On The Pore Size Distribution: Derivation And Testing Of A Stochastic Model To Analyze Pore-Size Data From Carbonate Reservoirs
Authors: Alhanai Waddah T. 
Published: December 2007

Abstract
The similarity in shape of pore-throat size distributions suggests the existence of a common underlying principle. This work proposes a stochastic model to fit the pore size distribution data. The computed population parameters are then used to find correlations for porosity, permeability and end-point saturations.

Introduction
The displacement efficiency of oil in a reservoir is controlled, among other things, by the geometry and topology of the pores in the rock. Reservoir rocks are composed of pores with highly variable shapes and sizes. The complex pore structure arose from infinitely many random events caused by several interacting factors in the geologic environment and history of the deposition (ex. compaction, cementation, leaching, oxidation, fracturing, dolomitization,… etc). Eventually, the results of these random events is a rock that contains a continuous spectrum of pore-size; from the smallest to the largest. The application of information about pore-size and pore-throat distributions is in a better understanding of the fluid flow process in reservoirs.
Capillary pressure data are used to assist in the interpretation of pore structure when the property of interest is the pore-throat size frequency distribution. Pore-throat sizes are calculated from Mercury-Injection capillary pressure data, under the assumption that pore-throats are equivalent to cylindrical capillaries. The Young-Laplace equation of capillarity is used to compute the pore-throat radius, r, from the equilibrium capillary pressure (Pc):
(1)
where is the Mercury/Air interfacial tension and is the contact angle measured in Mercury.
Although capillary pressure curves have long been used to interpret properties of porous media, a full evaluation of the information which could be derived from such curves is still lacking.
For example when one examines lab-reported plots of pore-throat size, calculated by Equation 1 versus percent pore volume invaded by Mercury, one immediately observes that the plots have all a stretched S-shape. This similarity in shape intuitively suggests the existence of a common underlying principle governing the pore-throat size distributions.
The objective of this work is to derive a model to fit the reported pore-throat distributions. A stochastic approach to the problem is proposed. The goodness-of-fit is tested by Hypothesis Testing, and Statistical Inference is used to infer the pore-throat population parameter. The population parameters of the accepted distributions are then correlated to the petrophysical properties of the lab rock samples.

Derivation of the Model
The grain-size problem of a reservoir rock is modeled mathematically via statistical techniques under the following assumptions:The grains have undergone a very large number of small random diagenetic processes after deposition.The diagenetic processes are extremely slow stepwise processes; each causing only a very small change in the rock at a macroscopic scale.The diagenetic processes are mutually-independent events.Although the successive diagenetic processes are responsible for the shape and size of each grain, a grain's scale at any instant is irrelevant to the diagenetic process that the grain happens to experience at that time.At any geologic instant, a single diagenetic change in a single grain has practically no effect on the surrounding grains.

Assumptions
In parallel to the so-called "grain size problem" it seems physically intuitive to assume that the following stochastic assumptions also hold for the pore-size problem:The pores have evolved from a statistically large number of small, random events,When averaged over overwhelmingly many pores, no single event should have left a large effect on the pores,The random events are all independent stochastically, i.e. the time to the next event is independent of the time since the last event.
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