A "semi-empirical" acoustics-based flow model is presented based on large scale flow loop testing using water, water thickened with a linear gel, and linear gel slurries containing various amounts of proppant. The flow model predicts volumetric flow rate through individual perforation clusters from calibrated high fidelity accelerometer measurements and knowledge of the fluid type and perforation cluster geometry. This model is applicable to the plug-and-perf completion often used in hydraulic fracturing of unconventional resources. The accelerometer data provides a benchmark for downhole fiber-optic distributed acoustic sensing (DAS).

Custom flow loops were constructed and tested to measure acoustic signals generated by fluid turbulently flowing through simulated downhole perforations using water, water with a linear gel, and slurries containing ceramic proppant. Flow rates, perforation-hole sizes, number of clusters, number of perforations per cluster, fluid viscosity, and proppant slurry density were varied in a controlled way. These parameters were adjusted to represent a downhole stimulation treatment. The data was evaluated using symbolic regression with a 1,000-core compute cluster. Approximately three trillion analytic models were evaluated and scored for prediction accuracy and model simplicity.

The data indicated that all of the fluids became significantly louder above a critical threshold. This threshold was observed to strongly depend on fluid viscosity, while no dependence was observed on the amount of proppant in the slurry. The discharge coefficient showed significant variability, even among repeated experiments when no erosion was present. Very different behaviors for the coefficient of discharge were observed between water and gelled water. The measurements and model provide a benchmark for wellbore monitoring with fiber-optic distributed acoustic measurements during stimulation. Accurate real-time flow allocation estimation allows for an immediate adjustment of treatment strategy using, for example, diverter spheres. Accurate flow allocation could possibly allow changes in stage designs to tune individual flow rates.

Analytic expressions were derived relating the root-mean-square of the acoustic signal to both the perforation pressure loss and the flow speed per perforation as a function of cluster injection rate, viscosity, perforation diameter, and number of perforations. The relative importance of each of these variables was investigated and is quantitatively described. Several models for flow and accuracy were investigated and are discussed.

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