A new mathematical model describing the effect of overburden pressure on porosity and permeability of Sandstone and Limestone rocks

Abstract

Measuring the petrophysical characteristics of the reservoir rocks (core) is an important measure in the reservoir studies because it gives a clear picture of the oil productivity.

Therefore, studying the factors that affect it is extremely important. One of the most important factors affecting the amount of porosity and permeability of rock is overburden pressure resulting from the weight of the ground layers. In this paper, we used the simple non-linear regression method to formulate several models of three families of mathematical functions (exponential, power and polynomial), which describe the relationship between overburden pressure and porosity/ permeability based on laboratory results of porosity / permeability measurements, we then subjected these models to study, analyze, validate and compare them using the coefficient of determination R² (which is one of the most important statistical criteria that measure the quality of regression models) and physical criteria to reach the optimal model.

All models were formulated using the famous statistics program Origin8 software. The results showed that there are several mathematical models that can be applied to represent the relation between overburden pressure and porosity/ permeability of sand and limestone rocks, because the value of R² for all mathematical models is very high (R² ≥ 98). For example, sandstone has shown that all models are a decrease in porosity ∅/∅_i with increasing overburdens pressure and that all of them give almost identical behavior.

The drop starts from a primary value of 100% to (95.25 - 94.75) %, (i.e. bandwidth of 0.5%) at a pressure equal to (P = 21 Pi) of primary pressure (Pi = 300 psi). After this pressure, the behaviour of mathematical models varies greatly, as this behaviour can be divided into three types: the rise in porosity again to an amount greater than its initial value, as described by the second-degree polynomial model. Continue to decrease in porosity and very slowly, as described by the power function models. Porosity stability, as described by steady state equation (SSE), and so on permeability and limestone rocks. SSE was chosen as the best mathematical model that describes the relationship between overburden pressure and porosity/ permeability for several factors, chemical factors and mineral properties of rocks, migration of granular particles and their transfer from one place to another due to fluid movement and fluid pressure within the pores. Excessive increase of Pressure will lead to physical forces and micro fractures between the components of the rock, all of which play an important role in resisting the effect of increasing overburden pressure, resulting in the death of pores and other new births.