Li, Qingchao http://orcid.org/0000-0001-7373-4046
Wu, Jingjuan
Funding for this research was provided by:
National Key Research and Development Program (2016YFC0304005)
Postdoctoral Program of Henan Polytechnic University (712108/210)
Article History
Received: 16 December 2020
Accepted: 10 February 2022
First Online: 2 April 2022
Declarations
:
: On behalf of all authors, they declare that there is no conflict of interests.
: Relationship between <i>σ</i><sub><i>j</i></sub>, <i>σ</i><sub><i>i</i></sub> and <i>σ</i><sub><i>r</i></sub> mentioned in Eqs. (11) may exhibits three possibilities: <i>σ</i><sub><i>j</i></sub> > <i>σ</i><sub><i>i</i></sub> > <i>σ</i><sub><i>r</i></sub>, <i>σ</i><sub><i>j</i></sub> > <i>σ</i><sub><i>r</i></sub> > <i>σ</i><sub><i>i</i></sub> or <i>σ</i><sub><i>r</i></sub> > <i>σ</i><sub><i>i</i></sub> > <i>σ</i><sub><i>j</i></sub>. Accordingly, derivation method for determining function of <i>f(P</i><sub><i>m</i></sub><i>)</i> was discussed in the following three cases.(1) When the case of <i>σ</i><sub><i>j</i></sub> > <i>σ</i><sub><i>i</i></sub> > <i>σ</i><sub><i>r</i></sub> occurs, σ<sub>1</sub> and σ<sub>3</sub> can be expressed as:Equation removedThe final form of <i>f(P</i><sub><i>m</i></sub><i>)</i> was obtained in this case by substituting Eqs. (A-1) into Eqs. (13):Equation removedTaking the first derivative of <i>f(P</i><sub><i>m</i></sub><i>)</i> function, we can get:Equation removedIn Eqs. (A-3), the six variables A, B, C, D, E and F were written asEquation removed(2) When the case of <i>σ</i><sub><i>j</i></sub> > <i>σ</i><sub><i>r</i></sub> > <i>σ</i><sub><i>i</i></sub> occurs, σ<sub>1</sub> and σ<sub>3</sub> can be expressed as:Equation removedFunction <i>f(P</i><sub><i>m</i></sub><i>)</i> and its first derivative can be written as Eqs. (A-6) and Eqs. (7) respectively.Equation removedEquation removed(3) When the case of <i>σ</i><sub><i>r</i></sub> > <i>σ</i><sub><i>i</i></sub> > <i>σ</i><sub><i>j</i></sub> occurs, σ<sub>1</sub> and σ<sub>3</sub> can be expressed as:Equation removedSimilarly, function <i>f(P</i><sub><i>m</i></sub><i>)</i> and its first derivative can be written as Eqs. (A-9) and Eqs. (10) respectively.Equation removedEquation removedBased on the function <i>f(P</i><sub><i>m</i></sub><i>)</i> and its first derivative derived above, the Newton–Raphson iteration was obtained.Equation removedwhere, <i>P</i><sub><i>m0</i></sub> is the pre-assumed fluid pressure in MPa. <i>P</i><sub><i>m1</i></sub> is the calculated result in MPa. Moreover, in order to facilitate readers to understand the role of each function in these Matlab codes submitted as “Supplementary Material”, Fig. A-1 gives the corresponding structure diagram.
: This work is supported by Program for the National Key Research and Development Program (Grant No. 2016YFC0304005) and the Postdoctoral Program of Henan Polytechnic University (Grant No. 712108/210).