There are a number of analytical solutions describing fluid flow through multi-frac horizontal wells presented in literature. This paper presents a way to extend the existing analytical solutions for multi-frac wells to the cases when completion properties (such as fracture conductivity and fracture length) are varying with pressure, and thus with time.

The paper provides a derivation of the suggested method; it uses the formulation of analytical solution of the diffusivity equation based on Green’s function. However it is demonstrated that the same methodology can also be applied to analytical solutions obtained without using Green’s function.

It is believed that in tight formations, production at high drawdown often results in an increased effective stress around the hydraulic fracture system (e.g. Okouma et al. 2011). Higher effective stress causes a decrease in conductivity and/or effective length of hydraulic fractures. Being able to model such a variation of fracture properties with pressure is essential for accurate history matching and production forecasting.

The suggested methodology was validated by comparing the derived solutions against numerical simulation. We found that results obtained by using the suggested methodology and numerical results are in good agreement for single phase fluid and single plane fractures; therefore the suggested method can be applied to analytical models for pressure-dependent multi-frac completions for such cases.

The proposed methodology allows extending available analytical solutions for multi-frac systems in a way that captures the variation of fracture properties with pressure.

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