#### 5.1. Sources of Deformation

The preferred best-fit model, including two dikes and a spherical reservoir, leads to reasonable residuals (

Figure 2). There are some residuals in the summit area, probably advocating for a more complex magma reservoir/northern dike geometry or a more complex rheology. However, it is difficult to further constrain the geometry of the summit deformation source due to the lack of coherence on the vegetated flanks. Similarly, some residuals are visible east of the southern flank eruptive fissure, possibly because of new ash deposits. The bottom of the southern dike extends beneath the southern flank of Nyamulagira where the opening and subsidence of a graben structure is visible [

20]

, as commonly observed at the upper tip of blade-shaped dikes [

42,

43,

44,

45]. Therefore, there is probably a pre-existing area of weakness close to the surface in which magma intrudes laterally along dikes to feed the SE flank eruptions from the shallow summit reservoir beneath the caldera. The upper southeastern flank of Nyamulagira may be developing a volcanic rift zone, similar to the well-developed volcanic rift zones observed on the volcanoes of Hawaii, La Réunion and the Canary Islands (e.g., [

46]).

As both fissures were fed by the same batch of magma [

10], the dikes and reservoir are probably connected, but our modeling method cannot resolve this connection. The connection between both dikes is probably via another dike, as this is the most efficient means of transporting magma through the lithosphere [

47,

48]. Such a dike may be too narrow to cause detectable InSAR displacements. Indeed, the data fit breaks down when we try to add a sizeable connecting dike (

Table 1), but if the connecting dike has a width smaller than 400 m at a 1-km depth, forward 3D-MBEM models show that it is undetectable. This dike width is similar to the eruptive fissure width, indicating that flow in the connecting dike could be the same as the flow at the surface. A similarly narrow path is also postulated for other volcanoes, such as Nyiragongo [

31] and Piton de la Fournaise [

39,

49].

The fissure that opened on the caldera floor follows an ~N45E orientation, which contrasts notably with the N155E orientation of the main rift zone and the southern flank eruptive fissure. However, this N45E orientation corresponds to a fracture system that opened during the 1938–1940 major eruption [

10]. Furthermore, during that same eruption, the SW caldera depression formed as the result of a collapse of the caldera. The magma reservoir modeled in this study is located in the same zone as the reservoir implied in the 1938–1940 eruption [

10]. In 2010, this reservoir could have induced the same stress field as in 1938–1940, reactivating N45E-trending pre-existing structures. The eruptive fissures are not located at the crest of the reservoir, as expected when a spherical reservoir fails under tension [

32], but away from the summit. This might be explained by host rock stresses inherited from previous eruptions.

There was no conclusively identified deflating reservoir during the 1996, 1998, 2000, 2001, 2002, 2004 and 2006 eruptions at Nyamulagira. The lack of an observed deflating reservoir at Nyamulagira can be explained by the following (not mutually exclusive) factors: (1) subtle deflation signals hidden by the deformation signals of large dike intrusions; (2) lack of coherent InSAR data in areas affected by reservoir deflation; (3) magma compressibility restoring pressure [

50,

51,

52,

53,

54]; and (4) a source of emitted magma too deep for inducing observable deformation [

55].

#### 5.2. Magma Reservoir Size

Assuming the reservoir is spherical, the maximum overpressure sustainable before tensile failure of the host rock is given as a function of the lithostatic stress,

P_{l}, and the tensile strength,

T_{0} [

39,

56], as:

where

${P}_{l}={\rho}_{r}gH$, with

${\rho}_{r}$ indicating host rock density,

$g$ indicating the acceleration of gravity and

$H$ indicating the depth to the top of the reservoir. If we assume

${\rho}_{r}$ = 2600 kg/m

^{3} [

57],

${T}_{0}$ = 0.5–5 MPa [

58] and use a value of

$H$ = 3.5 km from our inversions, the overpressure

$\Delta {P}_{r}$ = 179–183 MPa. If we consider that the pre-eruption excess volume

$\Delta {V}_{r}$, which induced the failure, corresponds to the pre-eruptive inflation detected by InSAR,

${u}_{z}=$ 5 cm [

21], we can estimate the radius of this reservoir. Assuming the reservoir volume change corresponds to that of a spherical reservoir in an infinite medium (the reservoir depth/radius ratio is >5), we can compute this radius from [

59]:

where

$E$and

$\upsilon $ are Young’s modulus and Poisson’s ratio, respectively, and

$\Delta {V}_{r}=\pi {u}_{z}{H}^{2}/\left(1-\upsilon \right)$. Because of the 1/3 power, the estimated volume has little sensitivity to the overpressure or the volume change. We find that the reservoir radius should be smaller than

R = 250 m in order for failure to occur. Since InSAR measurements are not continuous, this reservoir radius is likely larger. For instance, if pre-eruptive inflation corresponds to the co-eruptive deflation, then the reservoir should be smaller than

R = 315 m. These values are of the same order as the assumed reservoir radius, suggesting that our model is plausible. As surface deformation measurements are not sensitive to the reservoir radius for depth/radius ratios > 5, a change in the reservoir radius will not affect the depth and the reservoir volume changes determined from the inversions.

No reactivation of the ring fault around the summit caldera was observed. This occurs when magma reservoirs undergo under-pressure. Here, the depth of the eruptive fissure is at a higher elevation than the reservoir, and at the end of the eruption, degassed lava was emitted, implying that the reservoir is probably at lithostatic equilibrium at the end of the eruption.

#### 5.3. Magma Budget

The InSAR data covering the Nyamulagira 2010 eruption require a deflating source beneath the caldera, which we modeled as a spherical reservoir at ~3.5 km beneath the summit (

Figure 3) experiencing a volume decrease of ~5.1 × 10

^{6} m

^{3}. The intruded magma volume in the two dikes is ~4.9 × 10

^{6} m

^{3}, similar to the reservoir volume decrease. The estimated extruded lava flow and tephra volume is 45.5 ± 7.6 × 10

^{6} m

^{3} [

10] and 10 × 10

^{6} m

^{3} [

8], respectively, leading to an average dense rock equivalent volume of ~42.3 × 10

^{6} m

^{3} (assuming a lava, tephra and dense rock density of 2200, 1000 and 2600 kg/m

^{3} [

8], respectively). The total emitted volume is thus ~47.2 × 10

^{6} m

^{3} (= the sum of dike volumes + dense rock equivalent erupted volume), leading to a ratio (

r_{v}) between the total emitted volume and the reservoir volume decrease of ~9.3. This ratio is an intermediate value between commonly-reported values in the EARS, for instance of ~20–40 [

45] for the Dabbahu intrusion and of ~3 for the Dallol intrusion [

60]. Refill of the shallow reservoir from a deep reservoir is not necessarily required to explain this discrepancy, as the pressure in the reservoir could have been restored by gas exsolution in the magma chamber [

50].

We follow an approach similar to the one of Rivalta and Segall [

50] to express

r_{v}. We assume that mass is conserved between magma emitted from the reservoir and magma transmitted to the edifice and that magma is compressible. However, we additionally account for the lava erupted at the surface (see

Appendix A) and assume that pressure in the magma chamber before the eruption,

${P}_{r}$, corresponds to the maximum sustainable overpressure defined above,

${P}_{r}={P}_{l}+\Delta {P}_{r}=3{P}_{l}+{T}_{o}$, and that, after the eruption, pressure in the magma chamber returns to a lithostatic state. Thus, we obtain the following:

where

${\beta}_{m}$ and

${\beta}_{c}$ are the magma and reservoir compressibility,

${V}_{e}$ is the dense rock equivalent volume of emitted products,

$\Delta {V}_{r}$ is the reservoir volume change,

${\rho}_{e}$ is the dense rock equivalent density of emitted products (lava flows + tephra) and

${\rho}_{m}\left({P}_{l}\right)$ and

${\rho}_{m}\left({P}_{r}\right)$ are the densities of magma in the reservoir after and before the eruption. Reservoir compressibility can be computed by assuming a spherical reservoir and neglecting the influence of the ground surface [

50], leading to

${\beta}_{c}=3\left(1+\upsilon \right)/\left(2E\right)$.

Compressibility of magma,

${\beta}_{m}$, is derived from

${\beta}_{m}=1/{\rho}_{m}d{\rho}_{m}/dp$, assuming

${\beta}_{m}$ remains constant over the pressure interval, so that:

The variation of magma density with pressure depends on the amount of exsolved gas, given by:

where

${\chi}_{g}\left(P\right)$ is the mass fraction of gas in the magma and

${\rho}_{g}\left(P\right)$ and

${\rho}_{l}\left(P\right)$ are the densities of gas and liquid, respectively, all of which vary with pressure.

${\chi}_{g}\left(P\right)$ can be determined from the analytic expression derived from Henry’s law [

61] or derived from more complex models [

62,

63,

64]. The gas density is given by the ideal gas law

${\rho}_{g}\left(P\right)=PM/RT$, where

M is the gas molar mass,

R is the gas constant and

T is the absolute temperature, while the liquid density takes the liquid compressibility,

${\beta}_{l}=1-2\times {10}^{-10}P{a}^{-1}$, into account:

${\rho}_{l}={\rho}_{o}\left(1+{\beta}_{l}P\right)$, where

${\rho}_{o}=2500\text{kg}/{\text{m}}^{3}$ is the density of the liquid phase of magma at atmospheric pressure.

Here, the volatiles considered are CO

_{2} and H

_{2}O, and the magma compositions are estimated from recent studies of the volcano [

10,

65] (

Table 3). Using the online code for thermodynamics computation of Papale

et al. [

64], we find that for a 1200 °C magma at a 3.5-km depth, almost all CO

_{2} is gaseous and all H

_{2}O is liquid, so that

${\chi}_{C{O}_{2}}\left(P\right)=0.1\text{wt\%}$ and

${\chi}_{{H}_{2}O}\left(P\right)=0\text{wt\%}$. At the pressure corresponding to rupture of the reservoir,

${P}_{r}$, we find

${\chi}_{C{O}_{2}}\left({P}_{r}\right)=0.05\text{wt\%}$. Successively using Equations (5) and (6), we get a magma compressibility of

${\beta}_{m}$ = 1.9-3.2 × 10

^{9} Pa

^{−1}. If we consider a reservoir compressibility for a source in an infinite medium [

50] and take

${\rho}_{e}=$ 2600 kg/m

^{3} for the dense rock equivalent density and

${V}_{e}$ = 47.2 × 10

^{6} m

^{3}, using Equation (4), we obtain

r_{v} = 6.5–10.1, a value consistent with that determined from modeled volume changes. The contribution of the emitted lava flow to

r_{v} is 0.5, thus it accounts for approximately only 5% of

r_{v}. This value of

${r}_{v}$ indicates that the magma emitted can be entirely issued from the 3.5-km depth shallow magma chamber and that discrepancy between the emitted and the reservoir volume change are compensated by CO

_{2} exsolution.

**Table 3.**
Thermodynamics parameters and major and minor element composition considered for the 2010 lavas of Nyamulagira used to compute the weight percent of the exsolved gases.

**Table 3.**
Thermodynamics parameters and major and minor element composition considered for the 2010 lavas of Nyamulagira used to compute the weight percent of the exsolved gases.
T (°C) | P (MPa) | SiO_{2} | TiO_{2} | Al_{2}O_{3} | Fe_{2}O_{3} | FeO | MnO | MgO | CaO | Na_{2}O | K_{2}O | H_{2}O (wt%) | CO_{2} (wt%) |
---|

1200 | 87.5 | 45 | 3.5 | 15 | 13.1 | 12 | 0.2 | 4.5 | 11,5 | 3 | 2.2 | 1.25 | 0.1 |

#### 5.4. Stress Analysis

To understand the possible origin of the 2011–2012 eruption, we compute static stress changes induced by the modeled 2010 eruption deformation sources. Static stress changes can be used as an indicator for dike unclamping (e.g., [

31,

66,

67,

68,

69,

70]). Indeed, each dike intrusion changes the normal stress on pathways for potential diking. The sign of the induced normal stress change indicates whether subsequent intrusions are favored or prevented. If the normal stress changes are negative, subsequent dike emplacement is favored (unclamping); otherwise, subsequent dike emplacement is prevented (clamping). Note that we do not intend to forecast the shallow feeder dike orientation for the 2011–2012 eruption, but are interested in understanding the initiation of dikes. To quantitatively estimate the stress changes induced by the 2010 Nyamulagira eruption, we compute normal stress changes on five potential planes induced by the best-fitting deformation sources. From the distribution of eruptive vents, fissures and pre-existing rift-related structures (

Figure 1), five potential vertical dike surfaces were considered (

Figure 4): an ~N155E direction corresponding to the more active rift zone, an ~N45E direction crossing the summit caldera and an ~N45E south caldera (SC) direction, as well as an ~EW direction and an ~N65E direction connecting to the 2011–2012 eruptive fissure corresponding to a radial dike emplacement. Potential dike surfaces investigated range from the ground surface down to three kilometers below sea level (b.s.l.).

The 2010 eruption deformation sources induce unclamping in all potential dike surfaces (

Figure 6 and

Figure S3). Along the ~N45E surface crossing the caldera, the ~EW and ~N65E radial investigated surfaces, the unclamping is only significant in the shallowest part of the surfaces (<~2 km beneath the ground surface) and in the immediate proximity of the modeled reservoir. The ~N155E and the ~N45E SC surfaces are unclamped over large areas (66 and 111 km

^{2}, respectively) to greater depths. Stress changes from the 2010 intrusions and reservoir beneath the SSE flank could have thus encouraged the vertical ascent of magma from a deeper crustal reservoir along the ~N45E SC surface below the volcanic edifice. This interpretation is consistent with the hypothesis of a deep (~20–30-km depth) magma reservoir feeding the less common distal lower flanks’ eruptions [

19].

**Figure 6.**
Perspective plots from two orthogonal viewing angles showing the changes in the normal stress on the potential dike surfaces (

Figure 4) caused by the preferred model of the 2010 best-fitting deformation sources (black mesh). The color scale corresponds to normal stress change in MPa, with positive values (clamping) clipped to dark red and negative values (unclamping) from blue to red. SC stands for south of the caldera. Both the N155 and N45E SC dike surfaces are unclamped over large and deep areas beneath the volcano by the 2010 eruption sources, while only a small area of the N45E (caldera intersecting) surface is unclamped.

**Figure 6.**
Perspective plots from two orthogonal viewing angles showing the changes in the normal stress on the potential dike surfaces (

Figure 4) caused by the preferred model of the 2010 best-fitting deformation sources (black mesh). The color scale corresponds to normal stress change in MPa, with positive values (clamping) clipped to dark red and negative values (unclamping) from blue to red. SC stands for south of the caldera. Both the N155 and N45E SC dike surfaces are unclamped over large and deep areas beneath the volcano by the 2010 eruption sources, while only a small area of the N45E (caldera intersecting) surface is unclamped.

We also calculated the normal stress change induced by the 2010 Nyamulagira eruption on the ~NS potential dike surface at Nyiragongo, determined from the modeling of the 2002 Nyiragongo eruption [

31] (

Figure S7). The normal stress change is slightly negative close to Nyiragongo summit (the minimum value is −0.0015 MPa). Thus, dike intrusions in the SSE flank of Nyamulagira also slightly unclamp the northern part of the Nyiragongo dike surface. A single dike intrusion probably does not induce a normal stress change large enough to cause an eruption at Nyiragongo. However, repeated dike intrusions in Nyamulagira’s SSE flank may contribute to the failure of the shallow plumbing system of Nyiragongo.

#### 5.5. Influence of the Crustal Extension

The EARS extension probably plays a role in the preferred diking orientations and on the amount of intruded magma volumes. The tectonic rift extension, σt, in this part of the rift is oriented ~N110E [

11,

12]. Resolving this extension on the potential dike directions for host rocks under lithostatic stress,

${P}_{l}$, we get smaller amplitudes of host rock stresses on the N45E than on the N155E potential dike directions, with values of:

and:

respectively.

Next, we assume that failure of the reservoir corresponds to the same magma pressure (

P_{m}) whatever the intrusion orientation, which might be the case around a reservoir, as each intrusion changes the state of stress in the host medium. We find that the overpressure along the N45E surface

$\Delta {P}^{N45E}={P}_{m}-{P}_{l}+0.9{\sigma}_{t}$ is larger than the overpressure along the N155E surface

$\Delta {P}^{N155E}={P}_{m}-{P}_{l}+0.7{\sigma}_{t}$. If we consider that the length of the intrusion is a linear function of the overpressure, because the length is limited either by magma cooling [

69] or by the resistance to fracture at the dike tip [

70], dike intrusions are expected to be longer along the N45E direction than along the N155E direction.

#### 5.6. Magma Storage and Transport

From the detailed geodetic modeling of the 2010 Nyamulagira eruption and the results of [

10], we suggest a model of magma transfer and storage associated with a typical upper flank eruption on the SSE flank of Nyamulagira. The onset of the eruption may be marked by over-pressurized magma intruding upward in a dike from a shallow reservoir located at ~4 km beneath the SW caldera depression. Magma propagation is likely driven by the buoyancy of the gas pocket and gas-rich magma at the dike tip, as evidenced by the intense lava fountaining during the first stage of the eruption and the large

r_{v}. The magma overpressure keeps the dike open. Just after the eruption onset, the dike may grow laterally due to the decreased buoyancy corresponding to the increased density of the underlying, gas-poor magma, mainly in the direction of the slope, into a weak and fractured area beneath the SSE flank, as observed, for instance, at Piton de La Fournaise [

39] and Etna [

71]. As the overpressure in the reservoir is relaxed, the magma flow rate decreases; magma can no longer reach the summit caldera and erupts only at the lower flank vent. A few days after the eruption onset, the eruption is restricted to a small portion at the base of the fracture. Finally, after a few more days or weeks, the activity stops.

This study provides insights for the shallow magma plumbing system active during 2006–2012 (

Figure 7). In 2006, magma intruded from a depth into a shallow reservoir located at an ~1-km depth beneath the SW depression of the summit caldera [

20] as a sub-vertical dike beneath the volcano and probably grew laterally in the SSE flank of the volcano, where an eruptive fissure opened. This process of vertical ascent feeding lateral dikes is common for small volcanic edifices (e.g., [

39,

72,

73]). We also suggest that the 2010 sources unclamped the deep part of the potential dike surface oriented ~N45E, south of the caldera, in which magma could have intruded from a deep reservoir. Magma could have then migrated laterally following this ~N45E SC orientation, which may represent a plane of weakness due to pre-existing faults buried under the lava field [

13] toward the 2011–2012 eruption site location. Unusual seismic tremor data [

10] also suggest magma movement at a depth along a similar ~N45E orientation.

**Figure 7.**
Schematic shallow magma plumbing system for the period 2006–2012. Arrows represent possible magma migration paths. Red and blue sources experience inflation and deflation, respectively. Orange stars denote the locations of eruptive fissures and vents. SC stands for south of the caldera. The two dikes and magma reservoir inferred from the modeling of the 2010 eruption data are represented with black meshes.

**Figure 7.**
Schematic shallow magma plumbing system for the period 2006–2012. Arrows represent possible magma migration paths. Red and blue sources experience inflation and deflation, respectively. Orange stars denote the locations of eruptive fissures and vents. SC stands for south of the caldera. The two dikes and magma reservoir inferred from the modeling of the 2010 eruption data are represented with black meshes.