Numerical Investigations on the Effect of Fracture Length Distribution on the Representative Elementary Volume of 3D Discrete Fracture Networks

<jats:p>Determination of the representative elementary volume (REV) of fractured rock masses based on equivalent permeability (<jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>K</mi> </math> </jats:inline-formula>) is significantly dependent on the geometric characteristics of fractures. In this work, a series of numerical simulations were performed to analyze the relationship between geometric characteristics of fractures and the REV size, in which fracture length follows a power-law distribution. A method to evaluate the <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M2"> <mi>K</mi> </math> </jats:inline-formula> of a three-dimensional (3D) discrete fracture network (DFN) by extracting the equivalent pipe network (EPN) model from the DFN model was utilized and verified. The results show that <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M3"> <mi>K</mi> </math> </jats:inline-formula> of the 3D DFN model has an exponential relationship with the power exponent (<jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>a</mi> </math> </jats:inline-formula>) of fracture length distribution and the evaluation of <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M5"> <mi>K</mi> </math> </jats:inline-formula> agrees well with that reported in previous studies, confirming the reliability of the EPN model for calculating seepage properties of complex 3D DFN models. When the side length of submodels (<jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> </jats:inline-formula>) is small, the <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M7"> <mi>K</mi> </math> </jats:inline-formula> varies significantly due to the influence of random number seeds used to generate fracture length, location, and orientation. The <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M8"> <mi>K</mi> </math> </jats:inline-formula> holds a constant value after <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M9"> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> </jats:inline-formula> exceeds some specific value. The critical model scale is determined as the REV size, and the corresponding volume of the 3D DFN model is represented by <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M10"> <msub> <mrow> <mi>V</mi> </mrow> <mrow> <mtext>REV</mtext> </mrow> </msub> </math> </jats:inline-formula>. The <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M11"> <msub> <mrow> <mi>V</mi> </mrow> <mrow> <mtext>REV</mtext> </mrow> </msub> </math> </jats:inline-formula> varies within a narrow range when <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M12"> <mi>a</mi> <mo>≤</mo> <mn>4.0</mn> </math> </jats:inline-formula>. When <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M13"> <mi>a</mi> <mo>=</mo> <mn>4.5</mn> </math> </jats:inline-formula>, the <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M14"> <msub> <mrow> <mi>V</mi> </mrow> <mrow> <mtext>REV</mtext> </mrow> </msub> </math> </jats:inline-formula> rapidly increases to more than 3.4 times than that when <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M15"> <mi>a</mi> <mo>=</mo> <mn>4.0</mn> </math> </jats:inline-formula>. The fluid flow becomes more inhomogeneous due to the small nonpersistent fractures that dominate the preferential flow paths when <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M16"> <mi>a</mi> </math> </jats:inline-formula> exceeds a certain value (i.e., 4.5). The <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M17"> <mi>K</mi> </math> </jats:inline-formula> at the REV size decreases exponentially with the increment of <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M18"> <mi>a</mi> </math> </jats:inline-formula>. This tendency can be explained by the decrease of the average intersection length (<jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M19"> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </math> </jats:inline-formula>) with the increment of <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M20"> <mi>a</mi> </math> </jats:inline-formula>, which is a geometric parameter for reflecting the connectivity of the fracture network. </jats:p>