Self-Similar Composite Wave Solutions

• Self-similar composite wave solutions are analytical constructs that reduce nonlinear PDEs to a set of self-similar variables, capturing the essential dynamics of complex wave propagation.
• They are derived through asymptotic matching of inner and outer solutions, yielding uniformly accurate approximations across both marginally pressurized and critically stressed regimes.
• These solutions are pivotal in applications like hydraulic fracturing, geothermal and CO2 sequestration, offering benchmarks for numerical models and insights into rupture dynamics.

Self-similar composite wave solutions are exact or asymptotic solutions to nonlinear partial differential equations (PDEs) in which spatial and temporal dependences can be collapsed onto a reduced set of self-similar variables, enabling analytic or semi-analytic description of wave propagation, rupture, or fronts in complex media. In many systems—particularly those described by diffusive or elastodynamic equations—self-similarity arises when there are no intrinsic length or time scales, leading to profiles that scale as powers of time and space. Composite wave solutions are constructed by asymptotically matching inner and outer solutions or using perturbation expansions in multiple regimes, often yielding uniformly accurate descriptions even across singular or boundary-layer regions. In the context of frictional rupture driven by fluid injection at constant rate, self-similar composite wave solutions describe the axisymmetric expansion of rupture and fault slip, capturing both marginally pressurized and critically stressed regimes, with broad geophysical and modeling implications (Viesca, 24 Jan 2024).

1. Self-Similar Expansion of Fluid-Induced Rupture

The rupture induced by constant-rate fluid injection in a permeable fault plane is characterized by a circular, axisymmetric expansion that displays self-similar behavior for both the pore-fluid pressure and slip distribution. When the Poisson ratio is zero ($\nu=0$), diffusive self-similarity of the fluid pressure profile implies that the slip distribution will also evolve in a self-similar manner. The rupture radius $R(t)$ grows as: $R(t) = \lambda \sqrt{4\alpha_{hy} t}$ where $\alpha_{hy}$ is the hydraulic diffusivity, $t$ is the time since injection began, and $\lambda$ is a dimensionless prefactor determined by a stress-injection parameter $T$: $$T = (\sigma'/\Delta p) \left[1 - (\tau_b/\tau_p)\right]$$ Here, $\sigma'$ is the effective normal stress, $\Delta p$ is the pore pressure increase due to injection, $\tau_b$ is the initial background shear stress, and $\tau_p$ is the pre-injection fault strength. The prefactor $\lambda$ spans $(0,\infty)$, encapsulating the entire spectrum from marginally pressurized ($\lambda \ll 1$, $T \rightarrow \infty$) to critically stressed ($\lambda \gg 1$, $T \rightarrow 0$) states.

2. Mathematical Formulation and Similarity Variables

The axisymmetric problem is reduced to a single similarity variable via scaling: $\rho = \frac{r}{R(t)}$ where $r$ is the radial coordinate. The slip distribution $\delta(r, t)$ is then expressed as a function of $\rho$ and $t$, or further re-scaled depending on the asymptotic regime (see below). The entire spatiotemporal evolution is thus characterized by $\lambda$ and prescribed by the hydrodynamic parameters and stress state.

For slip, asymptotic solutions are sought in the form: $$\delta(r, t) = R f \frac{\Delta p}{\mu} \times S(\rho, \lambda, t)$$ where $f$ is the friction coefficient, $\Delta p$ is the overpressure, and $\mu$ is the shear modulus.

3. Perturbation Expansions: Marginally Pressurized and Critically Stressed Regimes

There are two principal asymptotic regimes:

A. Marginally Pressurized Limit ($\lambda \ll 1$):

• The rupture lags behind the pore-fluid diffusion front.
• Solution is obtained via a regular perturbation expansion in even powers of $\lambda$:

$$\frac{\delta_{mp}(r, t)}{R f \Delta p/\mu} = \frac{8}{\pi}\left[\sqrt{1-\rho^2} - \rho \arccos \rho \right] - \frac{16}{9\pi}\lambda^2 (1-\rho^2)^{3/2} + O(\lambda^4)$$

• Convergent and uniformly accurate across the rupture.

B. Critically Stressed Limit ($\lambda \gg 1$):

• The rupture front advances much faster than pressure diffusion, leading to boundary layers near injection.
• The perturbation expansion is singular; inner and outer expansions are required:
  • Inner Solution: Valid at $r = O(\sqrt{\alpha_{hy} t})$ near the wellbore, given in $s = r/\sqrt{\alpha_{hy} t}$:

  $$\delta_0(s, t) = 2\sqrt{\pi} e^{-s^2/2} \left[(1 + s^2) I_0(s^2/2) + s^2 I_1(s^2/2)\right] - 4s$$

  where $I_0$ and $I_1$ are modified Bessel functions. - Outer Solution: Valid at $r = O(R(t))$; constructed as a series in $1/\lambda$.

4. Matched Asymptotic Expansion and Composite Solutions

In the critically stressed regime, inner and outer solutions are matched in the overlap region to yield a uniformly accurate composite solution: $$\delta_{cs}(\rho, t) = \delta_{in}(\lambda\rho, t) + \delta_{out}(\rho, t) - \delta_{overlap}(\rho, t)$$ This composite approach resolves both the boundary layer near the injection point and the outer regime up to the propagating rupture front, ensuring that the solution converges uniformly throughout the rupture domain for all $\lambda$.

The expansion in the critically stressed case is divergent, but optimal truncation (cutoff at $j < \lambda^2 + 1/2$) ensures practical accuracy.

5. Applications and Implications

Self-similar composite wave solutions are directly applicable to several geophysical and engineering contexts:

• Hydraulic Fracturing: Describing the growth of aseismic slip fronts and induced seismicity.
• Geothermal and CO$_2$ Sequestration Operations: Quantitative modeling of fault activation triggered by subsurface fluid injection.
• Benchmarking and Verification: Providing analytical benchmarks for three-dimensional numerical methods (e.g., boundary-element simulations in the $\nu=0$ limit [aez et al., 2022]).
• Moment Release and Scaling Laws: Explicit scaling of rupture moment and peak slip as functions of $\lambda$ informs both seismic hazard forecasts and the interpretation of injection-induced slip events.

6. Comparison with and Validation by Numerical Models

Derived self-similar composite solutions serve to verify three-dimensional boundary-element solutions in the limit of vanishing Poisson ratio ($\nu=0$) [aez et al., 2022]. The analytical solutions, particularly the leading-order asymptotics in both marginally pressurized and critically stressed regimes, accurately reproduce the trends and scaling behaviors observed in full numerical simulations—confirming their physical relevance and mathematical robustness.

7. Significance in the Broader Context of Wave Propagation and Frictional Rupture

Composite self-similar wave solutions provide a fundamental analytical framework for understanding the complex interplay of injection-driven diffusion, elastic response, and frictional rupture in faults. The identification of distinct scaling regimes and the construction of uniformly valid solutions via matched asymptotics illuminate the mechanisms underlying rupture front propagation across stress states, establish connections to classical problems in heat conduction and moving boundary problems, and offer generalized strategies for tackling axisymmetric or two-dimensional moving-front phenomena in solids and fluids.