We consider a magma plumbing system in which two elastically deformable magma chambers are connected in series with non-deformable conduits (Fig. 1). A pressure change in each magma chamber is controlled by the balance between magma influx from the deeper conduit and outflux to the shallower conduit. In this case, the temporal changes in the chamber pressures are formulated as

$$\begin{aligned} \dfrac{dp_\text {1}}{dt} = C_\text {1} (Q_{\text {in1}}-Q_{\text {out1}}), \end{aligned}$$

(1)

and

$$\begin{aligned} \dfrac{dp_\text {2}}{dt} = C_\text {2} (Q_\text {in2}-Q_{\text {out2}}), \end{aligned}$$

(2)

where

$$\begin{aligned} C_ {1} = \dfrac{1}{\rho _{1} V_{1} (\kappa _\text {m1}+\kappa _{\text {ch1}})} \quad {\text {and}} \quad C_{2} = \dfrac{1}{\rho _{2} V_ {2} (\kappa _\text {m2}+\kappa _{\text {ch2}})}, \end{aligned}$$

(3)

*p* is the chamber pressure, *t* is time, \(Q_{\text {in}}\) and \(Q_{\text {out}}\) are the magma influx and outflux, respectively (in kg s\(^{-1}\)), \(\rho\) is the magma density, *V* is the chamber volume, \(\kappa _\text {m}\), is the magma compressibility in the chamber, and \(\kappa _{\text {ch}}\) is the compressibility of the magma chamber. The number subscripts “1” and “2” correspond to the deeper and shallower magma chambers, respectively (Fig. 1). Here, \(C_\text {1}\) and \(C_\text {2}\) are the parameters controlling a variability of the chamber pressure: a larger \(C_\text {1}\) or \(C_\text {2}\) leads to a larger magnitude of the chamber pressure change for given magma influx and outflux.

The magma flow in the conduit is modeled as a Poiseuille flow. To set the conduit geometry as general as possible, we consider a dyke-like conduit with an elliptical cross section (Costa et al. 2007, Fig. 1). Under the assumption that a temporal change in the magma density in the conduit is negligible, the mass conservation in the conduit between the two chambers is expressed as

$$\begin{aligned} \rho _\text {12} w_\text {12} \pi a_\text {12} b_\text {12} = Q_{\text {out1}} = Q_\text {in2}, \end{aligned}$$

(4)

where *w* is the ascent velocity, *a* and *b* are the semi-major and semi-minor axes of the ellipse, respectively, and the subscript “12” represents the conduit between the two chambers (Fig. 1). In the momentum conservation, we assume a linear pressure gradient and that the inertia term is negligible. In this case, we obtain:

$$\begin{aligned} \dfrac{p_\text {2}-p_\text {1}}{L_\text {12}} = - \rho _\text {12} g - \frac{4 \eta _\text {12} w_\text {12} (a_\text {12}^2 + b_\text {12}^2)}{a_\text {12}^2 b_\text {12}^2}, \end{aligned}$$

(5)

where *L* is the conduit length, *g* is the gravity acceleration, and \(\eta\) is the magma viscosity. Similarly, the mass and momentum conservations in the conduit between the shallower chamber and the surface are approximated as

$$\begin{aligned} \rho _\text {2s} w_\text {2s} \pi a_\text {2s} b_\text {2s} = Q_{\text {out2}}, \end{aligned}$$

(6)

and

$$\begin{aligned} \dfrac{p_\text {s}-p_\text {2}}{L_\text {2s}} = - \rho _\text {2s} g - \frac{4 \eta _\text {2s} w_\text {2s}(a_\text {2s}^2 + b_\text {2s}^2)}{a_\text {2s}^2 b_\text {2s}^2}, \end{aligned}$$

(7)

where \(p_\text {s}\) is the pressure at the surface, and “\(\text {2s}\)” represents the conduit between the shallower chamber and the surface (Fig. 1). From Eqs. (4)–(7), we obtain

$$\begin{aligned} Q_{\text {out1}} = \Omega _\text {12} (p_\text {1}-p_\text {2} -\rho _\text {12} g L_\text {12}), \end{aligned}$$

(8)

and

$$\begin{aligned} Q_{\text {out2}} = \Omega _\text {2s} (p_\text {2}-p_\text {s} -\rho _\text {2s} g L_\text {2s}), \end{aligned}$$

(9)

where

$$\begin{aligned} \Omega _\text {12} = \dfrac{\rho _\text {12} \pi a_\text {12}^3 b_\text {12}^3}{4 \eta _\text {12} L_\text {12}(a_\text {12}^2 + b_\text {12}^2)} \; {\text {and}} \; \Omega _\text {2s} = \dfrac{\rho _\text {2s} \pi a_\text {2s}^3 b_\text {2s}^3}{4 \eta _\text {2s} L_\text {2s}(a_\text {2s}^2 + b_\text {2s}^2)}. \end{aligned}$$

(10)

The parameters \(\Omega _\text {12}\) and \(\Omega _\text {2s}\) correspond to “conduit conductivity” (Slezin 2003): a larger \(\Omega _\text {12}\) or \(\Omega _\text {2s}\) leads to a higher magma flux for a given pressure gradient in the conduit due to lower viscous resistance. Here, the chamber pressure is expressed using the overpressure (\(\Delta p\)) with respect to lithostatic pressure as

$$\begin{aligned} p_\text {1} = \rho _\text {c} g (L_\text {12}+L_\text {2s}) + \Delta p_\text {1} \quad {\text {and}} \quad p_\text {2} = \rho _\text {c} g L_\text {2s} + \Delta p_\text {2}, \end{aligned}$$

(11)

where \(\rho _\text {c}\) is the crust density (\(\sim 2500\) kg m\(^{-3}\)).

From Eqs. (1), (2), (8), (9), and (11) and under the assumption that \(p_\text {s} \ll p_\text {2}\), we obtain

$$\begin{aligned} \dfrac{d \Delta p_\text {1}}{dt} = C_\text {1} \left[ Q_{\text {in1}}-\Omega _\text {12} \left\{ \Delta p_\text {1}-\Delta p_\text {2} +(\rho _\text {c} -\rho _\text {12})g L_\text {12} \right\} \right] , \end{aligned}$$

(12)

and

$$\begin{aligned} \dfrac{d \Delta p_\text {2}}{dt} = C_\text {2} \left[ \Omega _\text {12} \left\{ \Delta p_\text {1}-\Delta p_\text {2} +(\rho _\text {c} -\rho _\text {12})g L_\text {12} \right\} -\Omega _\text {2s} \left\{ \Delta p_\text {2} +(\rho _\text {c} -\rho _\text {2s})g L_\text {2s} \right\} \right] . \end{aligned}$$

(13)

Eqs. (12) and (13) are normalized by substituting \(t = \bar{t} t'\), \(\Delta p = \bar{p} \Delta p'\), and \(Q = \bar{Q} Q'\), where \(\bar{t}\), \(\bar{p}\), and \(\bar{Q}\) are

$$\begin{aligned} \bar{t} = \frac{1}{C_\text {1} \Omega _\text {12}}, \quad \bar{p} = (\rho _\text {c}-\rho _\text {12}) g L_\text {12}, \quad {\text {and}} \quad \bar{Q} = \Omega _\text {12} (\rho _\text {c}-\rho _\text {12}) g L_\text {12}. \end{aligned}$$

(14)

As a result, we get

$$\begin{aligned} \dfrac{d \Delta p'_\text {1}}{dt'} = - \Delta p'_\text {1} + \Delta p'_\text {2} - 1 + Q'_{\text {in1}} \end{aligned}$$

(15)

and

$$\begin{aligned} \dfrac{d \Delta p'_\text {2}}{dt'} = C' \left[ \Delta p'_\text {1}-(\Omega '+1) \Delta p'_\text {2} + 1 - \Omega ' \rho ' L' \right] , \end{aligned}$$

(16)

where

$$\begin{aligned} \rho ' = \dfrac{1-\dfrac{\rho _\text {2s}}{\rho _\text {c}}}{1-\dfrac{\rho _\text {12}}{\rho _\text {c}}}, \end{aligned}$$

(17)

and

$$\begin{aligned} L' = \frac{L_\text {2s}}{L_\text {12}}. \end{aligned}$$

(18)

It is noted that when \(\rho _\text {12}\) and \(\rho _\text {2s}\) in Eq. (17) are approximated as \((1-\phi _\text {12}) \rho _\text {lc}\) and \((1-\phi _\text {2s}) \rho _\text {lc}\), respectively, where \(\phi\) is the magma porosity in the conduit and \(\rho _\text {lc}\) is the liquid-crystals density which is close to \(\rho _\text {c}\), \(\rho '\) corresponds to the ratio of the porosity in the shallower conduit to that in the deeper conduit, \(\phi _\text {2s}/\phi _\text {12}\). Here, the non-dimensional numbers \(C'\) and \(\Omega '\) are the ratio of \(C_{2}\) to \(C_{1}\) and that of \(\Omega _{2s}\) to \(\Omega _{12}\), respectively, expressed as

$$\begin{aligned} C' \equiv \frac{C_\text {2}}{C_\text {1}} = \dfrac{\rho _\text {1} V_\text {1}}{\rho _\text {2} V_\text {2}} \kappa '^{-1}, \end{aligned}$$

(19)

and

$$\begin{aligned} \Omega ' \equiv \frac{\Omega _\text {2s}}{\Omega _\text {12}} = \dfrac{\rho _\text {2s} \eta _\text {12}}{\rho _\text {12} \eta _\text {2s}} \left( \dfrac{b_\text {2s}}{b_\text {12}} \right) ^{4} \dfrac{ r_\text {2s} (r_\text {12}^{-2} + 1 ) }{ r_\text {12} (r_\text {2s}^{-2} + 1 ) } L'^{-1}, \end{aligned}$$

(20)

where \(\kappa '\) is the ratio of the sum of the magma and the chamber compressibilities in the shallower chamber to that in the deeper chamber:

$$\begin{aligned} \kappa ' = \dfrac{\kappa _\text {m2}+\kappa _{\text {ch2}}}{\kappa _\text {m1}+\kappa _{\text {ch1}}}, \end{aligned}$$

(21)

and *r* is the ratio of the semi-major to semi-minor axes of the ellipse in the conduit cross section:

$$\begin{aligned} r_\text {12} = \dfrac{a_\text {12}}{b_\text {12}} \quad {\text {and}} \quad r_\text {2s} = \dfrac{a_\text {2s}}{b_\text {2s}}. \end{aligned}$$

(22)

From the simultaneous differential equations (15) and (16) and under the assumption that \(\rho '\), \(L'\), \(C'\), \(\Omega '\), and \(Q'_{\text {in1}}\) are constant during the pressure changes, we obtain an analytical solution for \(\Delta p'_\text {1}\) and \(\Delta p'_\text {2}\) as

$$\begin{aligned} \Delta p'_\text {1} = A_\text {+} \text {e}^{\lambda _\text {+}t'} + A_\text {-} \text {e}^{\lambda _\text {-}t'} -1 -\rho ' L' +\dfrac{\Omega ' + 1}{\Omega '} Q'_{\text {in1}}, \end{aligned}$$

(23)

and

$$\begin{aligned} \Delta p'_\text {2} = (\lambda _\text {+}+1) A_\text {+} \text {e}^{\lambda _\text {+}t'} +(\lambda _\text {-}+1) A_\text {-} \text {e}^{\lambda _\text {-}t'} - \rho ' L' +\dfrac{Q'_{\text {in1}}}{\Omega '}, \end{aligned}$$

(24)

where

$$\begin{aligned} \lambda _{\pm } = \frac{1}{2} \left[ - \left\{ C'(\Omega '+1)+1 \right\} \pm \sqrt{ \left\{ C'(\Omega '+1)+1 \right\} ^2 - 4 C' \Omega ' } \right] . \end{aligned}$$

(25)

When we use the boundary conditions that \(\Delta p'_\text {1}=\Delta p'_\text {1,0}\) and \(\Delta p'_\text {2}=\Delta p'_\text {2,0}\) for \(t'=0\), \(A_\text {+}\) and \(A_\text {-}\) are expressed as

$$\begin{aligned} A_{\pm } = \mp \frac{\lambda _{\mp } \left[ 1 +\rho ' L' -\dfrac{\Omega '+1}{\Omega '} Q'_{\text {in1}} + \Delta p'_\text {1,0} \right] + 1 - Q'_{\text {in1}} + \Delta p'_{1,0} - \Delta p'_\text {2,0}}{\sqrt{ \left\{ C'(\Omega '+1)+1 \right\} ^2 - 4 C' \Omega ' }}. \end{aligned}$$

(26)