Effective poroelastic properties of N-layered composite sphere assemblage: An application to oolitic limestone

Abstract

This article derives the effective poroelastic properties of N-layered composite assemblage. The derivation is based on the Hashin composite sphere assemblage (CSA) model and the linear elastic solution of n-layer coated inclusion-reinforced materials proposed by Hervé and Zaoui. The contribution of this study consists in the consideration of the poromechanical coupling to derive not only the bulk and shear drained moduli but also the Biot coefficient and the solid Biot modulus. The theoretical solutions are used for studying the oolitic limestone from Bourgogne (France), in which the microstructure exhibits generally an assemblage of oolite grains surrounded by a matrix. They are linked via interphase where most of the macropores locate. A two-step homogenisation scheme is proposed. The first step consists in upscaling the mesoscopic poroelastic properties of each porous phase by using the differential self-consistent scheme. In the second step, the three different porous constituents (oolite, ITZ and matrix) are homogenised using the CSA model. Results are validated against the data collected from the open literature.

Keywords:

• CSA
• differential self-consistent scheme; homogenisation
• N-layered composite sphere
• oolitic limestone
• poroelastic properties

Appendix. Poroelastic properties resulted from the four-phase CSA model

A1. Bulk modulus $k_{4ph}^{}$

(A1) $$k_{4ph}^{}=k_3-\frac{3k_3+4{\mu }_3}{3-\frac{f_1{\omega }_1+f_2{\omega }_2}{\left(f_1+f_1\right)\left(f_1{\omega }_3+f_2{\omega }_4\right)}}\mathrm{ }$$(A1)

with (A2) $${\omega }_1=\frac{3k_1+4{\mu }_3}{3k_1+4{\mu }_2}\mathrm{ }{\omega }_2=\frac{3k_2+4{\mu }_3}{3k_2+4{\mu }_2}\mathrm{ }{\omega }_3=\frac{k_1-k_3}{3k_1+4{\mu }_2}\mathrm{ }{\omega }_4=\frac{3k_2-k_3}{3k_2+4{\mu }_2}$$(A2)

A2. Shear modulus ${\mu }_{4ph}^{}$

The overall shear modulus ${\mu }_{4ph}^{}$ is the root of the following equation: (A3) $$\left|\begin{array}{cccccccccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}& a_{16}& 0& 0& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}& a_{26}& 0& 0& 0& 0& 0& 0\\ a_{31}& a_{32}& a_{33}& a_{34}& a_{35}& a_{36}& 0& 0& 0& 0& 0& 0\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}& a_{46}& 0& 0& 0& 0& 0& 0\\ 0& 0& a_{53}& a_{54}& a_{55}& a_{56}& a_{57}& a_{58}& a_{59}& a_{510}& 0& 0\\ 0& 0& a_{63}& a_{64}& a_{65}& a_{66}& a_{67}& a_{68}& a_{69}& a_{610}& 0& 0\\ 0& 0& a_{73}& a_{64}& a_{75}& a_{76}& a_{77}& a_{78}& a_{79}& a_{710}& 0& 0\\ 0& 0& a_{83}& a_{84}& a_{85}& a_{86}& a_{87}& a_{88}& a_{89}& a_{810}& 0& 0\\ 0& 0& 0& 0& 0& 0& a_{97}& a_{98}& a_{99}& a_{910}& a_{911}& a_{912}\\ 0& 0& 0& 0& 0& 0& a_{107}& a_{108}& a_{109}& a_{1010}& a_{1011}& a_{1012}\\ 0& 0& 0& 0& 0& 0& a_{117}& a_{118}& a_{119}& a_{1110}& a_{1111}& a_{1112}\\ 0& 0& 0& 0& 0& 0& a_{127}& a_{128}& a_{129}& a_{1210}& a_{1211}& a_{1212}\end{array}\right|=0$$(A3) in which, ${\mu }_{4ph}^{}$ is only appear in 4 components a₃₁, a₃₂; a₄₁ and a₄₂: (A4) $$a_{31}=\frac{3}{8}{a}_{32}=4a_{41}=-a_{42}=\frac{{\mu }_{4ph}^{}}{{\mu }_3}$$(A4) Other components of the matrix in Equation (A3)(A3) $$\left|\begin{array}{cccccccccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}& a_{16}& 0& 0& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}& a_{26}& 0& 0& 0& 0& 0& 0\\ a_{31}& a_{32}& a_{33}& a_{34}& a_{35}& a_{36}& 0& 0& 0& 0& 0& 0\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}& a_{46}& 0& 0& 0& 0& 0& 0\\ 0& 0& a_{53}& a_{54}& a_{55}& a_{56}& a_{57}& a_{58}& a_{59}& a_{510}& 0& 0\\ 0& 0& a_{63}& a_{64}& a_{65}& a_{66}& a_{67}& a_{68}& a_{69}& a_{610}& 0& 0\\ 0& 0& a_{73}& a_{64}& a_{75}& a_{76}& a_{77}& a_{78}& a_{79}& a_{710}& 0& 0\\ 0& 0& a_{83}& a_{84}& a_{85}& a_{86}& a_{87}& a_{88}& a_{89}& a_{810}& 0& 0\\ 0& 0& 0& 0& 0& 0& a_{97}& a_{98}& a_{99}& a_{910}& a_{911}& a_{912}\\ 0& 0& 0& 0& 0& 0& a_{107}& a_{108}& a_{109}& a_{1010}& a_{1011}& a_{1012}\\ 0& 0& 0& 0& 0& 0& a_{117}& a_{118}& a_{119}& a_{1110}& a_{1111}& a_{1112}\\ 0& 0& 0& 0& 0& 0& a_{127}& a_{128}& a_{129}& a_{1210}& a_{1211}& a_{1212}\end{array}\right|=0$$(A3) depends only on properties of three phases: (58) $$\begin{array}{c}{a}_{11}=1 a_{21}=1 a_{31}=\frac{{\mu }_e^{\Delta ph}}{{\mu }_3} a_{41}=\frac{{\mu }_e^{\Delta ph}}{4{\mu }_3}\\ a_{12}=-\frac{2}{3} a_{22}=1 a_{32}=\frac{8{\mu }_e^{\Delta ph}}{3{\mu }_3} a_{42}=-\frac{{\mu }_e^{\Delta ph}}{{\mu }_3}\\ a_{13}=\frac{2}{3}-\frac{7}{\left(6v_3\right)} a_{23}=-1 a_{33}=-\frac{7+2v_3}{\left(6v_3\right)} a_{43}=\frac{1}{8}\\ a_{14}=1 a_{24}=-1 a_{34}=-1a_{44}=-\frac{1}{4}\\ a_{15}=-1+\frac{3}{\left(5-4v_3\right)} a_{25}=-1 a_{35}=-\frac{2+2v_3}{\left(4v_3-5\right)} a_{45}=\frac{v_3-5}{8v_3-10}\\ a_{16}=\frac{2}{3} a_{26}=-1 a_{36}=-\frac{8}{3} a_{46}=1\\ a_{97}=\frac{f_1^{2/3}\left(7-4v_2\right)}{6v_2} a_{107}=f_1^{7/3} a_{177}=\frac{f_1^{2/3}\left(7+2v_2\right)}{6v_2} a_{127}=-\frac{f_1^{2/3}}{2}(A.18)\\ a_{98}=1 a_{108}=f_1^{5/3} a_{118}=1 a_{128}=1\\ a_{99}=\frac{2-4v_2}{f_1\left(5-4v_2\right)} a_{109}=f_1^{2/3} a_{119}=\frac{2+2v_2}{f_1\left(5-4v_2\right)} a_{129}=\frac{2v_2-10}{f_1\left(5-4v_2\right)}\\ a_{910}=-\frac{2}{3f_1^{\frac{5}{3}}} a_{1010}=1 a_{1110}=\frac{8}{3f_1^{\frac{5}{3}}} a_{1210}=\frac{-4}{f_1^{5/3}}\\ a_{911}=-\frac{f_1^{\frac{2}{3}}\left(7-4v_1\right)}{6v_1} a_{1011}=-f_1^{\frac{7}{3}} a_{1111}=-\frac{f_1^{\frac{2}{3}}\left(7+2v_1\right)}{6v_1}\frac{{\mu }_1}{{\mu }_2} a_{1211}=\frac{f_1^{2/3}}{2}\frac{{\mu }_1}{{\mu }_2}\\ a_{912}=1 a_{1012}=-f_1^{\frac{5}{3}} a_{1112}=-\frac{{\mu }_1}{{\mu }_2} a_{1212}=-\frac{{\mu }_1}{{\mu }_2}\\ a_{53}=\frac{{\left(f_1+f_2\right)}^{7/3}\left(7-4{\nu }_3\right)}{4{\nu }_3} a_{63}={\left(f_1+f_2\right)}^{7/3}\\ a_{53}=\frac{{\left(f_1+f_2\right)}^{7/3}\left(7-4{\nu }_3\right)}{4{\nu }_3} a_{63}={\left(f_1+f_2\right)}^{7/3}\\ a_{54}=\frac{3}{2}{\left(f_1+f_2\right)}^{5/3} a_{64}={\left(f_1+f_2\right)}^{5/3}\\ a_{55}=3\frac{{\left(f_1+f_2\right)}^{2/3}\left(2{\nu }_3-1\right)}{\left(4{\nu }_3-5\right)} a_{65}={\left(f_1+f_2\right)}^{2/3}\\ a_{56}=-1 a_{66}=1\\ a_{57}=\frac{{\left(f_1+f_2\right)}^{7/3}\left(-7+4{\nu }_2\right)}{4{\nu }_2} a_{67}=-{\left(f_1+f_2\right)}^{7/3}\\ a_{58}=-\frac{3}{2}{\left(f_1+f_2\right)}^{\frac{5}{3}} a_{68}=-{\left(f_1+f_2\right)}^{5/3}\\ a_{59}=-3\frac{{\left(f_1+f_2\right)}^{\frac{2}{3}}\left(2{\nu }_2-1\right)}{\left(4{\nu }_2-5\right)} a_{69}=-{\left(f_1+f_2\right)}^{2/3}\\ a_{510}=1 a_{610}=-1\\ a_{73}=\frac{{\left(f_1+f_2\right)}^{7/3}\left(7+2{\nu }_3\right)}{16{\nu }_3} { a}_{83}=-\frac{1}{8}{\left(f_1+f_2\right)}^{7/3}\\ a_{74}=\frac{3}{8}{\left(f_1+f_2\right)}^{5/3} a_{84}=\frac{1}{4}{\left(f_1+f_2\right)}^{5/3}\\ a_{75}=-3\frac{{\left(f_1+f_2\right)}^{\frac{2}{3}}\left({\nu }_3+1\right)}{4\left(4{\nu }_3-5\right)} a_{85}=\frac{{\left(f_1+f_2\right)}^{2/3}\left({\nu }_3-5\right)}{\left(10-8{\nu }_3\right)}\\ a_{76}=1 a_{86}=-1\\ a_{77}=-\frac{{\left(f_1+f_2\right)}^{\frac{7}{3}}\left(7+2{\nu }_2\right)}{16{\nu }_2} a_{87}=\frac{1}{8}{\left(f_1+f_2\right)}^{7/3}\frac{{\mu }_2}{{\mu }_3}\\ a_{78}=-\frac{3}{8}{\left(f_1+f_2\right)}^{\frac{5}{3}}\frac{{\mu }_2}{{\mu }_3} a_{88}=-\frac{1}{4}{\left(f_1+f_2\right)}^{5/3}\frac{{\mu }_2}{{\mu }_3}\\ a_{79}=3\frac{{\left(f_1+f_2\right)}^{2/3}\left({\nu }_2+1\right)}{4\left(4{\nu }_2-5\right)} a_{89}=-\frac{{\left(f_1+f_2\right)}^{2/3}\left({\nu }_2-5\right)}{\left(10-8{\nu }_3\right)}\frac{{\mu }_2}{{\mu }_3}\\ a_{710}=-\frac{{\mu }_2}{{\mu }_3} a_{810}=\frac{{\mu }_2}{{\mu }_3}\end{array}$$(58)

A3. Biot coefficient $b_{4ph}^{}$

(A5) $$\mathrm{ }{b}_{4ph}^{} =b_3^{}+\mathrm{ }\frac{\left(3k_3+4{\mu }_3\right)\left(f_1+f_2\right)\left(b_2-b_3\right)\left(4{\mu }_2\left(f_1+f_2\right)+3{\Gamma }_{12}\right)-f_1\left(\left(b_2-b_1\right)\left(3k_2+4{\mu }_2\right)\right)}{9\left(f_1+f_2\right)f_3{k}_1{k}_2+12{\mu }_2{\Gamma }_{21}{f}_3+\left(3k_3\left(f_1+f_2\right)+4{\mu }_3\right)\left(3{\Gamma }_{12}+4{\mu }_2\left(f_1+f_2\right)\right)}$$(A5)

where (A6) $$\begin{array}{c}{f}_3=1-f_1-f_2\\ {\Gamma }_{12}=f_1{k}_2+f_2{k}_1\mathrm{ }{\Gamma }_{21}=f_1{k}_1+f_2{k}_2\end{array}$$(A6)

A4. Biot modulus of solid skeleton $N_{4ph}^{}$

(A7) $$\frac{1}{N_{4ph}^{}}={\sum }_{i=1}^3{f}_i\frac{1}{N_i}+\frac{N{N}_{\mathit{hom}}}{D{N}_{\mathit{hom}}}$$(A7)

with (A8) $$\begin{array}{c}N{N}_{\mathrm{hom}}=3(-{(b_2-b_3)}^2{f}_3{(f_1+f_2)}^2(3k_1\\ +4{\mu }_2)+{(b_1-b_2)}^2{f}_1^2((f_1+f_2)(3k_3+4{\mu }_2)-4{\mu }_2+4{\mu }_3)\\ -f_1(f_1+f_2)(3b_3^2{f}_3(k_2-k_1)-2b_1{b}_3{f}_3(k_2-k_1)-2b_1{b}_3{f}_3(3k_2+4{\mu }_2)\\ +b_1^2(3f_3{k}_2+3\left(f_1+f_2\right)k_3+4{\mu }_3)\\ +b_2^2((f_1+f_2)(3k_3+8{\mu }_2)+4{\mu }_3-3f_3{k}_1-8{\mu }_2)\\ -2b_2(b_1((f_1+f_2)(3k_3+4{\mu }_2)-4{\mu }_2+4{\mu }_3)-b_3{f}_3(3k_1+4{\mu }_2))))\end{array}$$(A8)

and (A9) $$\begin{array}{c}D{N}_{\mathrm{hom}}=-9(f_1+f_2)(f_3{k}_1{k}_2+T_{12}{k}_3)-12(f_3{T}_{21}+{(f_1+f_2)}^2{k}^3){\mu }_2\\ -4(3T_{12}+4(f_1+f_2){\mu }_2){\mu }_3\end{array}$$(A9)

Additional information

Funding

This research was funded by National Foundation for Science and Technology Development (NAFOSTED), Vietnam under grant number 107.01-2017.307.