RRobert C. ViescaTufts University Self-similar fault slipin response to fluid injection Page 1 of 78 Feb. 2021
We present the self-similar solution to a problem of fault slip in response to fluid injection (Fig.1). Specifically, we consider a fault plane lying on y = 0 and a line fluid source of constant pressurelocated at x = 0, along the z axis. The medium containing the fault is linearly elastic and thedeformation may be in-plane or anti-plane. The in-plane case is illustrated in Fig. 1 (top left).The shear modulus of the medium is µ and the Poisson ratio ν . We define the effective elasticmodulus µ (cid:48) = µ/ [2(1 − ν )] for in-plane (mode-II) case and µ (cid:48) = µ/ τ (in-plane or anti-plane), the faultfriction coefficient f , the initial pore fluid pressure on interface p o , the initial total fault-normalcompressive stress σ , and the initial effective normal stress σ (cid:48) = σ − p o . The initial fault strength is τ p = f σ (cid:48) . This problem was presented in Bhattacharya and Viesca [2019] and is a simpler versionof one considered by
Garagash and Germanovich [2012], who examined the response to injection ofa fault whose friction coefficient weakens with slip. The simplicity of the problem at hand leads toclosed-form expressions of the solution for end-member scenarios and a concise numerical solutionof the intervening cases. Problem and solution details follow and provide a basis to verify numericalsolution procedures for quasi-static fault rupture.We consider one-dimensional diffusion of pore fluid pressure along the fault p t = α hy p xx where α hy is the hydraulic diffusivity of the fault core and where the pore pressure is subject tothe conditions of the initial state and injection at constant pressure ∆ p at x = 0, p ( x,
0) = p o , p (0 , t >
0) = ∆ p the solution to which is p ( x, t ) = p o + ∆ p erfc( | x | / √ αt )where we adopt a nominal diffusivity α = 4 α hy The fault obeys a Coulomb friction law: the local shear strength of the fault τ s is a constantproportion of the local effective normal stress, with a constant coefficient of friction fτ s ( x, t ) = f [ σ − p ( x, t )]Where sliding occurs, this strength must equal the shear stress on the fault. The shear stress can bedecomposed into a sum of the initial shear stress τ plus quasi-static changes due to a distributionof slip δ , such that the stress-strength condition is τ s ( x, t ) = τ + µ (cid:48) π (cid:90) a ( t ) − a ( t ) ∂δ ( s, t ) /∂ss − x ds where x = ± a ( t ) are the crack-tip locations.After non-dimensionalizing, the problem is found to have a sole parameter (cid:18) − ττ p (cid:19) σ (cid:48) ∆ p (1)that is bound between 0 and 1. The upper bound denotes a marginally pressurized fault, wherethe fluid pressure increase is just sufficient to initiate sliding: f [ σ − ( p o + ∆ p )] = τ . The lower a r X i v : . [ phy s i c s . f l u - dyn ] F e b obert C. ViescaTufts University Self-similar fault slipin response to fluid injection Page 2 of 78 Feb. 2021
10 10 10 10 a ( t ) = λ √ αt √ αtσ σ τ ττ τ xy µ, ν Fig. 1. Counter-clockwise: ( top left ) Unbounded elastic body containing a fault, loadedremotely with fault-normal and shear stress σ , τ . Fluid injection at x = 0 diffuses along fault as √ αt , inducing quasi-static slip out to a distance a ( t ). Fault has constant friction coefficient f .( bottom left ) Black: relation between rupture growth factor λ and a parameter reflecting theinitial state of stress and injection pressure, where σ (cid:48) = σ − p o and p o is pre-injection fault fluidpressure. Dashed: asymptotic behaviors, eqs. (2) and (5). ( bottom right ) Same as bottom left,with abscissa arranged to occupy a finite interval. ( top right ) Plot of self-similar slip distributionsat three instants in time after the start of injection, t = 1, 5, and 10 min., for the specific choices σ = 50 MPa, τ = 12 MPa, p o = 20 MPa, ∆ p = 12 MPa, f = 0 . α hy = 0 .
01 m / s, µ = 30 GPa, ν = 1 / µ (cid:48) = 20 GPa. For these choices, the parameter (1 − τ /τ p ) σ (cid:48) / ∆ p = 0 .
5. The correspondingself-similar slip distribution and factor λ are given in Table 1. obert C. ViescaTufts University Self-similar fault slipin response to fluid injection Page 3 of 78 Feb. 2021 Fig. 2. ( left ) Self-similar distributions of slip δ with distance from injection point x , which isscaled by the crack length a ( t ) = λ √ αt . Each distribution corresponds to one value of λ in therange λ = 10 − , − , ... , . To facilitate comparisons, slip is scaled differently for black andred curves. Black curves correspond to λ = 10 − , − , − , from top to bottom, with thefirst three indistinguishable on this scale; red curves correspond to λ = 10 , , from top tobottom. Cyan-dashed: solution for small λ , eq. (3). Blue-dashed: “outer” solutions for large λ , eq. (6). ( right ) For large values of λ , the distribution of slip is plotted over distances scaledby √ αt , which is much smaller than the crack length a ( t ). This “inner” behavior is describedby eq. (7), a single numerical solution shown here as black-dashed curves. Curves correspond to λ = 10 , , from bottom to top.bound denotes a critically stress fault, where the initial shear stress is equal to the initial shearstrength: τ = τ p .The solution consists of a self-similar distribution of slip, in which the crack front grows as a ( t ) = λ √ αt and the slip distribution can be written as δ (¯ x ), where the similarity coordinate is¯ x = x/a ( t )The factor λ , to be solved for, determines whether the crack lags ( λ <
1) or outpaces ( λ >
1) thediffusion of pore pressure, which stretches as √ αt . λ depends uniquely on the sole parameter (1),and that dependence is illustrated in Fig. 1b and tabulated at the top of Table 1. The self-similarprofile of slip, as it depends on | x | /a ( t ), is also presented in the bottom of Table 1 for several valuesof the parameter (1). Scaled plots of the self-similar profile for various values of λ are shown in Fig.2. In the limit that the parameter (1) approaches its end-member values, closed-form expressionsfor λ and δ are available and provided below. (i) Marginally pressurized faults: τ → f ( σ (cid:48) − ∆ p )In this limit, the parameter (1 − τ /τ p ) σ (cid:48) / ∆ p →
1, the factor λ (cid:28) − τ /τ p ) σ (cid:48) / ∆ p ≈ − π / λ − O ( λ ) (2)The slip distribution in this limit is δ (¯ x ) = λ √ αtf ∆ pµ (cid:48) π / (cid:34)(cid:112) − ¯ x − ¯ x ln (cid:32) √ − ¯ x | ¯ x | (cid:33)(cid:35) (3) obert C. ViescaTufts University Self-similar fault slipin response to fluid injection Page 4 of 78 Feb. 2021
10 10 10 10 Fig. 3.
The maximum slip, which occurs at x = 0, as it relates to the factor λ shown as ( top )linear ( left ) semi-log and ( right ) log-log plots. Red-dashed: end-member scalings at small λ , eq.(4), and at large λ , eq. (9). Blue-dashed: approximation of δ (0) for all λ , eq. (11).and the accumulation of slip at the center is δ (0) = 2 π / λ √ αtf ∆ pµ (cid:48) (4) (ii) Critically stressed faults: τ → τ p In this limit (1 − τ /τ p ) σ (cid:48) / ∆ p → λ (cid:29) − τ /τ p ) σ (cid:48) / ∆ p ≈ π / λ + O ( λ − ) (5)Similarly to the problem considered by Garagash and Germanovich [2012], the solution for slipcan be decomposed into an outer solution on distances comparable to the rupture distance a ( t ),and an inner solution on distances comparable to the diffusion lengthscale √ αt . The two solutionsare matched at an intermediate distance.The outer solution for the slip distribution is δ (¯ x ) = √ αtf ∆ pµ (cid:48) π / (cid:34) ln (cid:32) √ − ¯ x | ¯ x | (cid:33) − (cid:112) − ¯ x (cid:35) (6)where ¯ x is the similarity coordinate used above.The inner solution is given by the expression δ ( x/ √ αt ) = δ (0) + √ αtf ∆ pµ (cid:48) (cid:90) x/ √ αt (cid:20) π (cid:90) ∞−∞ erfc( | ˆ s | )ˆ x − ˆ s d ˆ s (cid:21) d ˆ x (7)The underlined portion is evaluated numerically and provided as a supplementary function f ( x/ √ αt )in Table 2 with the similarity coordinate ˜ x = x √ αt obert C. ViescaTufts University Self-similar fault slipin response to fluid injection Page 5 of 78 Feb. 2021 Fig. 4.
The peak slip as it relates to the problem parameter (1 − τ /τ p ) σ (cid:48) / ∆ p . Red-dashed:scalings as the parameter approaches its bounds, derived from eqs. (4) and (9) and the asymptoticrelations between the parameter and λ , eqs. (2) and (5).For large distances x/ √ αt , f behaves as f (˜ x ) ≈ − π / (cid:16) ln | ˜ x | + γ (cid:17) + O (cid:0) ˜ x − (cid:1) (8)where γ = 0 . ... is the Euler-Maraschoni constant. Using this asymptotic behavior tomatch the inner solution at large x/ √ αt with the outer solution at small x/a ( t ) provides the slipat the center δ (0) ≈ √ αtf ∆ pµ (cid:48) π / [ln(2 λ ) + γ/
2] (9)in the large λ limit.Other properties of f ( x/ √ αt ) include f (cid:48)(cid:48) (˜ x ) = 2 π / exp( − ˜ x )Ei(˜ x )where Ei( x ) = − (cid:82) ∞− x exp( − u ) /u du is the exponential integral, and in the limit that x/ √ αt is small, f behaves as f (˜ x ) ≈ − π / ˜ x (cid:18) ln 1 | ˜ x | + γ − (cid:19) + O (cid:18) ˜ x ln 1 | ˜ x | (cid:19) (10) (iii) Accumulation of slip at the injection point Figs. 3 and 4 show the solution for the peak slip, located at the injection point, as it dependson the parameter (1) or the factor λ . An approximation of peak slip at the injection point thatrespects the asymptotic behavior at both critically stressed and marginally pressurized limits—eqs.(4) and (9)—and is to within 5% error over the intervening range of λ , is δ (0) ≈ λ √ αtf ∆ pµ (cid:48) π / λ λ / [ln(6 + 2 λ ) + γ/
2] (11)
References
Bhattacharya, P., and R. C. Viesca (2019) Fluid-induced aseismic fault slip outpaces pore-fluidmigration,
Science , 364, 464–468, doi:0.1126/science.aaw7354Garagash, D. I., and L. N. Germanovich (2012), Nucleation and arrest of dynamic slip on a pres-surized fault,
J. Geophys. Res. , 117, B10310, doi:10.1029/2012JB009209 obert C. ViescaTufts University Self-similar fault slipin response to fluid injection Page 6 of 78 Feb. 2021
Table 1.
Tabulation of the dependence of both factor λ and self-similar slip distribution δ (¯ x ) on the problem parameter (1 − τ/τ p ) σ (cid:48) / ∆ p . Note abbreviation: T = (1 − τ/τ p ) σ (cid:48) / ∆ p .(1 − τ/τ p ) σ (cid:48) / ∆ p → → λ (2 /π / ) /T π / / − T ) | x | /a ( t ) δ/ ( √ αtf ∆ p/µ (cid:48) ) δ/ ( λ √ αtf ∆ p/µ (cid:48) ) δ/ ( λ √ αtf ∆ p/µ (cid:48) )0 (2 /π / )[ln(2 λ )+ γ/
2] 0.22235814 0.28884483 0.29158012 0.26854750 0.23366202 0.19225196 0.14690288 0.099114755 0.049891607 0.35917424.05 0.96600080 0.21293697 0.28287092 0.28705167 0.26495909 0.23081724 0.19005163 0.14529019 0.098055692 0.049366583 0.35541318.10 0.71771469 0.19406542 0.26977998 0.27684966 0.25676095 0.22426165 0.18495186 0.14153779 0.095585199 0.048140087 0.34662298.15 0.57320909 0.17141546 0.25241461 0.26293875 0.24543311 0.21513089 0.17781153 0.13626551 0.092106171 0.046410685 0.33422335.20 0.47146496 0.14804366 0.23237418 0.24641194 0.23179150 0.20404685 0.16909871 0.12980996 0.087836899 0.044285816 0.31898215.25 0.39336448 0.12579403 0.21080933 0.22806232 0.21642674 0.19145812 0.15914994 0.12241251 0.082933625 0.041842292 0.30144813.30 0.33039758 0.10566200 0.18860371 0.20851874 0.19981014 0.17772295 0.14823375 0.11426559 0.077520767 0.039141242 0.28205787.35 0.27803088 0.088034290 0.16644630 0.18830074 0.18233746 0.16314401 0.13657762 0.10553231 0.071703840 0.036234507 0.26118168.40 0.23356530 0.072898864 0.14486896 0.16784610 0.16435133 0.14798653 0.12438207 0.096356965 0.065576337 0.033168065 0.23914800.45 0.19526763 0.060023208 0.12427123 0.14752660 0.14615421 0.13248893 0.11182922 0.086871368 0.059223968 0.029984140 0.21625871.50 0.16196334 0.049088021 0.10494029 0.12765847 0.12801685 0.11686999 0.099088536 0.077199247 0.052727610 0.026722691 0.19279968.55 0.13282499 0.039772618 0.087069279 0.10851017 0.11018427 0.101334019 0.086321205 0.067459621 0.046165603 0.023422573 0.16904926.60 0.10725384 0.031798805 0.070774899 0.090309083 0.092880849 0.086075209 0.073683848 0.057769761 0.039615810 0.020122586 0.14528583.65 0.084810067 0.024945699 0.056114600 0.073248018 0.076314980 0.071281873 0.061332237 0.048248183 0.033157715 0.016862563 0.12179568.70 0.065170463 0.019048185 0.043103256 0.057492242 0.060684332 0.057141201 0.049425558 0.039018155 0.026874910 0.013684652 0.098882348.75 0.048103357 0.013988831 0.043103256 0.043187811 0.046182579 0.043845559 0.038132126 0.030212417 0.020858460 0.010635068 0.076879363.80 0.033456068 0.0096894840 0.021971389 0.030472711 0.033009402 0.031602114 0.027638158 0.021980444 0.015212073 0.0077667764 0.056169753.85 0.021154984 0.0061062150 0.013816179 0.019494061 0.021387701 0.020649782 0.018163273 0.014501247 0.010061206 0.0051442208 0.037220404.90 0.011226140 0.0032311841 0.0072858109 0.010440840 0.011599155 0.011294788 0.0099929633 0.0080100955 0.0055720081 0.0028531253 0.020653307.95 0.0038742471 0.0011124332 0.0024976528 0.0036291987 0.0040812723 0.0040083095 0.0035674726 0.0028713219 0.0020027229 0.0010270500 0.00743832391 0 0 0 0 0 0 0 0 0 0 0
Table 2.
Tabulation of f ( x/ √ αt ). Note discretization: | x | / √ αt = tan( π n/ n | x | / √ αt − f ( x/ √ αt ) n | x | / √ αt − f ( x/ √ αt ) n | x | / √ αt − f ( x/ √ αt ) n | x | / √ αt − f ( x/ √ αt )0 0 0 16 0.41421 0.12611 32 1 0.39580 48 2.4142 0.767601 0.024549 0.0010645 17 0.44327 0.13934 33 1.0503 0.41649 49 2.5924 0.794992 0.049127 0.0036609 18 0.47296 0.15307 34 1.1033 0.43756 50 2.7948 0.823563 0.073764 0.0074562 19 0.50336 0.16730 35 1.1593 0.45900 51 3.0270 0.853594 0.098491 0.012279 20 0.53451 0.18203 36 1.2185 0.48078 52 3.2966 0.885425 0.12334 0.018014 21 0.56649 0.19725 37 1.2814 0.50289 53 3.6135 0.919446 0.14834 0.024577 22 0.59938 0.21296 38 1.3483 0.52532 54 3.9922 0.956147 0.17352 0.031903 23 0.63324 0.22915 39 1.4199 0.54803 55 4.4532 0.996198 0.19891 0.039941 24 0.66818 0.24583 40 1.4966 0.57102 56 5.0273 1.04049 0.22456 0.048648 25 0.70427 0.26298 41 1.5792 0.59429 57 5.7631 1.090110 0.25049 0.057990 26 0.74165 0.28060 42 1.6684 0.61783 58 6.7415 1.146911 0.27674 0.067940 27 0.78041 0.29868 43 1.7652 0.64168 59 8.1078 1.213612 0.30335 0.078475 28 0.82068 0.31723 44 1.8709 0.66585 60 10.153 1.294713 0.33035 0.089573 29 0.86261 0.33622 45 1.9867 0.69043 61 13.557 1.398814 0.35781 0.10122 30 0.90635 0.35566 46 2.1143 0.71549 62 20.355 1.545015 0.38574 0.11340 31 0.95208 0.37552 47 2.2560 0.74117 63 40.735 1.7943 obert C. ViescaTufts University Self-similar fault slipin response to fluid injection Page 7 of 78 Feb. 2021
10 10 10 Fig. 5.
Spatio-temporal component, f ( x/ √ αt ), of “inner” solution for slip δ in the criticallystressed limit, eq. (7). Shown on linear ( top ), log-linear ( left ), and logarithmic ( right ) axes. Red-and blue-dashed curves: outer and inner asymptotic behavior of ff