The optimal semi-analytical modeling for the infinite-conductivity horizontal well performance under rectangular bounded reservoir based on a new instantaneous source function

The main objective of this study is to develop the optimal semi-analytical modeling for the infinite-conductivity horizontal well performance under rectangular bounded reservoir based on a new instantaneous source function. The available semi-analytical infinite-conductivity models (ICMs) for horizontal well under rectangular bounded reservoir in literature were developed by applying superposition of pressures in space (SPS). A new instantaneous source function (i.e., instantaneous uniform-flux segmentary source function under bounded reservoir) is derived to be used instead of SPS to develop the optimal semi-analytical ICM. The new semi-analytical ICM is verified with ICM of Schlumberger [1] and with previous semi-analytical ICMs in terms of bottom hole pressure (BHP) profile and inflow rate distribution along the wellbore. The model is also validated with real horizontal wells in terms of inflow rate distribution along the wellbore. The results show that the developed model gives the optimal semi-analytical modeling for the infinite-conductivity horizontal well performance under rectangular bounded reservoir. Besides that, high computational-efficiency and high-resolution of wellbore discretization have been achieved (i.e., wellbore segment number could be tens of hundreds depending on solution requirement). The results also show that at pseudo-steady state (PSS) flow regime, inflow rate distribution along the wellbore by previous semi-analytical ICMs is stabilized U-shaped as performance of inflow rate distribution at late radial flow regime. Therefore, the previous semi-analytical ICMs are incorrectly modeling inflow rate distribution at PSS flow regime due to the negative influence of applying SPS. The optimal semi-analytical ICM is in a general form and real time domain, and can be applicable for 3D horizontal well and 2D vertical fracture well under infinite and rectangular bounded reservoirs, of uniform-flux and infinite-conductivity wellbore conditions at any time of well life.

The novelties in this study are as follows:

1. At PSS flow regime:

(1) Inflow rate distribution along the wellbore is stabilized uniform-flux which was verified mathematically.

(2) Primary pressure derivative (PPD) (i.e., PPD = ∂P_Dt/∂t_DA) is equal to (2π/m_t) for any well and reservoir configurations and depends only on half-length wellbore segments number (m_t).

2. The new ICM gives different trend of Bourdet derivative for the first three flow regimes (i.e., early radial, early linear, late radial) and gives the same trend of Bourdet derivative for PSS flow regime, to their counterparts by uniform-flux model (UFM). The trend of pressure derivatives by UFM for any flow regime is well studied in literature, while the counterparts by ICM are new and need detailed study.