Variable rate and pressure integral solutions to the nonlinear gas diffusivity equation in unconventional systems

Unconventional gas resources, including tight gas and shale gas, are becoming major sources of natural gas production in U.S. Commercial production from tight or ultra-tight formations requires horizontal wells and massive hydraulic fracturing treatment. These multi-fractured horizontal wells (MFHWs) can exhibit long-term (typically years) early-transient behavior prior to pressure transient reaching the reservoir boundary. Analytical models that have been traditionally developed for transient flow in conventional gas reservoirs face significant challenges when applied to unconventional tight and shale gas, particularly when variable-rate and variable-pressure data needs to be analyzed. This paper presents a rigorous and direct semi-analytical integral solution to the governing nonlinear gas diffusivity equations that are directly applicable to realistic unconventional production scenarios of MFHWs. The validity of proposed solution is verified by matching against numerical simulation results using synthetic case studies for both constant and varying rate production conditions. Its applicability to real-life production scenarios is showcased by a field case study using production data from a MFHW in Marcellus Shale.