Long-term cavity closure in non-linear rocks

Abstract

The time dependent closure of pressurized cavities in viscous rocks due to far-field loads is a problem encountered in many applications like drilling, cavity abandonment and porosity closure. The non-linear nature of the flow of rocks prevents the use of simple solutions for hole closure and calls for the development of appropriate expressions reproducing all the dependencies observed in nature. An approximate solution is presented for the closure velocity of a pressurized cylindrical cavity in a non-linear viscous medium subjected to a combined pressure and shear stress load in the far field. The embedding medium is treated as homogeneous, isotropic, and incompressible and follows a Carreau viscosity model. We derive analytical solutions for the end-member cases of the pressure and shear loads. The exact analytical solution for pressure loads shows that the closure velocity vR is given by the implicit expression |$\frac{{\Delta p}}{{2{\mu _0}D_{II}^*}} = - \frac{1}{2}B( {\frac{{v_R^2}}{{RD_{II}^* + v_R^2}};\frac{1}{2}, - \frac{1}{{2n}}} )$|⁠, where Δp is the pressure load, R is the hole radius, B is the incomplete beta function, and μ0, |$D_{II}^*$|⁠, n are, respectively, the threshold viscosity, transition rate and stress exponent of the Carreau model. The closure velocity is dominated by the linear mechanism under pressure loads smaller than |$1.8{\mu _0}D_{II}^*$| and by the non-linear one under large pressure loads. In the non-linear regime, pressure variations support an increasing part of the load with increasing degree of non-linearity. The decay of the stress perturbation in the non-linear zone varies as r− 2/n where r is the radial distance to the hole. A solution for the maximum closure velocity at the cavity rim vRmax under far-field shear is given: |${v_{R\max }} = ( {1 + {{\overline {{M_s}} }^{ - {1 {{/}} 2}}}} )R\overline {{D_{II}}} $|⁠, where |$\overline {{M_s}} = {{( {{{1 + {{\overline {{D_{II}}} }^2}} {{\big/}} {nD{{_{II}^*}^2}}}} )} {{\big/}} {( {1 + {{{{\overline {{D_{II}}} }^2}} {{\big/}} {D{{_{II}^*}^2}}}} )}}$| and |$\overline {{D_{II}}} $| is the second invariant of the far-field deformation rate. The solution remains valid in the limit of ideal power-law fluid when |$\overline {{M_s}} = {1 {{/}} n}$|⁠. The solution is based on an approximation which transforms the non-linear isotropic constitutive law into a linear anisotropic one in the far field. The proposed approximate solution for closure velocity for general far-field load is based on the two end-member analytical solutions. They are additively combined after replacing the material threshold viscosity μ0 by the apparent background viscosity due to the far-field shear. Benchmarking the solution shows that there is a 50 per cent misfit at most between the analytical and the numerical solution for closure rate. Comparing the closure velocities obtained with a Carreau viscosity model with the ones from a power-law material shows an increase of several orders of magnitude for pressure loads smaller than |$2{\mu _0}D_{II}^*$|⁠. Far field shear can also increase hole closure rate by several orders of magnitude. Compared with other available solutions, the approximate solution presented here ensures that both speed ups are taken into account making it well suited for actual underground conditions where both diffusion creep and shear stresses occur. The additional closure mechanisms considered here can potentially explain the fast closure rates observed underground without referring to transient mechanisms.