Constant flow, rate solutions are presented for a fractured reservoir with transient interporosity flow in a convolution form, considering matrix, microfractures, vugs and fractures flow.
New solutions are presented for two cases, where there is no primary flow through the microfractures and where the compressive and distensive strength process has created an interconnected system of microfractures. In both cases there is an interaction between matrix, microfractures, and fracture systems.
The numerical inversion was carried out with Stehfest's algorithm. In addition, approximate analytical solution for short and long dimensionless time are obtained and compared with the solution calculated by numerical inversion, providing satisfactory results. The values of the numerical inversion were used to generate the results, presented in terms of the dimensionless groups derived from the approximate analytical solution.
Introduction
In areas without cores, open-hole wireline logs may be used to help identify vugs zones; however, vugs are not always recognized by conventional logs because of their limited vertical resolution. Vugs porosity is common in many reservoirs and its importance in the petrophysical and productive characteristics of a rock has been recognized in several works. Vugs porosity can be subdivided into small and large types. Vugs effect on permeability is related to their connectivity3. One purpose of our work is to present a technique to identify high secondary porosity, mainly vugs porosity.
When dealing with NFR, it is important to study its origin and main characteristics, such as the regularity of their distribution, and morphology, aperture, width, etc. Figs. 1 and 2 presents photographs of whole cores taken in highly productive naturally fractured carbonate vuggy and fractured formations. Fig. 1.a shows two fracture system, one being vertical and the other inclined, connected through vugs. Fig. 1.b illustrates a fracture system that has a main vertical direction. Fig. 1.c shows a system of vertical and conjugate fractures; some vugs can be observed and an open fracture of aperture of about 3mm.
Fig. 2 shows three additional images of carbonate whole cores. Fig. 2.a illustrates the case of fractures filled by calcite. Fig 2.b presents the important physical condition of fracture planes affected by dissolution, which increases both secondary porosity and bulk permeability. Finally, Fig. 2.c exhibits vertical fractures, enhanced by dissolution vugs, which improve the formation bulk conductivity.
It has been observed in the literature that vugs zones strongly influence production performance3, 4.
Recently two papers have discussed results strictly related to the presents study. Camacho etal1. developed a new way to model the secondary porosity of naturally reservoirs (NFR), mainly vuggy porosity the authors derived solutions for two cases, one where there is no primary flow through the vugs (which is an improvement of the Warren and Root model), and second in which the dissolution process of the pore volume has resulted in an interconnected system of vugs and caves. Thus, this is a triple-porosity / dual permeability model.
Rodriguez et al 2. developed a nested - triple -porosity single - permeability model for the pressure transient behavior of a well producing in a NFR, there porosity systems, acting at different scales, are assumed to coexist in these reservoirs: matrix, small scale and large scale secondary porosity media, flow in series among these media is considered; the matrix exchanges fluids whit the small - scale secondary porosity, which in turn feeds the large - scale secondary porosity. The first two media are assumed to have a local effect on fluid flow, and are considered as discontinuous, while the third, the large - scale secondary porosity medium, is considered to affect fluid flow at the reservoir scale and it is consequently continuous.
