An analytical form of the pore-emanating crack model in 3-D with implications for the uniaxial compressive strength of sandstone

SUMMARY The uniaxial compressive strength ${\sigma _c}$ of rocks is a key material property in a wide range of applications. Models for ${\sigma _c}$ typically either require numerical solutions, restricting their wide utility, or are empirical and therefore confined to a specific case. Here, we study the theoretical pore-emanated crack model and provide an analytical emulator function that matches the 2-D and 3-D solutions to a high degree of accuracy over all porosities, $\phi $. A key input to both the full solution and to our emulator functions is the pore radius, assumed in the model to be circular or spherical, in a porous rock. In most porous lithologies, including sandstone, the notion of a pore radius is poorly defined since they are built from compacted or lithified grains. And so here we explore statistical methods to find a characteristic pore length scale, ${l_2}$, from an initial particle radius; this method is provided as an easy-to-use supplementary tool. We advocate for the use of our 3-D function ${\sigma _c} \approx 1.57{K_{{\bf{Ic}}}}/( {{\phi ^{0.43}}\sqrt {\pi {l_2}} } )$, where ${K_{{\bf{Ic}}}}$ is the fracture toughness of the solid matrix. A compilation of ${K_{{\bf{Ic}}}}$ values for minerals and rocks allows us to explore the effect of this parameter and to make recommendations for appropriate values in the model. We compare our simple emulator function for ${\sigma _c}$ with existing data sets across a wide range of sandstones to demonstrate the utility of this law for applied cases. We find that our function performs particularly well for relatively low porosity sandstones ($\phi \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} 0.15$) representative of mature basin systems from a diagenetic point of view; we discuss alternative models that are more appropriate for higher porosity sandstones.