# TSO Description for fracDesign

At first, in design and simulation modes was added a control for TSO effect. It initiates then dry-to-wet width ratio became larger then TSO_ratio: propped_width>TSO_ratio*hydraulic_width

So in design procedure at first we calculate slurry concentration that is injected at last stage and hydraulic volume at last stage. Then we found dry-to-wet width ratio and compare it with TSO_ratio. If TSO was not initiated and max concentration was not reached then common and TSO design are equivalent.

If TSO design is not used we stop pumping then TSO_initiated. It is the same behavior as if injected concentration became larger then defined max concentration.

If TSO design is used and TSO initiated or max concentration reached algorithm of pumping schedule became the following:

1. Calculate concentration then TSO effect occurs TSO_conc.

2. Perform steps 1-11 from fracDesign algorithm.

3. Use ce = min(TSO_conc, max_concentration) instead of step (12).

4. Perform later steps.

5. Calculate mass of proppant pumped to this stage Mass_before_TSO.

6. Calculate remained mass of proppant Mass_remain.

7. Calculate time for injection of remained mass t_TSO = Mass_remain/(TSO_conc*q).

8. Calculate total time of pumping t_total = t_common + t_TSO.

9. Calculate final width for known t_total and xf achieved on stages (1)-(4) by mass balance equation (use the same open-time distribution factor Kl as founded early).

10. Calculate final fracture volume, net pressure and fluid efficiency.

11. Add additional stage to pumping schedule, on which TSO effect occurs. Stage lasts from t_common to t_TSO with injection of TSO_conc.

In general TSO design can be described as follows. At first use the same schedule as for common design, but with final concentration found at step 3. As a result of these stages we will obtain a fracture with target half-length, but with less (compared to common design) injected proppant. Then add additional stage during which half-length is not changing. The injected concentration is the same as on previous step (found on step 3). And the duration is found in order to place all the available proppant.

In simulation mode we don’t take into account options TSO design and max concentration. But we control for TSO effect according to TSO_ratio. In each stage we examine for the TSO initiation. If it was detected that during some stage TSO effect was initiated we seek for the moment inside stage then the start of tip screen-out occurred. Fixing the half-length at this moment we calculate next parameters with account for TSO effect.