Viscoplastic analysis of salt caverns under time-asymmetric pressure cycling: definition and validation of a dimensionless operational rule with real case data

Abstract Operational asymmetry is a characteristic feature of underground gas storage systems—particularly in salt caverns—where injection phases are typically longer and steadier than withdrawal phases, which are shorter and more abrupt in response to fluctuating energy demand. Although large-scale hydrogen storage in salt caverns remains under development, similar operational patterns are expected once such facilities reach commercial maturity. Understanding how this temporal imbalance influences the mechanical behavior of the cavern is therefore important for assessing long-term performance and supporting the development of robust operational strategies for future hydrogen storage systems. This study introduces a dimensionless parameter, $$\chi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> </mml:math> , to quantify the temporal asymmetry of cyclic operation. $$\chi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> </mml:math> accounts not only for the residence times at high and low pressure but also for the different durations required to reach and leave these states. It is defined as the ratio between the effective durations of the high-pressure and low-pressure stages, including both the holding and ramp phases. The numerator represents the total effective time under high-pressure conditions, which includes half of the filling ramp ( $$0.5\tau _{\textrm{fill}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.5</mml:mn> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>fill</mml:mtext> </mml:msub> </mml:mrow> </mml:math> ), the full high-pressure holding phase ( $$\tau _{\textrm{WGP}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>WGP</mml:mtext> </mml:msub> </mml:math> ), and half of the withdrawal ramp ( $$0.5\tau _{\textrm{withd}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.5</mml:mn> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>withd</mml:mtext> </mml:msub> </mml:mrow> </mml:math> ). The denominator represents the complementary effective time under low-pressure conditions, consisting of half of the withdrawal ramp ( $$0.5\tau _{\textrm{withd}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.5</mml:mn> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>withd</mml:mtext> </mml:msub> </mml:mrow> </mml:math> ), the full low-pressure holding phase ( $$\tau _{\textrm{CGP}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>CGP</mml:mtext> </mml:msub> </mml:math> ), and half of the filling ramp ( $$0.5\tau _{\textrm{fill}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.5</mml:mn> <mml:msub> <mml:mi>τ</mml:mi> <mml:mtext>fill</mml:mtext> </mml:msub> </mml:mrow> </mml:math> ). This formulation expresses the relative exposure of the rock mass to stabilizing (high-stress) versus creep-promoting (low-stress) conditions. A viscoplastic numerical model is developed to reproduce the time-dependent deformation of salt under cyclic pressure loading. Two measurable indicators are analyzed: the vertical displacement at the cavern roof and the volumetric shrinkage associated with viscoplastic mechanical closure. Results show that temporal asymmetry exerts a dominant control on the long-term mechanical response. When $$\chi &gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>χ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> —indicating longer effective residence under high pressure—the cavern evolves toward a steady-state equilibrium; when $$\chi &lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>χ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , viscoplastic deformation accelerates, promoting mechanical cavern closure. Comparison with operational data from three underground natural gas storage facilities in the United States is consistent with the $$\chi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> </mml:math> ranges associated with the stability trends identified numerically. The proposed parameter $$\chi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> </mml:math> thus provides a physically grounded and practical first-order indicator for interpreting long-term mechanical behavior in future hydrogen storage caverns, within the scope of a viscoplastic mechanical framework that does not explicitly account for damage, fracture, or permeability evolution.