Remarks on the derivation of the paleostress system from inferred faults on deep seismic reflection records

275

Tectonophysics, 168 (1989) 275-282 Elsevier Science Publishers B.V.. Amsterdam - Printed in The Netherlands

Remarks on the derivation of the paleostress system from inferred faults on deep seismic reflection records J.H. MCBRIDE Shell Offshore Inc., P.O. Box 61933, New Orleans, L.A 70161 (U.S.A.)

(Received June 24,198s; revised version accepted January 31,1989)

Abstract McBride, J.H., 1989. Remarks on the derivation of the paleostress reflection records. ~ectonophysjcs, 168: 275-282.

system from inferred faults on deep seismic

Geological dip-slip faults have become a common feature of interpreted deep crustal seismic reflection records. The applicability of Anderson’s theory of faulting to these cases, as argued by previous workers, would allow the derivation of the paleostress system from the dip angle of the fault. Knowledge of paleostress would obviously be important in constraining tectonic reconstructions. Two well-constrained examples of crustal dip-slip faults inferred from deep seismic reflection data (from the Wind River Mountains, Wyoming and the ~ddle~e buried Triassic-Jurassic basin, Georgia, U.S.A.) suggest that in certain cases the approach can yield reasonable estimates of the coefficient of static friction and the paleostress system from deep, reflection-defined faults. This study, however, also notes that the sensitivity of the calculation of the coefficient of static friction (and thus stress) to the fault-dip angle, together with ambiguities often involved in measuring angles of deep seismic reflectors, may make such estimates meaningless in cases where true dip angles cannot be precisely determined.

Introduction Major crustal-scale dip-slip faults have been inferred from deep seismic reflection records (e.g., O-20 s, reflection time) on the basis of projecting a deep reflection or reflection pattern up to a surface mapped fault (Brewer et al., 1981; Oliver, 1982). Faults that do not project to the surface also have been interpreted, with a lesser degree of certainty, from a dipping reflection(s) which truncates adjacent reflections above or below. For example, deep seismic reflection data across the Wind River range Laramide uplift in Wyoming revealed that the southwestern boundary of the uplift approximately projects down along a nearly planar reflector which was initially observed to have an average dip angle of 30 O-40 Q to a depth of at least 24 km (Brewer et al., 1980; Brewer and Turcotte, 1980) and is traceable for over 80 km ~-1951/89/$03.50

@ 1989 Eksevier Science Publishers B.V.

along the survey line. Similarly, a gently dipping reflector recorded on a deep reflection line from the Sevier Desert in the eastern Basin and Range of Utah extends from more than 120 km and has been interpreted as a low-angle normal fault with an apparent dip of 12O to about 8 km depth (Allmendinger et al., 1983). Less than 50 km to the west in the same region, Hauser et al. (1987) interpret a major east-dipping fault from reflection data beneath the Northern Snake Range to at least as deep as 15 km on the basis of inferred fault-plane reflections extending down from the Schell Creek normal fault. For a last example, reflection data collected over the buried Riddleville basin, part of the early Mesozoic South Georgia (U.S.A.) rift (e.g., McBride et al., 1987), delineate a steeply dipping (50 “) fault-plane reflector imaged as deep as lo-12 km and projecting to the northern border of the basin (Petersen et al., 1984).

276

J H McBKIDf-

Many other similar

examples

could be mentioned

(e.g., White and Bretan, 1985). A remarkable aspect of all these is that all or part of the inferred fault is planar deep

brittle,

Bretan,

at least in the upper simple

shear,

1985; Jackson,

ble applicability

faults”).

ity of deep planar earthquake

dip-slip

of the brittle

crust

data

theory

and

of faulting

of the possibil-

geodetic

levelling

and

can be interpreted

faults that extend (Jackson,

Furthermore,

measurements

(White

and thus the possi-

In support

faults,

seismicity

infer planar 1988).

1987)

of Anderson’s

(as “Coulomb

crust, suggest

behavior

1987;

underground

from areas of active

cate that laboratory-derived

to

to the base Stein

et al.,

in situ

stress

faulting

indi-

coefficients

Fig. 1. Orientation

of principal

tangential

7, respectively)

illustrating surface

(u, and

Anderson’s

gravitational

theory

(‘~1,~and ox,) and normal stresses

of faulting.

acceleration

and

on a dip-slip

fault

Bulk density

= p;

= g; Au,, = oxx - 0, v. devia-

toric or “tectonic”

stress.

of static crust

oriented parallel and perpendicular to the earth’s surface with principal vertical stress equal to litho-

In such cases of a deep, planar fault-plane reflection, the stress system under which the fault formed may be deduced (Brewer and Turcotte,

equal to lithostatic pressure plus a depth-dependent deviator& (“tectonic”) stress (Fig. 1). The deviatoric stress (Au,, = a,, - Q,) is useful for

1980) but only if the angle of reflector dip can be accurately determined. Knowledge of the paleo-

understanding tectonic problems (e.g., Bott and Kusznir, 1984) and for calculating the pre-existing normal and shear stress system. Applying Amon-

friction

apply

to natural

used to estimate (Zoback

stress

and

stress in the brittle,

and Healy,

system

faults

thus can be upper

static

1984).

would

obviously

be important

for

pressure,

understanding tectonic history. Deep seismic reflection data afford a unique opportunity for in-

ton’s law (cf. Ranalli, 1987):

vestigating the state of paleostress on a regional, crustal scale. The purpose of this study is to (1)

171 =dJn-PJ

reevaluate

the role of fault-dip

angle in the calcu-

lations, somewhat confused in practice, and (2) use two well-constrained case studies of fault-plane reflections, where the analysis can be more reliably applied, in order to test whether results consistent with laboratory tained.

measurements

can be ob-

Previous investigations The derivation of the calculation of the stress system for a brittle fault according to Anderson’s theory of faulting (Anderson, 1951) is given by, for example, Sibson (1974), Jaeger and Cook (1979), and Turcotte and Schubert (1982). Here, I briefly outline the main results of this derivation, without proof, in order to review the role of the fault dip angle. In applying Anderson’s theory of faulting, let us require an assumption of plane strain and that the two principal stresses were

and

pressure,

horizontal

Coulomb-Navier

criterion,

stress

e.g.,

(1)

where p is coefficient pore

principal

of static friction

it has been

shown

and p,

is

(e.g., Turcotte

and Schubert, 1982) that the dip angle, j3, for a brittle shear fault minimizing deviatoric stress, 1Auxx 1, (i.e., initiating failure) is determined by: tan 2p = &-l/p

(2)

(+ for reverse, - for normal). The relation between j3 and p is given graphically in Fig. 2 for reverse and normal faults. Figure 3 shows an example of deviatoric stress for a “wet” crust (i.e., with pore pressure)

plotted

as a function

of dip

angle and shows a strong dependence, particularly for a reverse fault, of deviatoric stress on the dip angle. Obviously, the reflector dip angle must be measured very accurately (e.g., for j? = 30 O, say at 1%) in order to be of any value in estimating the coefficient of static friction and thus the paleostress system. A number of workers have attempted to implement this approach for deducing the paleostress

PALEOSTRESS

SYSTEM

FROM

FAULTS

ON

DEEP

SEISMIC

REFLECTION

RECORDS

700 -

I007

600

-

500-

400-

loI&77’~ , ,

j

40’

45”

ov :

35O

_ 30’

25O

20”

I

45O

40°

35O

30”

25O

200=

45O

50”

55O

60”

65’

700

Fig. 2. Graph of fault

j.3+ 45 O. tion range

of coefficient

dip angle,

P(Reverse fault) =P(Narmal fault)

of static friction,

Also shown is example of inaccuracy

of p from

j3 = 35 o (55 o ) f 5 “. Stippled

of p = 0.60-0.85

( > 200 MPa)

and

p, as a function

observed

by Bye&e

low ( < 200 MPa)

V-vermiculite,

normal

Fig.

3. Example

of curves

function

of dip angle

in determina-

pressure,

p,)

region

For /3=45”,

/3. For p = 45O, p( 8) is in the Limit

(1978)

as

shows for high

stresses

(a,).

M-montmorillonite.

involved

at y =lO a(&

km depth, Aa,,

where

stress crust

fault.

of Au,,

as a

p is as in eqn. (2).

as B + 45 O. Note

from Turcotte

error

in j3 = 35 ’

Coefficient

= p( = 2700 kg/mm3),

= g. Derivation

(Au,,)

(i.e., with pore

for a f 5 o inaccuracy

for a reverse

= p, bulk density

acceleration

for deviatoric for a “wet”

is in the limit

in determining

(or = 55O), especially friction

(8)

of static

gravitational and Schu-

bert (1982).

system from the dip angle measured from a deep seismic reflection record (e.g., Brewer and Turcotte, 1980; Turcotte and Schubert, 1982; Meissner, 1986). From unmigrated deep seismic data across the Wind River Thrust in Wyoming, Brewer and Turcotte (1980) choose a fault-plane reflector dip of /3 = 35 o as the midpoint of the observed range of 30 “-40 o in order to calculate a coefficient of friction and from that the paleostress system as outlined above. For this case a value of p = 0.36 was derived. Using the same data set (and data processing), Meissner (1986) uses an angle of fi = 30” to obtain /.L= 0.6. Meissner (1986) interprets p = 0.6 as consistent with laboratory experiments for brittle shearing. On the other hand, Brewer and Turcotte (1980) and Turcotte and Schubert (1982) interpret p = 0.36 as suggesting the presence of phyllosilicate minerals along the fault, or some other effect. Brewer and Turcotte (1980) also go on to calculate a paleostress system. As can be seen from Fig. 2, the

source of the contradiction in interpretation is the extreme sensitivity of p on /3, which the above workers apparently did not notice. The midpoint of a range of p such as 30”~40” cannot be used to represent the actual fault-plane dip. In general, the accuracy of the dip angle on a seismic section depends on (1) delineating clearly a fault-plane reflection, (2) projecting the stacked section onto a vertical plane normal to the strike of the fault, (3) knowledge of the average compressional wave velocity, and (4) proper seismic migration (e.g., Yihnaz, 1987). In the next section, the foregoing approach is applied to two case studies which attempt to take these factors into account to derive an accurate fault-plane reflector dip. Case studies

The Wind River Mountains of the Rocky Mountain foreland have been an area of consider-

27X

.I H

McBRIDI-

NE

Wind

River

Basin

km

401 0

10

I

Fig. 4. Seismic reflection Wind River mountains

20km

I

line-drawing (Wyoming,

fault-plane

able interest for studies patterns (e.g., Couples,

interpretation

showing

U.S.A.) modified

reflector,

P,-base

of Paleozoic

of paleostress and fault 1977). The Wind River

seismic line discussed above has been intensively reprocessed by Sharry et al. (1986) in order to better define the fault geometry work, the data were corrected

major

from Sharry

at depth. In their for crooked line

features

of a reprocessing

et al. (1986). No vertical

section reflector.

of the COCORP

exaggeration.

WRM-Wind

survey

WRT-Wind

across

the

River thrust

River Mountains.

image the structure and border fault of the basin (Fig. 5). A main conclusion of this study was that the northern border fault defined an overall halfgraben and is fairly planar down to about 8 km, below which it appears to flatten into an interpre-

geometry, effect of apparent dip, and for lateral velocity contrasts, and were depth migrated. Fig-

ted shallow-dipping thrust detachment underlying it at lo-12 km. Petersen et al. (1984) determine the dip of the normal fault-plane reflector, apply-

ure 4 shows

ing seismic

the results,

which

indicate

that

the

dipping reflector associated with the Wind River thrust is planar to at least as deep as 6.5 km with a

liqueness

migration

and correcting

of the survey

line

constant true dip angle of fi = 30 O. Using eqn. (2) above, p = 30” gives a p = 0.58 (Fig. 2). This particular

value may be nonunique

migration

son, 1981). Early Mesozoic

(see also Zawislak

geologic

Plam

Coastal

km

and Smith0

rift basins

to surface

since, for ex-

ample, Lynn et al. (1983) estimate fi to be 32O, apparent, on the basis of an earlier preliminary seismic

for the ob-

,

are a characteristic

feature of the eastern U.S.A. Appalachian Piedmont and Coastal Plain related to the early formation of the Atlantic Ocean (e.g., McBride et al., 1987). These basins are typically bounded by a single master normal fault (Lindholm, 1978). Seismic reflection data for the buried Biddleville Triassic-Jurassic basin in eastern Georgia (Cook et al., 1981) were reprocessed by Petersen et al. (1984) for the upper 5 s (reflection time; approximately 15 km in depth) in order to better

IO

20 km

8

Fig. 5. Seismic

reflection

major

of the COCORP

features

Coastal

line-drawing

Georgia

(U.S.A.)

Petersen

et al. (1984). No vertical

Plain modified

ville Basin, AR-interpreted -border

fault-plane

reflector Dashed

interpretation

survey

across

from reprocessing

exaggeration.

Augusta

fault-plane

of the Riddleville where inferred.

showing the eastern by

RR-Riddlereflector, Triassic

BF basin.

PALEOSTRESS

SYSTEM

FROM

FAULTS

ON

DEEP

SEISMIC

REFLECTION

trends, to be 50 O. This value, which falls within the usual range for normal faults, yields a p J 0.18 (Fig. 2). Discussion The value of p = 0.58 for the Wind River thrust is at the low end of the range, O-6-1.0, obtained by, for example, Byerlee (1978) and Morrow et al. (1982) for laboratory experiments of brittle shearing of various rock types at various pressures. This result is therefore consistent with a hypothesis of the fault forming under conditions for brittle failure at least to the maximum depth of the planar fault reflector (i.e., 6.5 km) (Fig. 4). Furthermore, Byerlee (1978) has shown that coefficient of friction loses its dependence on rock type with increasing normal stress (> 200 MPa) converging to an experimental value of 0.6. The close correspondence between this value and that deduced for the Wind River thrust gives additional credence for brittle behavior along the deeper, planar portions of the fault. In contrast to the Wind River thrust, the Riddleville basin normal fault angle yields a relatively low coefficient of friction. A ,u = 0.18 is typical for that of vermiculite and montmorillonite (Byerlee, 1978; Meissner and Strehlau, 1982; Morrow et al., 1982). From laboratory results, Byerlee (1978) has shown that these minerals being present in the gouge zone of a fault can substantially lower the coefficient of static friction. Although it is tempting to associate a deduced low coefficient of friction with the presence of clay minerals (e.g., Brewer and Turcotte, 1980), the concentration of these minerals would seemingly post-date the initial formation of the fault zone itself. Tectonic stress in the upper brittle crust of cont~ental interiors is thought to be typically on the order of lo-100 MPa (Bott and Kusznir, 1984; Ranalli, 1987). Laboratory measurements of rock strength suggest a tectonic stress equal to 100 MPa to be common for the upper crust (Hanks, 1977; Brace and Kohlstedt, 1980). For this study, calculating tectonic stress from fault-dip angle (Fig. 3) gives, for the Riddleville basin border fault, - 40 MPa at 8 km and, for the Wind River thrust, - 220 MPa at 6.5 km using a “wet” rheol-

RECORDS

279

ogy for both. Using a “dry” (i.e., pw = 0) rheology gives considerably larger (and probably unreasonable) values of tectonic stress. Evidence from conductivity measurements and from mines and drillholes indicates water-filled interconnected pore spaces at least as deep as 5 km (Brace, 1972; Brace and Kohlstedt, 1980). Deep resistivity measurements suggest this Iimit may extend to 20 km (Nekut et al., 1977). Thus, the tectonic stress estimates from this study would be consistent with independent estimates of tectonic stress and with a wet rheology for the upper crust. An obvious question is the depth limit of brittle faulting in the crust and thus the limit of applicability of this analysis. The limit of brittle faulting is often taken to be the base of the seismogenic layer as defined by earthquakes. This depth is variable but typically lo-15 km (Eaton et al., 1970; Bollinger et al., 1985) for “average” continental crust. For convergent areas of low heat flow, this maximum could be as deep as 20 km (e.g., Dewey et al., 1986) and for some areas (e.g., East Af~ca-Shudofsky, 1985) as deep as near the base of the continental crust. Below this depth, rocks would be influenced more by plastic flow making the analysis of this study inappropriate (Ranalli, 1987). For the above two case studies, planar fault reflectors with constant dip angle are restricted to the upper 10 km (of the present-day crust) (Figs. 4 and 5). The thrust fault case (Fig. 4) shows a flattening of the fault to near zero dip, whereas for the normal fault (Fig. 5) the degree of flattening is much less. Perhaps a more difficult question is the meaning of listric geometry with respect to rheology for both the normal and thrust fault case. Can the listric shape of the faults be explained in terms of a change in coefficient of friction with depth, as at least a contributing factor? On the Wind River line, reflections associated with the thrust become listric below 6.5 km and gradually approach the horizontal in the depth range of 20-25 km. 1f brittle conditions were maintained through the zone of shallowing fault dip, the app~cation of Anderson’s theory of faulting (i.e., using eqn. 2) would imply p to be increasing with depth, possibly suggesting increasing shear strength with depth (Fig. 6). Models of shear strength as a

280 Reverse Fault

~~!~_I~

tOkm

Fig. 6. Idealized represen~tion of a reverse and a normal fault (cf. Figs. 4 and 5, respectively) showing the relation between coefficient of static friction (p) and fault dip angle (/3) given by eqn. (2) (i.e., if only p were the dominating property). As emphasized in the text, many other effects combine to complicate this simpb picture so that changing coefficient of static friction with depth is probably at most a eun~~jb~~jngfactor. No vertical exaggeration. Modified from Meissner (1986).

function of depth in the crust (e.g., Meissner and Strehlau, 1982) predict that strength should increase from the surface to a m~mum near the base of the brittle upper crust, below which strength decreases as ductile (creep) processes increasingly dominate. For example, Meissner (1986) uses this inference to relate the listric shape of thrusts at deeper crustal depths to increasing p down to the base of the brittle crust (Fig. 6). The transition from brittle to plastic behavior in the crust is a broad region of semi-brittle response in which both fracturing and plastic flow are thought to occur (Kirby, 1980). Such a broad transitional region may explain the appearance of numerous splays off the interpreted Wind River fault where it flattens (see also Lynn et al., 1983; Sharry et al., 1986). For the normal fault case (Fig. 5), a listric shape at depth is also seen although much less flattened. Using the above reasoning, a decrease in dip for a normal fault would give a trend opposite to that for a reverse fault, decreasing p with depth (Fig. 6). Such a decrease may be a consequence of reduced strength due to crustal extension and rift magmatism (Meissner, 1986); however, as pointed

out by Meissner (1986), the expected rise is crustal temperature accompanied by the onset of ductile conditions would tend to invalidate the above analysis for areas having undergone significant extension.

Some objections

Anderson’s theory, strictly speaking, equates geological faults and “Coulomb” fractures (e.g., Zoback and Healy, 1984). This involves several assumptions that may be violated in nature: (1) The mathematical formulation relating p to the dip angle, eqn. (2), is valid only for isotropic rock bodies. Most rock bodies are anisotropic due to fractures, faults, joints, foliations, and lithic contacts. However, depending on the scale, a crustal region of rock may be argued to be large enough to include a statistically random distribution of dis~on~uities so that a favorably-o~ented plane of weakness will be utilized by a stress field (e.g., Brewer and Turcotte, 1980). (2) A similar problem is the reactivation of a pre-existing fault or zone weakness by later stress, invalidating the above approach (e.g., BlQ and Feuga, 1986). For example, the Sevier Desert detachment in the eastern Basin and Range, a lowangle (apparent dip = 12”) fault, appears to have experienced both normal and reverse motion (Allmendinger et al., 1983). As noted by Al~en~nger et al. (1983), extension on such a low-angle fault seems mechanically impossible (i.e., not Coulomb failure), although a growing amount of evidence suggests that low-angle normal faults do occur. (3) Anderson’s theory requires that the principal directions be oriented with respect to a constant reference frame as the fault evolves. If the history of fault displacement is complex or if stress refraction with depth was important, then a reorientation of the principal stress directions may have occurred (i.e., as a function of time and/or depth) (e.g., McKinstry, 1953; Bradshaw and Zoback, 1988). (4) A last concern is the possibility that observed fault dips are not primary features, especially for low-angle normal faults (e.g., Buck, 1988; Davis, 1988; Wernicke and Axen, 1988). An im-

PALEOSTRESS

SYSTEM

FROM

FAULTS

ON

DEEP

SEISMIC

REFLECTION

phcit assumption in the present type of study is that the original fault dip is preserved. This assumption is probably more warranted for areas that were stable following the tectonic event in question. For example, faults that were originally steep may have been rotated in areas having experienced si~fi~t and continued extension (Jackson, 1987; Buck, 1988). Conclusions

(1) Inferred, planar upper crustal faults are a typical feature of interpreted deep seismic reflection profiles. (2) Anderson’s theory of faulting would imply that the coefficient of static friction and paleostress can be estimated from planar faults (with assumed stress directions) imaged on the profiles for the upper crust (above the brittle-ductile transition). (3) Application of this approach is highly dependent on several assumptions including accurate measurement of fault dip from deep seismic sections. Automatic seismic migration is always necessary. (4) The Wind River thrust fault (Wyoming, U.S.A.) has a deduced coefficient of friction consistent with experimental results for high stress. (5) The Riddleville buried Triassic basin (Georgia, U.S.A.) has a deduced coefficient of friction consistent with a clay mineralogy; however, caution must be used for areas where heat flow was high and the depth of the brittle-ductile transition was shallow at the time of fault fo~ation. Rotation after initial faulting could produce an apparently low coefficient of friction. (6) Tectonic stresses derived in this study are of the order for expected values based on independent reasoning,‘and consistent with a “wet” crust rheology. (7) The contributing effect of the coefficient of friction changing with depth should not be ignored in explaining the listric and flattening fault geometry seen on deeper portions of crustal faults. Acknowledgments

Appreciation is expressed to J.W. Gephart, T. Reston, and M. Barazangi for reading the

RECORDS

281

manuscript and providing very useful criticism. R.W. Allmendinger is thanked for discussion on the topic of this study. The illustrations were drafted by L. Angell. Reviews by R. Meissner and an anonymous referee improved the paper. References Allmendinger, R.W., Sharp, J.W., Von Tish, D., Serpa, L., Brown, L., Kaufman, S., Oliver, J. and Smith, R.B., 1985. Cenozoic and Mesozoic structure of the eastern Basin and Range from COCORP seismic reflection data. Geology, 11: 532-536. Anderson, E.M., 1951. The Dynamics of Faulting and Dyke Formation with Applications to Britain. Oliver and Boyd, Edinburgh, 191 pp. BlBs, J. and Feuga, B., 1986. The Fracture of Rocks. Elsevier, New York, 131 pp. Boihnger, G.A., Chapman, MC., Sibol, MS. and Costain, J.K., 1985. An analysis of earthquake focal depths in the southeastern U.S. Geophys. Res. Lett., 12: 785-788. Bott, M.H.P. and Kusznir, N.J., 1984. The origin of tectonic stress in the lithosphere. Tectonophysics, 105: l-13. Brace, W.F., 1972. Pore pressure in geophysics. In: H.C. Heard, I.Y. Borg, N.L. Carter and C.B. Raleigh (Editors), Flow and Fracture of Rocks. Geophys. Monogr., Am. Geophys. Union, 16: 265-273. Brace, W.F. and Kohlstedt, D.L., 1980. Limits on lithospheric stress imposed by laboratory experiments. J. Geophys. Res., 85: 6248-6252. Bradshaw, G.A. and Zoback, M.D., 1988. List& normal faulting, stress refraction, and the state of stress in the Gulf Coast basin. Geology, 16: 271-274. Brewer, J.A. and Tumotte, D.L., 1980. On the stress system that formed the laramide Wind River mountains, Wyoming. Geophys. Res. Lett., 7: 449-452. Brewer, J.A., Smithson, S.B., Oliver, J.E., Kaufman, S. and L.D. Brown, L.D., 1980. The Laramide orogeny: evidence from COCORP deep crustal seismic profiles in the Wind River mountains, Wyoming. Tectonophysics, 62: 165-189. Brewer, J.A., Cook, F.A., Brown, L.D., Oliver, J.E., Kaufman, S. and Albaugh, D.S., 1981. COCORP seismic reflection profiling across thrust faults. In: K. McClay and N. Price (Editors), Thrust and Nappe Tectonics. Geol. Sot. London, Spec. Publ., 9: 501-511. Buck, W.R., 1988. Flexural rotation of normal faults, Tectonics, I: 959-974. Byerlee, J., 1978. Friction of rocks. Pure Appl. Geophys., 116: 615-626. Cook, F.A., Brown, L.D., Kaufman, S., Oliver, J.E. and Petersen, T.A., 1981. COCORP seismic profifing of the Appalachian orogen beneath the Coastal Plain Georgia. Geol. Sot. Am. Bull., 92: 738-748. Couples, G., 1977. Stress and shear fracture (fault) patterns resulting from a suite of complicated boundary conditions

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