are: • Well log spatial fluctuation power spectra: S(k) ~ 1/k, 1/km < k < 1/cm; • Well core poroperm spatial fluctuation correlation: δφ ~ δlog(κ); • Well flow lognormality due to fracture-connectivity: κ ~ exp(αφ), α >> 1; • Critical density ncrit grain-scale cement-bond-defect interactions leading to fluid percolation via long-range critical-state fracture-connectivity pathways; • Seismic wave emission from dislocation slips at pressure-sensitive instabilities in large-scale flow-system fracture-connectivity structures. Surface-seismic-array detection/mapping of large-scale in situ flow-system fracture-connectivity dislocation structures has been proven for the current generation of producing shale reservoirs. The same observational technology deployed at geothermal reservoirs can sample/map in situ flow structures at the spatial scales relevant to effective flow models and drill-site risk management.