Question

Hydraulic fracturing method is used to determine the major principal stress (\( \sigma_1 \)) in an underground rock strata having tensile strength of 6 MPa. The minor principal stress (\( \sigma_3 \)) in the strata is 8 MPa. If fluid pressure of 10 MPa is required to fracture the vertical borehole in that strata, the magnitude of \( \sigma_1 \), in MPa, is:

Hint:

In hydraulic fracturing, the major principal stress \( \sigma_1 \) is determined using the equation: \[ \sigma_1 = p_f + T + \sigma_3, \] where \( p_f \) is the fluid pressure, \( T \) is the tensile strength, and \( \sigma_3 \) is the minor principal stress.

Answer 1

Step 1: Understanding the hydraulic fracturing equation. In hydraulic fracturing, the major principal stress (\( \sigma_1 \)) is related to the fluid pressure (\( p_f \)), tensile strength (\( T \)), and the minor principal stress (\( \sigma_3 \)) by the equation: \[ \sigma_1 = p_f + T + \sigma_3. \] Step 2: Substituting the given values. From the problem, we are given the following values: - \( p_f = 10 \, {MPa} \), - \( T = 6 \, {MPa} \), - \( \sigma_3 = 8 \, {MPa} \). Substituting these values into the equation: \[ \sigma_1 = 10 + 6 + 8 = 24 \, {MPa}. \] Thus, the magnitude of \( \sigma_1 \) is \( \boxed{24.00} \, {MPa} \).