A magma reservoir inflates until the rock fails, a conduit opens and the pressure drains. The top curve shows the ground uplift a GPS network would see (Mogi model).
Magma flows into a chamber of volume V at rate Qin. Magma compressibility βm and the elastic compliance of the surrounding rock, βc = 3/(4μ) for a sphere in an elastic medium of shear modulus μ, store the extra volume as overpressure: C dP/dt = Qin - Qout, C = (βm + βc) V. While the conduit is closed P = Qin t / C.
When P reaches the tensile strength T0 the roof fractures and a conduit opens. Flow in a cylindrical conduit of radius a, length L and viscosity η is Poiseuille flow, Qout = k P, k = π a4/(8 η L). The overpressure then relaxes as P(t) = P∞ + (P0 - P∞) e-kt/C with P∞ = Qin/k until it falls to the closure pressure Pc = f T0, the conduit seals and the cycle repeats. This hysteresis gives periodic eruptions.
The ground deforms as in Mogi's point-source model: with source volume change ΔV = βc V P at depth d, the vertical displacement is uz(r) = (1 - ν) ΔV d / (π (d² + r²)3/2), whose integral over the surface equals 2(1 - ν) ΔV.
Simplifications: a spherical source in a homogeneous elastic half-space, constant material properties, no gas exsolution, crystallisation or thermal effects, one fixed closure fraction, and a cylindrical conduit. The drawn uplift is exaggerated 5000 times.