Backflow from a model fracture network: an asymptotic investigation

Abstract

We develop a model for predicting the flow resulting from the relaxation of pre-strained, fluid-filled, elastic network structures. This model may be useful for understanding relaxation processes in various systems, e.g. deformable microfluidic systems or by-products from hydraulic fracturing operations. The analysis is aimed at elucidating features that may provide insight on the rate of fluid drainage from fracturing operations. The model structure is a bifurcating network made of fractures with uniform length and elastic modulus, which allows for general self-similar branching and variation in fracture length and rigidity between fractures along the flow path. A late-time $t^{-1/3}$ power law is attained and the physical behaviour can be classified into four distinct regimes that describe the late-time dynamics based on the location of the bulk of the fluid volume (which shifts away from the outlet as branching is increased) and pressure drop (which shifts away from the outlet as rigidity is increased upstream) along the network. We develop asymptotic solutions for each of the regimes, predicting the late-time flux and evolution of the pressure distribution. The effects of the various parameters on the outlet flux and the network’s drainage efficiency are investigated and show that added branching and a decrease in rigidity upstream tend to increase drainage time.

Non‐Newtonian Backflow in an Elastic Fracture

“…Consequently, their model predicts an asymptotic −4∕3 time scaling for the flow rate exiting the fracture system. The model was further generalized by Dana et al (2019), allowing for variations in fracture length and elasticity among different orders of fractures, but the time scaling exponent did not change significantly. A second class of approaches represents explicitly the backflow phenomenon via detailed numerical models (de Borst, 2017;Jia et al, 2019;Medina et al, 2018).…”

Viscous backflow from a model fracture network: influence of a permeable boundary

“…The dynamic boundary condition to account for the squeezing force driving the flow was introduced and discussed by Dana et al. (2018, 2019). For simplicity of notation, we define the position variables The force balance on the upper plate of each generation of the network is written accordingly as where , similar to , is a modified Young's modulus of the foundation divided by its original thickness.…”

“…Balancing all terms in (2.4) and (2.6), we define dimensionless variables by Since this non-dimensionalization involves a balance between lubrication flow and permeation leakage, it is different from that presented in the previous studies of impermeable networks (Dana et al. 2018, 2019). We will use the impermeable scaling, given by (3.3) below, in some comparisons with previous results.…”

“…Dana et al. (2018, 2019) provided complete solutions for such systems. When is small, permeation is the main drainage mechanism.…”

“…2018), and later, power-law varying properties (Dana et al. 2019). In this model, each fluid-filled channel was bounded by two thin rigid plates pressed together in elastic response to a pre-strained state.…”