A multi-fractured well performance model for unconventional reservoirs.

Due to the ultra-low permeability of unconventional reservoirs, transient state prevails for a considerable period. Despite this, fracture interference can cause an apparent no-flow boundary. Consequently, the Duong's model, which was developed for transient-state period, yields unreliable estimates during the late-time period. In this paper, the Duong's model is modified to account for boundary effects caused by fracture interference and/or unstimulated reservoir regions that serve as no-flow boundaries. Specifically, an empirical correction function, which assumes an exponential decline, has been used as a "modifier" to extend the Duong's model to boundary-dominated flow period. The correction function ensures that during boundary-dominated flow period, an exponential-decline behaviour dominates. The proposed rate-decline model encompasses a gamma function, which converges at large times. Results show that a fractured-well production behaviour is characterised by a decaying power-law during early-time period and tends to exponential decline during late-time period. The results also suggest that although the conventional Duong's model gives good estimates during the transient-state period, it yields optimistic estimates during the boundary-dominated flow period. The proposed model gives a good match and estimates not only in the transient-state period, but also in the boundary-dominated flow period. A major advantage of the proposed model is that it converges to estimated ultimate recovery at large times without imposing any rate and time limits. A good agreement of the estimated ultimate recovery with analytical and semi-analytical models was obtained. Also, results suggest that the proposed model gives conservative estimates. The proposed model will be useful for analysing and predicting both the early- and late-time production performance of a multi-fractured well producing from an unconventional reservoir.

Similarity solutions for early-time constant boundary flux imbibition in foams and soils.

The foam drainage equation and Richards equation are transport equations for foams and soils, respectively. Each reduces to a nonlinear diffusion equation in the early stage of infiltration during which time, flow is predominantly capillary driven, hence is effectively capillary imbibition. Indeed such equations arise quite generally during imbibition processes in porous media. New early-time solutions based on the van Genuchten relative diffusivity function for soils are found and compared with the same for drainage in foams. The moisture profiles which develop when delivering a known flux into these various porous materials are sought. Solutions are found using the principle of self-similarity. Singular profiles that terminate abruptly are obtained for soils, a contrast with solutions obtained for node-dominated foam drainage which are known from the literature (the governing equation being now linear is analogous to the linear equation for heat transfer). As time evolves, the moisture that develops at the top boundary when a known flux is delivered is greater in soils than in foams and is greater still in loamy soils than in sandstones. Similarities and differences between the various solutions for nonlinear and linear diffusion are highlighted.