**4. History matching of field injection pressure — Uncoupled geomechanical injection models**

The concept used for approximation of geomechanical effects in an uncoupled model and equations developed for production modeling [1, 2] can be used also for injection modeling. For injection cases change in pressure is always positive and consequently the effective mean stress is always larger when poroelastic effects are considered. For uncoupled modeling it can

**Figure 1.** BHIP, Wellhead pressure and Proppant concentration - Well A

be assumed that the stress changes have stabilized and fracture opening or closing pressure is equal to the minimum (adjusted) horizontal total stress during the treatment.

**4.1. Effects of fracture permeability reduction factor (Rfa)**

**Figure 2.** BHIP, Wellhead pressure and Proppant concentration - Well B

**4.2. Effects of limiting length of fracture propagation**

Reservoir parameters used in simulation runs are same as in base case.

permeability reduction factor.

Fracture permeability given by Equation (2) becomes an intrinsic permeability of smooth open fracture if the reduction factor is taken as 1. Fracture permeability in reality is much smaller than this value due to tortuosity, asperities interlocking, rock chipping at fracture face, unequal and rough surface of rock faces, and fracture degradation. Three injection cases were run to match field injection pressure with simulation injection pressure by varying only the fracture

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The initial runs produced a flat injection pressure while the field pressure is steadily increasing. A mechanism that would create larger pressure increase with time is required. One method is to restrict or confine fracture propagation in length (half length), which can be achieved by modifying transmissibility of grid blocks in fracture plane only within an assumed fracture half length. Possible justification is the scale-dependence of effective fracture toughness that was proposed theoretically and indicated by matching data [14]. Several simulation cases were run both for each well using different values of pre-determined maximum fracture half length.

In uncoupled history matching, any changes in stresses due to poroelastic effects must be incorporated manually in reservoir simulator for permeability multipliers and transmissibility calculation. Modified stresses are used in uncoupled model assuming that the hydraulic fracture increases in-situ stresses near wellbore and around fractures due to poroelasticity and permeability enhancement in that region must be predicted by using these modified stresses. Fracture height of 50 ft, Poisson's ratio of 0.125 and Elastic modulus of 7.99 E6 psia is used for fracture transmissibility calculation. The method of SRV permeability multiplier calculation is explained in references [1, 2]; the strength of the nonlinearity is given by "stress factor" S. Value of S=6.0 was used for all simulation runs. Injection model was setup in reservoir simulator and run for given injection period by using the actual injection rate, subject to maximum bottom‐ hole injection pressure of 10,000 psi.

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**Figure 2.** BHIP, Wellhead pressure and Proppant concentration - Well B

#### **4.1. Effects of fracture permeability reduction factor (Rfa)**

be assumed that the stress changes have stabilized and fracture opening or closing pressure is

In uncoupled history matching, any changes in stresses due to poroelastic effects must be incorporated manually in reservoir simulator for permeability multipliers and transmissibility calculation. Modified stresses are used in uncoupled model assuming that the hydraulic fracture increases in-situ stresses near wellbore and around fractures due to poroelasticity and permeability enhancement in that region must be predicted by using these modified stresses. Fracture height of 50 ft, Poisson's ratio of 0.125 and Elastic modulus of 7.99 E6 psia is used for fracture transmissibility calculation. The method of SRV permeability multiplier calculation is explained in references [1, 2]; the strength of the nonlinearity is given by "stress factor" S. Value of S=6.0 was used for all simulation runs. Injection model was setup in reservoir simulator and run for given injection period by using the actual injection rate, subject to maximum bottom‐

equal to the minimum (adjusted) horizontal total stress during the treatment.

**Figure 1.** BHIP, Wellhead pressure and Proppant concentration - Well A

314 Effective and Sustainable Hydraulic Fracturing

hole injection pressure of 10,000 psi.

Fracture permeability given by Equation (2) becomes an intrinsic permeability of smooth open fracture if the reduction factor is taken as 1. Fracture permeability in reality is much smaller than this value due to tortuosity, asperities interlocking, rock chipping at fracture face, unequal and rough surface of rock faces, and fracture degradation. Three injection cases were run to match field injection pressure with simulation injection pressure by varying only the fracture permeability reduction factor.

#### **4.2. Effects of limiting length of fracture propagation**

The initial runs produced a flat injection pressure while the field pressure is steadily increasing. A mechanism that would create larger pressure increase with time is required. One method is to restrict or confine fracture propagation in length (half length), which can be achieved by modifying transmissibility of grid blocks in fracture plane only within an assumed fracture half length. Possible justification is the scale-dependence of effective fracture toughness that was proposed theoretically and indicated by matching data [14]. Several simulation cases were run both for each well using different values of pre-determined maximum fracture half length. Reservoir parameters used in simulation runs are same as in base case.

Results of all uncoupled cases are not shown here except best matched case (see Figure 3 and 4) as history matching was not achieved. Only their effects are discussed here (detailed description of the results is provided in reference [2]). Results show that decreasing the *Rfa* factor pushes the injection pressure upward. Smaller reduction factor means smaller trans‐ missbility mutlipliers and hence larger pressure drop down in the fracture. However, fracture propagation confinement did not improve the rising trend of injection pressure in uncoupled simulation.

**Property – Well A Case - 1 Case - 2 Case - 3 Case – 4**

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The effect of fracture permeability reduction was discussed in detail in uncoupled simulation section; decreasing its value shifts pressure injection curve upward which can be observed in Figure 3. Although simulation results of Case – 4 of well A do not exactly match field injection pressure, it represents a reasonable history match. It is concluded that injection history match requires some mechanism to constrain fracture propagation at a late stage. This issue was not pursued further; however, the coupled cases show much improvement compared to the

Biot's constant (α) 0.65 0.75 Permeability reduction factor (*Rfa*) 0.00001 0.0000052 0.0000052 Fracture half length, ft Not restricted 100 130 130

**Table 1.** Parameters varied in coupled injection Cases 1 – 4 – Well A

**Figure 3.** Comparison of simulation results and field BHIP - Well A

uncoupled simulations as shown in Figures 3 and 4.
