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An ElrodAdamsmodelbased method to account for the fluid lag in hydraulic fracturing in 2D and 3D
An efficient method to model the fluid lag in hydraulic fracturing has been developed based on the ElrodAdams model. The main feature of this method is the absence of the need to explicitly track the free end of the fracturing fluid, but rather,the fluid front is obtained by solving the pressure field (zero for the lag) and an auxiliary field for the entire fracture. The main idea of the ElrodAdams model is to introduce a liquid fraction field θ: Γ →[0,1], whose physical meaning is the percentage of liquid occupying the fracture at s$ More precisely, θ(s)=1 means that liquid is completely filled at s, while the region with θ(s)=0 corresponds to the part of the fracture with no fluid. Correspondingly, the pressure p(s) is positive only when θ(s)=1 (Figure 1). With these definitions, the governing equations of the liquid inside the fracture reads Equation (1a) is the main equation of the ElrodAdams model, which is merely a restatement of mass conservation. Equation (1b) describes the condition for the pressurized zone, while (1c) both the transition zone {0< θ <1} and the dry zone {θ=0}. Equation (1d) enforces massconservation at the fluid front boundary.Equation eqref{EAFlowRate} models the flow injection as a discontinuity in the flow rate, and Equation (1e) is the condition for the withdrawal stage.
Fig1: The essence of the ElrodAdams model is to employ a liquid fraction field θ defined over the entire fracture Γ that assumes value between zero and unity. It represents the percentage of liquid occupying the fracture width at s. In particular, θ(s)=0 means that there is no liquid at s (the dry zone), while θ(s)=1 means that the liquid is completely filled at s. 