In this work, we developed two models of fracture conductivity in the presence of proppant-pack damage. The models are derived from the analysis of experiments relating dynamic fracture conductivity to flowback rate, reservoir temperature, polymer loading, presence of breaker, closure stress, and proppant concentration.
The first model is purely empirical, and it is modeled after fracture-conductivity studies conducted in previous work. The experimental design and planning table were derived from fractional-factorial designs. This meant that we could uniquely quantify the effect of all the dependent variables and also ensure that the factors are not correlated. We tested the empirical model for adequacythat is, ensuring the model residuals are normally distributed with no obvious trend and approximately constant variance. Because empirical models are generally constrained by the data on which they are built, we developed a semiempirical model dependent on dimensional analysis and nonlinear regression. Dimensionless products from the dimensional analysis procedure form the backbone of this semiempirical model.
We used subset-regression analysis to determine the optimal regression model and found that model was adequate. We also found two dimensionless groups and used them to develop the semiempirical model. We thereafter corrected the semiempirical model for the effect of temperature. Both the purely empirical model and the semiempirical model match experimental data with reasonable accuracy.
These models can be used as a first approximation for the short-term fracture conductivity of field hydraulic fractures. We also present methodology by which engineers can develop empirical models, assess uncertainty, and test for regression-model adequacy given noisy data.