Elevated fluid pressure in compression facilitates flexural reverse faulting

Abstract

Elastic flexure of tectonic plates shapes large-scale topographic features and can induce notable slip on lithosphere-scale faults. At mid-ocean ridges, recent evidence shows that up to 50% of extensional fault offsets formed within axial valleys can be reversed by compressional slip across valley flanks. While standard models predict the location of this faulting, they consistently underpredict the depth-extent of compression and related surface offsets. Here, using numerical models, we show that elevated pore fluid pressures in compression allow increased slip on flexure-driven reverse faults. Quantitative agreement between modeled and observationally inferred reverse fault–bending strain is achieved only when pore-fluid pressures in reverse faults approach lithostatic levels. These findings highlight the critical role of inherited structural heterogeneities and high pore-fluid pressures in facilitating deep-seated reverse faulting during lithospheric unbending. This mechanism may also amplify flexure-induced seismicity in continental forebulges, such as the 2001 Bhuj earthquake (moment magnitude, 7.6), and underscores pore-pressure modulation as a fundamental control on global tectonic hazards.

INTRODUCTION

Lithospheric flexure governs how Earth’s surface responds to geological loads by generating bending stresses that peak at the upper and lower surfaces of flexed plates (1). Where these stresses exceed the yield strength, elastic bending gives way to inelastic deformation: At shallow levels, this manifests as brittle failure that nucleates or reactivates faults capable of producing substantial earthquakes, whereas in the lower part of the envelope, it activates ductile flow (1–3). Flexure-related earthquakes are well known at sites of plate convergence (4) but have only recently been detected in regions of plate spreading (5–8).

Plate convergence produces downward bending that puts the top of a down-going plate in extension. Normal faults produced by such extension at the outer rise of subduction zones have long been recognized because of the ∼50- to 100-m offsets clearly seen in bathymetric and seismic reflection data (9–10). In places where preexisting faults are close to parallel to the axis of bending those faults are seen to be reactivated (9). Such outer-rise normal faulting can trigger substantial tsunamigenic events, as exemplified by the 2009 Samoa-Tonga sequence. This sequence comprised a moment magnitude (M_W) 8.1 normal-faulting mainshock followed by two M_W 7.8 thrust earthquakes and produced run-ups of up to 12 m that resulted in 192 fatalities (3). Bending-related faulting related to plate convergence also occurs in continental areas where they likewise pose notable hazards. For instance, the stress field associated with the ∼670 km–wavelength India-Tibet forebulge likely facilitated the reactivation of a rift-related normal fault, resulting in the devastating 2001 Bhuj reverse-faulting earthquake (M_W 7.6), which claimed more than 20,000 lives (11, 12).

While the structural and stress complexity of continental lithosphere makes interpreting such flexural faulting difficult, the comparatively simpler structure and history of oceanic lithosphere near mid-ocean ridges (MORs) provides an ideal natural laboratory to study the development of bending-related faults and earthquakes. The amplitude of normal faults formed at fast spreading ridges are consistent with the strains produced by bending as lithosphere flattens out adjacent to axial highs (13–15). In the same way that the “unbending” of lithosphere moving away from an axial high should produce surface extension, unbending of lithosphere moving away from an axial valley should produce surface compression. Researchers in the early years after the acceptance of the plate tectonic theory suggested that surface-cutting reverse (i.e., compressional) faults should produce earthquakes adjacent to axial valleys (16, 17). No such reverse-faulting earthquakes were found (18) until 2022 during an earthquake sequence on a segment of the Mid-Atlantic Ridge at ∼54°N (6–8). The sequence began on 26 September 2022 with normal-faulting earthquakes, which migrated ∼45 km southward along the ridge axis within 3 days. On 29 September 2022, an M_W 5.1 reverse-faulting earthquake occurred ∼15 km east of the axis, followed by 11 additional reverse-faulting events, the last detected on 4 January 2023. Notably, these reverse-faulting earthquakes nucleated symmetrically in two narrow zones located ∼15 km from the ridge axis on both sides (Fig. 1).

Fig. 1. Reverse faulting at the 54°N segment of the Northern Mid-Atlantic Ridge.

(A) and (B) show fault scarp identification and tectonic strain analysis from across-axis bathymetry profiles. Fault throw data are from Olive et al. (7); the locations of the across-axis profiles are shown in (C). (C) displays the spatial distribution of the earthquake cluster that occurred at 54°N segment of the Northern Mid-Atlantic Ridge in 2022.

The reverse-faulting earthquakes occurred at the same distance from the spreading axis as a marked transition in seafloor deformation patterns (7). Quantitative bathymetric analysis shows that the slope of cumulative fault horizontal offset versus distance from the ridge axis (Fig. 1B) exhibits a pronounced decrease at approximately ∼15 km from the axis, coinciding with the onset of reverse faulting (Fig. 1C). Approximating this slope as the apparent tectonic strain T (19), we observe a decrease from T = 0.13 in the near-axis segment to T′ = 0.07 in the far-field region (Fig. 1B). Considering a fault spacing of ∼3 km, this reduction in extensional strain represents a decrease in fault horizontal offset of approximately 200 m.

The discovery of these flexural reverse-faulting events prompted a comprehensive examination of global datasets on near-axis seismicity and bathymetry to evaluate whether such compressional phenomena are widespread. Through precise earthquake relocations, Jackson and McKenzie (6) identified 72 reverse-faulting events close to mid-ocean ridges since the advent of digital seismic recording in 1977. Concurrently, Olive et al. (7) analyzed bathymetric profiles across diverse ridge segments with axial valleys, revealing analogous patterns of off-axis compressional deformation. Collectively, these findings indicate that approximately half of the ocean floor, equivalent to more than one-third of Earth’s solid surface, is marked by fault-bounded abyssal hills that have experienced notable offset reduction due to bending-induced compression.

Previous models suggest that lithospheric unbending may be the mechanism responsible for generating symmetric compressional deformation on the rift valley shoulders of mid-ocean ridges. As newly formed lithosphere moves away from the rift center, it transitions from downward bending without flexural stress to flattening, generating substantial compressional stresses in the upper 2 km of the plate (20–22). However, these models predict a compressional stress zone that is too shallow by a factor of 2 or more to explain the magnitude of seafloor-bending strain inferred from bathymetric data.

RESULTS AND DISCUSSION

Pore pressure weakening enables deep reverse faulting

We attribute this mismatch to an assumption of equal pore-fluid pressures in compressional and extensional domains. Most standard models of lithospheric deformation (23, 24), including our own previous simulations of plate-spreading centers (20, 21), assume that compressional (reverse) faults and extensional (normal) faults experience the same pore pressures.

Geologic observations strongly support the possibility of different pore pressures in zones of compression and extension. The structure of compressional mountain belts and accretionary prisms have long been taken to require pore pressures that approach the pressure of the overlying rock (i.e., the lithostatic pressure) as noted in the classic studies of Hubbert and Rubey (25) and Davis et al. (26) and many subsequent studies. The argument is that the shear stress needed for fault slip is proportional to the normal stress across the fault minus the pore pressure. Pore pressure essentially pushes the fault faces apart, allowing faults to slip at lower shear stresses. Also, increasing compression should reduce pore space, possibly leading to an increase in pore pressure. Measurements in boreholes show very high pressures in convergent settings (27). By contrast, normal faults typically maintain hydrostatic pore pressures because overpressure exceeding the minimum principal stress would induce fracturing and pressure release (28).

A simple analytic model, detailed in Materials and Methods, illustrates the possible effects of pore pressure variations on lithosphere flexure. Unbending an initially concave-down layer (Fig. 2, A and C) compresses the top and extends the bottom, while not changing the average stress in the layer [e.g., Bodine et al., 1981 (29)]. The surface bending strain depends linearly on the neutral depth where the bending stresses are zero (Eq. 2). Thus, we focus on effect of pore pressures on the neutral depth Z_n for a given lithospheric thickness, elastic and plastic parameters. Figure 2 (B and D) shows depth profiles of bending stress, brittle yield stress, and the resulting neutral depth for an 8-km-thick lithosphere with compressional (reverse-fault) pore-pressure ratio ${\mathrm{\lambda }}_C$ = 0.33 (near hydrostatic) and 0.99 (near lithostatic). Under near-hydrostatic conditions, Z_n ≈ 2.4 km, as ${\mathrm{\lambda }}_C$ approaches lithostatic values, Z_n deepens to ≈4.0 km.

Fig. 2. Analytical models demonstrate that pore-pressure weakening enables deep reverse faulting.

(A) to (D) illustrate cases in which the compression zone features suprahydrostatic and hydrostatic pressures, respectively. (A) and (C) are schematic diagrams depicting a newly formed, unflexed, concave-down lithosphere that generates compressional stresses in the shallow lithosphere and extensional stresses in the deep lithosphere during flattening of the axial valley. (B) and (D) show the relationships among bending stress, compressional and extensional brittle yield stresses, and depth during unbending.

Modeled unbending strain at MOR flanks

While the analytical model above indicates that elevated pore-fluid pressures can deepen the bending neutral surface, it fails to quantify the magnitude of bending-induced reverse-faulting strain that develops during the formation of an axial valley. To address this limitation, we use two-dimensional (2D), elasto-viscoplastic, cross-axis models of a slow-spreading mid-ocean ridge. Our approach substantially builds on the methodology of previous studies (20, 21, 30, 31), incorporating the effects of magmatic emplacement over long timescales. Here, we assume in all our models that the ratio of magmatically accommodated extension to total plate separation, i.e., the M factor, is 0.85. This choice is motivated by the fact that M typically ranges between 0.7 and 0.9 along magmatically robust segments of slow-spreading mid-ocean ridges (19, 30, 32). Previous bathymetric analyses of the Mid-Atlantic Ridge at 54°N have yielded an M value of 0.865 (7), which directly supports our chosen value. The key refinement is that the yield stress calculation is now explicitly a function of pore-fluid pressure. This pressure is, in turn, controlled by the accumulated plastic strain and the ambient stress state (Materials and Methods). This modification allows inherited abyssal hill faults to self-consistently develop varying pore pressures as they enter the compressional stress zone. Consequently, we are able to quantitatively analyze the relationship between pore-fluid pressure in compression and the resulting flexural reverse-faulting strain.

Figure 3 presents two end-member numerical models defined by their pore-pressure assumptions. Figure 3 (A to C) assumes a uniform, near-hydrostatic pore-pressure ratio in both the compressional and extensional domains (${\mathrm{\lambda }}_C$ = ${\mathrm{\lambda }}_E$ = 0.33). Figure 3 (D to F), in contrast, tests the effect of differential pore pressure, maintaining near-hydrostatic conditions in extension (${\mathrm{\lambda }}_E$ =0.33) while setting the compressional domain to near-lithostatic pressure (${\mathrm{\lambda }}_C$ = 0.99). For simplicity, we fix the thermal structure such that the 350°C isotherm shoals from a depth of 8 km in the off-axis region (H₀) to 4 km at the ridge axis over a distance of 5 km from the axis. This simplification enables direct comparison with analytical plate-flexure solutions, which assume uniform plate thickness, and reflects the fact that the off-axis (∼5 to 60 km) brittle lithospheric structure of slow-spreading ridges remains poorly constrained. Resolving it in detail would require additional assumptions about hydrothermal circulation and its coupling to lithospheric deformation (21, 33), introducing complexity unwarranted by the available data. Despite this simplification, the adopted thermal structure, combined with continuous axial magma emplacement, produces an axial valley with relief (h) of ∼2 km and abyssal hill–bounding faults spaced ∼3 km apart (Fig. 3, A and D), consistent with observations at the 54°N segment of the Mid-Atlantic Ridge and indicating that the model captures the first-order mechanics of the system.

Fig. 3. Numerical modeling results show reverse-faulting strain at mid-ocean ridge flanks.

(A) to (C) depict a case in which all regions maintain hydrostatic pressures, whereas (D) and (E) illustrate a case in which suprahydrostatic pressures approaching lithostatic levels can form as inherited faults enter the compression zone and accumulate a certain amount of shortening strain.

Assuming that the brittle layer thickens gradually away from the axis would modify these results in two ways. First, the axial valley would be shallower: A smaller off-axis thickness gradient places the strong flanking lithosphere farther from the axis, reducing its lateral mechanical support and allowing yielding to occur under a smaller isostatic restoring force and, thus, at a shallower valley depth (34). Second, abyssal hill–bounding faults would be more widely spaced: A smaller off-axis thickness gradient causes the stress required to maintain slip on an active fault to increase more slowly with distance from the axis so that individual faults remain active longer before being abandoned in favor of a newly formed axial fault, resulting in larger fault spacing (30, 35).

We calculate the cumulative horizontal shortening strain (${{\mathrm{\epsilon }}_{\mathit{xx}}}^C$) by time-integrating the horizontal strain rate exclusively in regions experiencing horizontal deviatoric compression (Fig. 3, C and F). This shortening strain develops on the rift shoulders as newly formed lithosphere unbends and flattens during off-axis migration (fig. S1 and movie S1). When reverse faults are subjected to moderate pore pressures (Fig. 3C), the shortening strain is diffused, manifesting as low-amplitude bands confined to shallow depths. Conversely, when reverse faults are subjected to superhydrostatic pore pressures (Fig. 3F), the high-pressure fluids facilitate fault weakening, thereby localizing the shortening strain into narrow, high-amplitude bands that dip toward the axis and extend to depths of ∼4 km. The depth extent of the modeled shortening strain is consistent with the hypocenter depths for the 2022 Mid-Atlantic Ridge earthquake swarm, which were constrained to range between 2.7 and 4.2 km below the ocean surface (7).

Elevated pore pressure reconciles modeled and observed unbending strain

To enable a direct quantitative comparison of our results with the shortening strain derived from bathymetric analysis, we calculated the cumulative horizontal shortening at the seafloor as a function of off-axis distance (Fig. 3, B and E). Ideally, we would have used the model’s free surface topography to do this analysis, but this is subject to artificial diffusion that is endemic to most numerical methods. Thus, the cumulative shortening is derived by integrating the surface shortening strain (${{\mathrm{\epsilon }}_{\mathit{xx}}}^C$) across the model grid elements. The slope of the cumulative shortening versus distance curve serves as a robust proxy for the reverse-faulting strain (Fig. 3, B and E) that is not affected by numerical diffusion.

In our simulations, the onset of seafloor shortening typically occurs ∼20 km off-axis on both sides of the ridge. An increasing rate of cumulative seafloor shortening with increasing distance betrays active reverse fault slip between ∼20 and ∼30 km off-axis. Beyond ∼30 km, cumulative seafloor shortening scales linearly with distance, indicating that no additional shortening takes place (steady state). By calculating the slope in the steady-state range (30 to 60 km off-axis), we obtain a reverse-faulting strain of 0.036 for the low-pressure case (where ${\mathrm{\lambda }}_C$ = ${\mathrm{\lambda }}_E$ = 0.33; Fig. 3B). This is compared to a strain of 0.073 for the overpressurized case (where ${\mathrm{\lambda }}_C$ = 0.99; Fig. 3E). This doubling of flexural strain is consistent with our analytical solution (Materials and Methods), which predicts that flexural strain scales proportionally with the depth of the neutral surface (Zn). As Zn deepens from 2 km (Fig. 3C) to 4 km (Fig. 3F), the resulting strain increases accordingly. To validate the robustness of these results, we performed additional sensitivity tests, which included models with intermediate pore-fluid pressures (between hydrostatic and lithostatic values; fig. S2). Furthermore, two additional series of simulations were conducted with varying axial valley relief (h) generated by modifying the thermal structure. For each series, we analyzed the influence of compressional pore-fluid pressure on the resulting reverse-faulting strain (figs. S3 to S8 and table S2).

To evaluate these model predictions against global data, we refined the bathymetric analysis dataset from Olive et al. (7) by excluding ultraslow-spreading ridges and asymmetric segments (table S3 and Fig. 4), where lithospheric flexure is likely influenced by large-offset detachment faulting (22), a process that is not accounted for by our simulations of magmatically robust symmetric Mid-Ocean Ridge sections (36). The comparison between modeled and observed reverse-faulting strains yields two key insights (Fig. 4). First, both the simulated strain and the strain derived from bathymetric analysis exhibit a consistent positive correlation with axial relief (h). This trend is expected, as axial relief is a primary control on the curvature of lithospheric flexure (Materials and Methods) (13). Second, and crucially, the simulated reverse-faulting strains only approach the magnitude of the observed bathymetric strain when pore pressures within the reverse faults are near-lithostatic (Fig. 4).

Fig. 4. Comparison of modeled and observed unbending strains suggests elevated pore pressures within the compression zone.

Both the modeling results and observations demonstrate that the unbending strain increases proportionally with axial relief. Furthermore, the simulated unbending strains align with the observations only when pore pressures in the compression zone approach lithostatic levels. In this plot, the circular symbols with error bars represent the observational data derived from seafloor bathymetry (details are provided in table S3), and the various colored squares denote the results of the numerical simulations where the color gradient signifies the different pore-fluid pressures assumed within the compression zone. The differently colored dashed lines illustrate the results obtained from the analytical solution, with the color representing variations in pore-fluid pressure.

Notably, our simulated flexural strains fall at the lower edge of the range observed at mid-ocean ridge segments worldwide with comparable seismic events, even when ${\mathrm{\lambda }}_C$ = 0.99. Two factors likely explain this underestimation. First, the rifts in our simulations (X_c^M; table S2) are wider than those inferred from observations (X_c; table S3). Because surface flexural strain scales inversely with the square of rift width (Eqs. 1 and 2), this geometric mismatch alone can account for the offset. Rift width is governed by lithospheric thickness (thermal structure) and rheology (34), so modest changes to these parameters bring the predictions closer to observations. For example, setting H₀ = 8 km and halving Young’s modulus produces a narrower rift than the reference model in Fig. 3, yielding a flexural strain of ∼0.07 at a rift depth of ∼1.5 km (fig. S9) and shifting the results toward the upper left of Fig. 4. We note, however, that rift morphology across mid-ocean ridges reflects controls too complex to reproduce fully by varying H₀ alone within a fixed thermal structure. Second, our models assume steady deformation at slow strain rates (<10⁻¹³ S⁻¹), whereas reverse-faulting earthquakes are high strain-rate transients in nature, potentially facilitated by rapid loading from axial dike intrusion (7). Given the viscoelastic rheology of the lithosphere, its effective thickness under such transient high strain-rate conditions may exceed the time-averaged value resolved by our steady-state models, permitting a deeper neutral plane and consequently larger compressive strains at the seafloor than our simulations predict.

Summary and global implication

Integrated analytical and numerical modeling, constrained by seismic and bathymetric observations, indicate that reverse faulting along the flanks of mid-ocean ridge axial valleys requires near-lithostatic pore fluid pressures. Such overpressures substantially weaken compressional faults, focusing flexural unbending strain into discrete shear bands that reach the neutral surface, reconciling observed reverse-faulting seismicity and reduced abyssal hill amplitude with plate-bending mechanics. This localization hinges on inherited structural weaknesses—preexisting normal faults and fractures in which fluids that were well drained at the axis become trapped and overpressurize as they move off-axis. Where such fabrics are pervasive, overpressures persist and reverse fault localizes on preexisting normal faults during flexure; where they are scarce, as in the footwalls of oceanic core complexes at slow-spreading ridges, unbending strain is accommodated diffusely without persistent shear bands (22).

The same mechanical framework developed here to treat flexural faulting at mid-ocean ridges may prove useful in studies of devastating intraplate earthquakes, such as the 2001 Bhuj (M_w 7.6) and 1993 Latur (M_w 6.3) events, both of which occurred within the compressional belt south of the Indian Plate’s flexural bulge imposed by Tibetan Plateau loading. Specifically, the Bhuj mainshock nucleated at 20- to 30-km depth within the Mesozoic Kutch Rift Basin, where fault reactivation and weakening were likely amplified by trapped high-pressure fluids. This interpretation is consistent with aftershock tomography, which images seismic-velocity anomalies (37, 38). Collectively, these observations suggest that elevated compressional pore-fluid pressures are a key factor in facilitating flexure-induced reverse-faulting earthquakes.

MATERIALS AND METHODS

Analytic model

To quantify how different pore-fluid pressures in compressional and extensional parts of a bending plate affect the depth of the neutral surface ($z_n$), we use thin-plate flexure approximations (1). Thin-plate theory describes the bending of a layer that is thin relative to the horizontal scale of the deformation.

We first need to determine the curvature of the plate generated during the axial valley formation. An approximate scaling (13) yields the initial curvature of the plate at the ridge axis as

$$k\approx 2h_0/w^2$$ (1)

where $h_0$ is the depth of the axial valley relative to the flanks and $w$ is the half-width of the valley. We use $h_0$ to denote the axial valley depth in this analytical context to distinguish it from h, which represents the axial relief in our numerical models (Fig. 3). The difference between $h_0$ and h primarily stems from the short-wavelength topographic amplitude generated by individual fault offsets. If the plate is flattened out, the change in curvature is approximately $∆k\approx -k$. Thin-plate theory predicts that the horizontal strain, ${\mathrm{\epsilon }}_{\mathit{xx}}$, in a layer experiencing a change in curvature, $∆k$, varies with vertical distance from the neutral surface ($z_n$) as

$${\mathrm{\epsilon }}_{\mathit{xx}}=(z-z_n)∆k$$ (2)

where $z$ is the depth below the surface, measured positively downward from the top.

We first consider the case of a plate of thickness $H_0$ with uniform elastic properties and demonstrate that the bending stresses are sufficiently large to cause yielding by faulting in parts of the plate. The horizontal stress produced by bending is

$${\mathrm{\sigma }}_{\text{Bend}}=\frac{E}{(1-{\mathrm{\nu }}^2)} {\mathrm{\epsilon }}_{\mathit{xx}}=-\frac{E}{(1-{\mathrm{\nu }}^2)}(z-z_N) \frac{2h_0}{w^2}$$ (3)

where $E$ is Young’s modulus and $\mathrm{\nu }$ is Poisson’s ratio for the lithosphere (1). Here, we adopt the geological convention where compressive stresses are designated as positive. For the unbending of axial-valley lithosphere of thickness $H_0$, the maximum compressional stress occurs at the top of the lithosphere. Using a lithospheric thickness $H_0$ = 5 km, valley depth $h_0$ = 1 km, $w$ = 15 km, and reasonable elastic moduli for rock (E = 7.5 × 10¹⁰ Pa and $\mathrm{\nu }$ = 0.25) yields a maximum compressive stress of 10¹⁰ Pa. This value exceeds the brittle yield stress for faulting at a few kilometers depth by more than two orders of magnitude. Under such elastic bending stresses, the topmost part of the plate would reach the yield stress for reverse faulting, while the lowermost part would reach the yield stress for extensional faulting. Thus, calculating the neutral depth requires defining these yield stresses.

We follow numerous studies in defining the brittle yield stress as the differential stress required to initiate slip on an optimally oriented fault within an Andersonian stress field (23). This approach neglects the ductile portion of the yield strength envelope—assumed to be greatly reduced by strong thermal gradients—and thus implies a sharp transition from a brittle lithosphere to a near-inviscid asthenosphere. The optimal fault orientation minimizes the differential stress for slip on a fault characterized by a friction coefficient μ and cohesion $C_0$. A detailed derivation of the stresses required for slip (or yielding) on such optimally oriented faults is provided in Turcotte and Schubert (39). For compressional (reverse) faulting, the maximum principal stress is horizontal, whereas for extensional (normal) faulting, it is vertical. Pore pressure is commonly expressed as

$$P_W=\mathrm{\lambda }\mathrm{\rho }gz$$ (4)

where λ is the ratio of pore pressure to vertical stress. We denote this ratio as ${\mathrm{\lambda }}_E$ in an extensional setting and as ${\mathrm{\lambda }}_C$ in a compressional setting. ρ is the lithospheric density, and $g$ is gravitational acceleration. Consequently, the brittle yield stress (differential stress) for normal faulting, ${\mathrm{\sigma }}_{\text{Yield}}^E$, is

$${\mathrm{\sigma }}_{\text{Yield}}^E=-\frac{2{\mathrm{\mu }}_E(1-{\mathrm{\lambda }}_E)\mathrm{\rho }\mathit{gz}+C_0}{\mathit{\text{sin}}2{\mathrm{\theta }}_E+\mathrm{\mu }(1+\mathit{\text{cos}}2{\mathrm{\theta }}_E)}$$ (5)

and for reverse faulting, ${\mathrm{\sigma }}_{\text{Yield}}^C$, is

$${\mathrm{\sigma }}_{\text{Yield}}^C=\frac{2{\mathrm{\mu }}_C(1-{\mathrm{\lambda }}_C)\mathrm{\rho }\mathit{gz}+C_0}{\mathit{\text{sin}}2{\mathrm{\theta }}_C-\mathrm{\mu }(1+\mathit{\text{cos}}2{\mathrm{\theta }}_C)}$$ (6)

where ${\mathrm{\theta }}_E$ and ${\mathrm{\theta }}_C$ represent the dip angles of normal faults and reverse faults, respectively. Given the occurrence of normal faulting within axial valleys, we infer that the average horizontal tectonic stress (${\mathrm{\sigma }}_{\mathit{Tec}}$) is extensional such that ${\mathrm{\sigma }}_{\mathit{Tec}}<0$. Thus, by considering the plate unbending and flattening process as it migrates away from the axial valley, the stress-depth relationship for the compressional zone above the neutral surface can be expressed as

$${\mathrm{\sigma }}_H^C=\text{min}({\mathrm{\sigma }}_{\text{Bend}},{\mathrm{\sigma }}_{\text{Yield}}^C)$$ (7)

Conversely, for the extensional zone below the neutral surface, the stress-depth relationship can be expressed as

$${\mathrm{\sigma }}_H^E=\text{max}({\mathrm{\sigma }}_{\text{Bend}},{\mathrm{\sigma }}_{\text{Yield}}^E)$$ (8)

The neutral depth is defined as the depth separating the extensional and compressional portions of the lithosphere such that the integral of the bending stresses over each region perfectly compensate

$${\displaystyle \underset{0}{\overset{Z_n}{\int }}}({\mathrm{\sigma }}_H^C-{\mathrm{\sigma }}_{\mathit{Tec}})\mathit{dz}+{\displaystyle \underset{Z_n}{\overset{H_0}{\int }}}({\mathrm{\sigma }}_H^E-{\mathrm{\sigma }}_{\mathit{Tec}})\mathit{dz}=0$$ (9)

By simultaneously solving Eqs. 3 to 8 through 9, we can determine the depth of the neutral surface (Z_n) for various pore-fluid pressures, given the rift geometry and the elastic and plastic properties of the plate (Fig. 2). Subsequently, substituting Z_n back into Eq. 2 allows us to calculate the magnitude of the flexural strain at the surface (Fig. 4).

Numerical model

We use GeoFlac for simulating the development of reverse faults in 2D oceanic spreading and lithospheric unbending scenarios. This method extends from various models of spreading centers (21, 30, 31), which used GeoFLAC (Fast Lagrangian Analysis of Continua), an explicit technique combining finite-element and finite-difference methods, to address the conservation equations for mass, momentum, and energy within a viscoelastic-plastic medium (40).

Brittle-plastic deformation is modeled using the Mohr-Coulomb failure criterion. We define the shear yield function ($f_s$) in terms of principal stresses as

$$f_s ({\mathrm{\sigma }}_1,{\mathrm{\sigma }}_3)={\mathrm{\sigma }}_1-{\mathrm{\sigma }}_3\frac{1+\text{sin}\mathrm{\phi }}{1-\text{sin}\mathrm{\phi }}-2P_w\frac{\text{sin}\mathrm{\phi }}{1-\text{sin}\mathrm{\phi }}+2C_0\frac{\text{cos}\mathrm{\phi }}{1-\text{sin}\mathrm{\phi }}$$ (10)

When $f_s$  = 0, the material has reached the yield limit and undergoes brittle-plastic deformation; when $f_s$  < 0, it exhibits viscoelastic deformation. σ₁ and σ₃ denote the maximum and minimum principal stresses, respectively. φ is the internal friction angle. $P_{\mathrm{W}}$ is the pore fluid pressure, which, as per Eq. 4, is scaled by lithostatic pressure. The ratio is denoted as ${\mathrm{\lambda }}_E$ in an extensional regime and ${\mathrm{\lambda }}_C$ in a compressional regime. As in our analytic model, we adopt the geological convention where compressive stress is positive and tensile stress is negative. However, the underlying GeoFLAC code uses the engineering convention (tensile stress is positive, and compressive stress is negative).

In all models, we set the fluid pore pressure coefficient under horizontal extensional stress states to hydrostatic levels, i.e., ${\mathrm{\lambda }}_E$ = 0.33. The magnitude of ${\mathrm{\lambda }}_C$ is simultaneously controlled by the stress state and accumulated plastic strain. This formulation enables abyssal hill faults, initially formed in the rift valley, to develop elevated pore pressure when they enter the compressional stress regime at the rift shoulder. ${\mathrm{\lambda }}_C$ is expressed as

$${\mathrm{\lambda }}_C=\left\{0.33+\begin{array}{cc}0.33& {\mathrm{\epsilon }}_p<{\mathrm{\epsilon }}_p^1 \text{or} {\mathrm{\sigma }}_{\mathit{xx}}<0\\ \frac{{\mathrm{\epsilon }}_p-{\mathrm{\epsilon }}_p^1}{{\mathrm{\epsilon }}_p^2-{\mathrm{\epsilon }}_p^1}({{\mathrm{\lambda }}_C}^0-0.33)& {\mathrm{\epsilon }}_p^1<{\mathrm{\epsilon }}_p<{\mathrm{\epsilon }}_p^2 \text{and} {\mathrm{\sigma }}_{\mathit{xx}}>0\\ {{\mathrm{\lambda }}_C}^0& {\mathrm{\epsilon }}_p>{\mathrm{\epsilon }}_p^2 \text{and} {\mathrm{\sigma }}_{\mathit{xx}}>0\end{array}\right.$$ (11)

where ${\mathrm{\epsilon }}_p$ represents the accumulated plastic strain at a given grid point in the model, ${\mathrm{\epsilon }}_p^2$ and ${\mathrm{\epsilon }}_p^1$ are two characteristic thresholds for cumulative strain, and ${\mathrm{\sigma }}_{\mathit{xx}}$ is horizontal deviatoric stress. The corresponding parameters are listed in table S1.

Model setup

Our analysis examines deformation within a 2D model domain spanning 150 km in width and 15 km in depth (Fig. 3 and fig. S1). The upper boundary is a free surface, while the lower boundary uses a Winkler foundation to replicate isostatic buoyancy on an inviscid substrate with mantle density. The lateral boundaries are driven outward at a constant total velocity (2 U = 25 mm/year). Plate separation is achieved by incorporating crustal material through diking at the model center. This process facilitates plate separation along the spreading axis at a consistent average opening rate, quantified by the M parameter [the proportion of plate spreading accounted for by diking, as outlined in Buck et al. (31)], resulting in a dike opening rate defined as 2UM. We adopt a constant value of M = 0.85 for all models in our calculations. Periodic regridding of the deformed Lagrangian mesh restores the base and sides of the model domain to their initial positions. During the remeshing process, mantle material is added to or removed from the base as required. The model domain is discretized using a Lagrangian mesh with a uniform resolution of 200 m. Remeshing is triggered when any corner angle within the triangular elements formed by the grid diagonals and boundaries falls below 15°.

For simplicity, the thermal structure was held constant in the reference model (Fig. 3). The seafloor temperature was set to 20°C. The 350°C isotherm is positioned at a depth of 4 km at the ridge axis and deepens to H₀ (8 km in the reference model; Fig. 3) at a distance of 5 km from the axis. Values of H₀ used in sensitivity tests can be found in table S2. Temperature increases linearly with depth from the seafloor to the 350°C isotherm. It continues to rise linearly between the 350° and 1300°C isotherms but with a vertical temperature gradient 10 times steeper. Temperature remains constant at 1300°C throughout the deeper portions of the model.

The viscous rheology follows a diabase flow law (41). The base of the brittle lithosphere approximately corresponds to the 350°C isotherm. Tectonic stresses can build-up elastically in the region above this isotherm, whereas below it, viscous flow rapidly relaxes any deviatoric stress. Building on prior studies [e.g., Buck et al. (31)], we incorporate two phases of strain weakening. The initial cohesion and coefficient of internal friction are 48 MPa and 0.753, respectively. Cohesion is reduced in two linear phases: a 10-MPa decrease as accumulated plastic strain rises to ${\mathrm{\epsilon }}_p^1$ = 0.02, followed by an additional 10-MPa reduction as accumulated plastic strain increases to ${\mathrm{\epsilon }}_p^2$ = 0.2.

Supplementary Materials

The PDF file includes:

Tables S1 to S3

Figs. S1 to S9

Legend for movie S1

Other Supplementary Material for this manuscript includes the following:

Movie S1