The diffusivity equation for naturally fractured reservoirs represented by dual-porosity systems has been solved as a function of dimensionless times (t0) and dimensionless water influx (Q (t))for various ratios of aquifer to reservoir size.
Results are presented in a similar format to the one used by Van Everdingen and Hurst,1 in their classic solution for conventional single-porosity reservoirs.
Unrestricted (transient) and restricted (pseudo steady) interporosity flow have been taken into account.
Results indicate that the use of conventional water influx equations in naturally fractured reservoirs can lead to significant errors.
Some naturally fractured reservoirs are partially or totally bounded in their peripheries by large or small bodies of water known as aquifers. The aquifers might contain natural fractures of a tectonic, regional or contractional type. Parts of the water can be stored in the matrix and parts in the system of natural fractures.
The aquifer might be limited by an impermeable lithology and thus the oil reservoir and the aquifer can be contained within a closed volumetric unit. In other cases, the aquifer might outcrop at one or more places and is replenished by surface waters. Yet in other cases the aquifer might rise structurally over the hydrocarbon reservoir giving origin to artesian flow of water towards the reservoir.
Craft and Hawkins2 have presented a lucid treatment of water influx in conventional single-porosity reservoirs, based on Laplace transformation solutions published by van Everdingen and Hurst.1
In their approach the aquifer is considered as an independent unit which encroaches water in the hydrocarbon reservoir as a response to time variations in the average pressure along the water-hydrocarbon contact.
This paper presents a similar approach for analyzing water influx from a naturally fractured aquifer. Values of dimensionless water influx Q (t), have been calculated as a function of dimensionless time (tD) for the case of infinite aquifers and for various ratios of aquifer to hydrocarbon reservoir size.
The solutions presented in this paper have been validated by running calculations using a value of omega equal to 1.0 and comparing results against those published by Van Everdinqen and Hurst. 1 Comparisons are highly satisfactory.
Fig. 1 shows a schematic of the idealized model considered in this study. The aquifer is naturally fractures. The hydrocarbon reservoir might or might not contain natural fractures. The reservoir and the aquifer are horizontal and have radial configuration. Water movement towards the reservoir occurs only through fractures in the aquifer. The upper and lower boundaries are impermeable. Fracture permeability is independent of pressure and exceeds the matrix permeability by at least one order of magnitude. The storage capacity of the matrix is large compared with the storage capacity of the fractures. The matrix blocks are uniformly distributed throughout the aquifer. The matrix and fracture compressibilities can be equal or they can be different provided that pon decompression they do not affect fluid flow in a significant way.
Single phase flow of a slightly compressible fluid is modelled with Darcy's law. Viscosity is constant. Density of fluids in matrix and fractures is equal.
