A Fractal Approach to the Modelling and Simulation of Heterogeneous and Anisotropic Reservoirs

New reservoirs are increasingly more heterogeneous and more anisotropic. Unfortunately, conventional reservoir modelling has a resolution of only about 50 m, which means it cannot be used to model heterogeneous and anisotropic reservoirs effectively when such reservoirs exhibit significant inter-well variability at scales less than 50 m. This paper describes a new fractal approach to the modelling and simulation of heterogeneous and anisotropic reservoirs. This approach includes data at all scales such that it can represent the heterogeneity of the reservoir correctly at each scale.

Three-dimensional Advanced Fractal Reservoir Models (AFRMs) can be generated easily with the appropriate code. This paper will show: (i) how 3D AFRMs can be generated and normalised to represent key petrophysical parameters, (ii) how these models can be used to calculate permeability, synthetic poro-perm cross-plots, water saturation maps and relative permeability curves, (iii) the effect of altering controlled heterogeneity and anisotropy of generic models on fluid production parameters, and (iv) how AFRMs which have been conditioned to represent real reservoirs provide a much better simulated production parameters than the current best technology.

Results of generic modelling and simulation with AFRMs show how total hydrocarbon production, hydrocarbon production rate, water cut and the time to water breakthrough all depend strongly both on heterogeneity and anisotropy. The results also show that in heterogeneous reservoirs, the best production data is obtained from placing both injectors and producers in the most permeable areas of the reservoir – a result which is at variance with common practice. Modelling with different degrees and directions of anisotropy shows how critical hydrocarbon production data depends on the direction of the anisotropy, and how that changes over the lifetime of the reservoir.

We have developed a method of fractal interpolation to condition AFRMs to real reservoirs across a wide scale range. Comparison of the hydrocarbon production characteristics of such an approach to a conventional krigging shows a remarkable improvement in the modelling of hydrocarbon production when AFRMs are used; with AFRMs in moderate and high heterogeneity reservoirs returning values always within 5% of the reference case, while the conventional approach often resulted in systematic underestimations of production rate by over 70%.