**6. Numerical approach description**

The treatment of permeable natural fractures is a part of UFM model. It is possible to model leakoff from HF into the NF form UFM in different ways, depending on the importance of required accuracy, numerical stability and CPU time.

The most computationally expensive but at the same time most accurate is a fully coupled numerical approach. The fully coupled approach means to discretise different parts of the invaded NFs and numerically solve the pressure as a part of total system of equations for the fracture network using iterative solution scheme. Another, more CPU efficient approach, is a decoupled numerical approach, where pressure and width along the invaded but closed part of NF are calculated separately based on the results from previous time step calculations along the HFN and corresponding pressure at HF/NF intersections (inlets).

Two approaches have been considered to model interaction of hydraulic fracture with permeable natural fractures.

#### **6.1. Decoupled numerical approach**


**•** Save elements information (volume, pressure, flow rates) for the next time step

#### **6.2. Fully coupled numerical approach**

If some elements in closed not disturbed part of NF are sliding then the corresponding shear stress is involved in the stress shadow calculations and thus influences simulations results.

The treatment of permeable natural fractures is a part of UFM model. It is possible to model leakoff from HF into the NF form UFM in different ways, depending on the importance of

The most computationally expensive but at the same time most accurate is a fully coupled numerical approach. The fully coupled approach means to discretise different parts of the invaded NFs and numerically solve the pressure as a part of total system of equations for the fracture network using iterative solution scheme. Another, more CPU efficient approach, is a decoupled numerical approach, where pressure and width along the invaded but closed part of NF are calculated separately based on the results from previous time step calculations along

Two approaches have been considered to model interaction of hydraulic fracture with

**•** When NF is intercepted by HF, create elements along whole NF to be used at next time step **•** Evaluate initial guess of flow rate into the NF based on the pressure at the intersection and

**•** Check for a possibility of frictional sliding along the closed parts of NFs to evaluate pressure-

**•** Iteratively calculate pressure, hydraulic width, flow rate and length of each zone along NF with using corresponding equations (for filtration and pressurized zones) by marching from intersection till the end of NF until condition (27) is satisfied indicating final results and final

**•** Track the end of filtration zone for each pressure and flow rate iterations by checking the

**•** Save invaded volume for volume balance, influx for mass balance, pressure at intersection,

**•** Track the pressure at intersections and/or tips of opened HF part in NF to capture opened

**•** If intersection is opened, use the pressure at the tip of opened HFN part along invaded NF

**6. Numerical approach description**

304 Effective and Sustainable Hydraulic Fracturing

permeable natural fractures.

positions of zones' fronts

zones.

**6.1. Decoupled numerical approach**

old pressure profile along closed NF

dependent permeability and conductivity along NF

volume of fluid injected into NF at given time step

for calculations along corresponding closed NF part

**•** Apply rules to treat special situations (intersecting NFs, etc)

and time step to be used at the next time step

required accuracy, numerical stability and CPU time.

the HFN and corresponding pressure at HF/NF intersections (inlets).


**Figure 5.** The proposed algorithm to account for leakoff from HF into the permeable NFs (*i=N* indicates the end of NF)

Figure 5 The proposed algorithm to account for leakoff from HF into the permeable NFs (*i=N* indicates the end of NF)

12

**•** Calculate the flow rate and volume of fluid injected to NFs at each pressure iteration, and update/calculate pressure along HFN using existing scheme for opened elements and new equations (described above) for elements in closed invaded and pressurized NF parts

[2] Chuprakov, D, Melchaeva, O, & Prioul, R. Hydraulic Fracture Propagation Across a

Hydraulic Fracturing in Formations with Permeable Natural Fractures

http://dx.doi.org/10.5772/56446

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The fully coupled NF modeling approach is heavier and more CPU expensive than decoupled approach. The Decoupled Numerical Approach has been selected as basic approach and it is described schematically on Figure 5 as a part of total solution
