ona model simulatinglack of hydraulicconnection betweena

On a model simulating lack of hydraulic connectionbetween a man-made reservoir and the volume

of poroelastic rock hosting the focus of apost-impoundment earthquake

Ramesh Chander1 and S K Tomar

2,∗

1No. 290 Sector – 4, Mansa Devi Complex, Panchkula 134 114, India.2Department of Mathematics, Panjab University, Chandigarh 160 014, India.

∗Corresponding author. e-mail: [email protected]

The idea that a direct hydraulic connection between a man-made reservoir and the foci of post-impoundment earthquakes may not exist at all sites is eminently credible on geological grounds. Our aimis to provide a simple earth model and related theory for use during investigations of earthquakes nearnew man-made reservoirs. We consider a uniform circular reservoir which rests on the top surface of ano-hydraulic-connection earth model (NHCEM). The model comprises a top elastic (E) layer, an inter-mediate poroelastic (P) layer, and a bottom elastic half space. The focus of a potential earthquake in theP layer is located directly under the reservoir. The E layer disrupts the hydraulic connection betweenthe reservoir and the focus. Depth of water in the reservoir varies as H ′ + h cos(ωt). Expressions forreservoir-induced stresses and pore pressure in different layers of the NHCEM are obtained by solvingthe boundary-value problem invoking full coupling between mean normal stress and pore pressure in theP layer. As an application of the derived mathematical results, we have examined and found that earth-quakes on 60◦ normal faults may occur in the P-layer of a selected NHCEM at epochs of low reservoirlevel if the reservoir lies mostly in the footwall of the fault. The exercise was motivated by observationsof such earthquakes under the man-made Lake Mead after it was impounded.

1. Introduction

The hazard and safety concerns as well as goodengineering and scientific practice require that eachnew episode of earthquakes near man-made reser-voirs (ENMRs) be investigated. Such investigationsat present are carried out using assumed simpleearth models because adequate information aboutsubsurface geology up to earthquake depths undera reservoir is not available in general. We dis-tinguish here between a full-hydraulic-connectionearth model (FHCEM) and a no-hydraulic-connec-tion earth model (NHCEM). In the two main

examples of FHCEM in current use, it is assumedthat porous-elastic rock(s) (Roeloffs 1988) or asystem of fractures in rock (Snow 1972) providehydraulic connection between the reservoir and thefocus of a post-impoundment earthquake. Here, wedescribe a procedure to evaluate the response ofa three-layered NHCEM to the load of a circularreservoir with oscillating water level.The top E layer of the NHCEM (figure 1a) is

elastic and much thicker than the intermediateporoelastic (P) layer. The bottom layer, an elas-tic half space, is called the H layer. The focus of apotential earthquake is located in the P layer. We

Keywords. Earthquake hazard; man-made reservoirs; reservoir-influenced seismicity; tectonic earthquakes; fault stability;

Lake Mead seismicity.

J. Earth Syst. Sci., DOI 10.1007/s12040-016-0751-5, 125, No. 8, December 2016, pp. 1543–1555c© Indian Academy of Sciences 1543

1544 Ramesh Chander and S K Tomar

Figure 1. (a) A schematic cross-sectional representation ofan NHCEM under a circular reservoir. The origin O andthe z-axis of an (r, θ, z) cylindrical coordinate system areshown. The plane of the figure is called ‘reference axial plane’(RAP). (b) The concept of reservoir mostly on the foot-wall (RMOFW) and reservoir mostly on the hanging wall(RMOHW) of a normal fault is indicated. (c) When thestrike of a fault is normal to RAP, the line of symmetry(LOS) of the fault is the line of its intersection with the RAP.

concentrate on positions of the focus directly underthe reservoir.So far, all earthquakes observed near man-made

reservoirs have been tectonic earthquakes thatoccur most commonly due to renewed slip on pre-existing faults (Richter 1957). Hubbert and Rubey(1959) argued that pore fluids in rocks on two sidesof a fault facilitate slip by reducing friction. TheP layer represents water-filled rock around earth-quake focus. The E layer, being non-porous andimpermeable, disrupts the hydraulic connectionbetween the reservoir and the earthquake focus.

The simplest example of a NHCEM is an idealelastic half space. The response of this model tosurface loads is well known due to the work ofBoussinesq (Love 1944; Jaeger and Cook 1969).Response of an NHCEM in which a pocket ofporoelastic rock is embedded in an elastic halfspace deserves inquiry. But we consider a poroelas-tic layer embedded in an elastic half space becausethe analytical treatment is vastly simpler.Reservoir weight and hydraulic pressure at its

bottom both influence stress and pore pressureinside a FHCEM (e.g., Roeloffs 1988), while onlythe weight of the reservoir exerts influence withinthe NHCEM.Beginning with Haskell (1953), numerous multi-

layered models have been used to explain observa-tions in different disciplines of earth science. Wang(2000) discusses instances where models with oneor more poroelastic layers have been considered ingeo-mechanics, hydrogeology and petroleum engi-neering. Segall’s (1985) model is the closest toours in that both involve deformation of a three-layered earth. However, Segall simulated groundsubsidence and induced seismicity due to with-drawal of fluid from the P layer. We examine howthe surface reservoir influences the occurrence ofan earthquake in the P layer. Moreover, the Eand H layers in Segall’s (1985) model were ‘fluid-infiltrated, impermeable’, while in our case, theyare dry and impermeable.The main contribution of this paper, namely,

analytical estimation of stress and pore pressurefields in the above NHCEM due to oscillatory andsteady loads is placed in Appendix A because ofits mathematical content. The sample results dis-cussed in sections 3–7 enable us to interpret theobservations of earthquakes under Lake Mead atepochs of low reservoir level (Roeloffs 1988). Theearthquakes occurred on steeply dipping normalfaults at 5000 m depth. We shall refer to theseearthquakes as Lake Mead earthquakes (LMEQs).

2. Preliminaries

2.1 Earth model: ONHCEM

Properties of the different layers of the modelshown in figure 1(a) are listed in table 1. Poro-elastic properties of Ohio sandstone given by Wang(2000, table C1) have been adopted for the P layer.We will refer to this specific model as the OhioNHCEM (ONHCEM).

2.2 The reservoir model

We consider a uniform circular reservoir witha radius of 4000 m. Its water level oscillates

Hydraulic connection between a man-made reservoir and post-impoundment earthquake 1545

Table 1. Properties of ONHCEM.

Layer Thickness (m) Elastic properties Poroelastic properties

E 4950 νE = 0.18, GE = 6.8 GPa Nil

P 100 νdP = 0.18, GP = 6.8 GPa νuP = 0.28, αP = 0.74,

cP = 3.9× 10−2 m2/s

H 5050 to +∞ νH = 0.18, GH = 6.8 GPa Nil

Table 2. Stresses and pore pressure at the point (3000 m, 0◦, 5000 m) in the P layer.

Full poroelastic

Contribution due contribution due to h = 1 m

to H ′ = 1 m Amp Phase

(Pa) (Pa) (◦)

σzzP 3818 4609 0.05

τrzP 1354 1354 0.0

σrrP 304 1083 0.16

σθθP −25 751 0.21

pP 0 1063 −179.8

harmonically with an angular frequency ω of 2 ×10−7 radians per second, corresponding to an yearof 364.7 days approximately. Since values of tbetween 0 and π × 107 s could be required, weuse, instead of time t, phase ωt of reservoir levelexpressed in degrees (◦). Reservoir level corre-sponding to water depth H ′ − h at ωt = 180◦ willbe referred hereafter as LRL (lowest reservoir levelduring routine annual operation). Similarly, reser-voir level corresponding to water depth H ′ + h atωt = 0◦ or 360◦ will be referred to as HRL (highestreservoir level during routine annual operation).

2.3 Reference axial plane RAP

The origin O of the (r, θ, z) cylindrical coordinatesystem used in Appendix A (figure 1a) lies atthe base of the circular reservoir and coincideswith its center. The z-axis points down into theONHCEM. The concept of a reference axial plane(RAP) defined by the z-axis and the initial line ofthe θ coordinate is convenient.Without loss of generality, the strike of every

normal fault considered in the following sectionsis normal to the RAP. Then, the intersection ofthe fault and the RAP, shown schematically infigure 1(c), is a dip-parallel line of symmetry (LOS)in the fault. Maximum influence of the circularreservoir on a buried fault is along the latter’s LOS.

3. Reservoir influence at a pointin the P layer

The theory of Appendix A allows us to computereservoir influence in terms of stress componentsσzz, σrr, σθθ, τrz = τzr and pore pressure p at anypoint in the ONHCEM. The remaining two pairs of

shear stress components are zero because of axialsymmetry. We present in table 2, as an example,estimates of the stress components and pore pres-sure at (3000 m, 0◦, 5000 m) in the P layer. Thevalue of a stress component at such a point is thesum of: (1) a time independent elastic contributiondue to H ′ (column 2 of table 2) and (2) an oscil-latory porous-elastic contribution due to h cos(ωt)(column 3). Pore pressure at the point arises dueto h cos(ωt) only. The total reservoir influence atthe point (3000 m, 0◦, 5000 m) is obtained bycombining results from second and third columnsof table 2 using specific values of H ′ and h. Thus,for example,

σzzP (3000 m, 0◦, 5000 m)=H ′(3818) + h(4605)

× cos(ωt+ 0.2◦) Pa.

The small values of phases listed in the thirdcolumn of table 2 indicate that stresses are verynearly in phase and the pore pressure is similarlyin opposite phase to oscillating water level.

4. Reservoir influence on an extended areaof a fault in ONHCEM

The reservoir influence on a fault in the ONHCEMwill be in the form of normal stress (σ) in E and Hlayers and as effective-normal stress (σ′ = σ − αp,see Appendix B) in the P layer. There will also beshear stress components τd and τs along dip andstrike directions of the fault for points in everylayer. These stress components may be computedfor each point on a fault using such stress and porepressure data as given in table 2. The 3D formulasfor transformation of stress components (e.g., Love1944) are required.


Figure 2. A schematic view of a dipping fault plane when we look down on it normally. See section 4 for full details. Thearrows near the points 1 to 9 show dip and strike components of the reservoir induced shear stress along the fault. Exactvalues given in table 3(c and d) are shown here schematically. The rock in the hanging wall of the fault at a point will tendto move in the direction of an arrow under reservoir influence.

Table 3(a). Coordinates of points 1–9 shown schematically in figure 2.

1 (6647 m,−37.0◦, 1000 m) 2 (5309 m, 0◦, 1000 m) 3 (6647 m, 37.0◦, 1000 m)

4 (5767 m,−43.9◦, 3000 m) 5 (4155 m, 0◦, 3000 m) 6 (5767 m, 43.9◦, 3000 m)

7 (5000 m,−55.1◦, 5000 m) 8 (3000 m, 0◦, 5000 m) 9 (5000 m, 55.1◦, 5000 m)

Table 3(b). Values of σ and σ′ (MPa) at points 1–9 offigure 2.

1 (σ = 0.049) 2 (σ = 0.172) 3 (σ = 0.049)

4 (σ = 0.185) 5 (σ = 0.386) 6 (σ = 0.185)

7 (σ′ = 0.153) 8 (σ′ = 0.252) 9 (σ′ = 0.153)

We show in figure 2, a 3 × 3 grid of points on a60◦ fault in the E and P layers. The points, num-bered 1–9, span an area of 8 km along fault strikeand about 4.6 km along fault dip. Points 2, 5 and8 lie on the LOS of the fault. The remaining sixpoints lie in symmetrical positions on either sideof LOS. Two rows of points lie in the E layer andone row lies in the P layer. The reservoir is in thehanging wall of the fault as far as these points areconcerned. The coordinates of the points are listedin table 3(a). For the sake of continuity in discus-sion, point 8 of the grid is also the point for whichstresses and pore pressure are listed in table 2.We list in table 3(b, c and d), numerical values

of resolved stress components acting on the faultat the nine grid points. Positive values of entriesin these tables indicate respectively that σ or σ′ iscompressive, τd promotes thrusting and τs promotesleft-lateral strike-slip effect on the fault plane.A schematic view of the shear stress components

at the grid points is shown in figure 2. We learnfrom this figure and the tables that σ′ or σ andτd have maximum magnitudes and τs is zero alongthe LOS of the fault. This reservoir influence interms of stresses on the dipping fault is the sameregardless of whether the fault is dip-slip normalor dip-slip reverse/thrust, etc.

Table 3(c). Values of τd (MPa) at points 1–9 of figure 2.

1 (+0.003) 2 (−0.024) 3 (+0.003)

4 (−0.075) 5 (−0.224) 6 (−0.075)

7 (−0.129) 8 (−0.235) 9 (−0.129)

Table 3(d). Values of τs (MPa) at points 1–9 of figure 2.

1 (+0.023) 2 (0.0) 3 (−0.023)

4 (−0.089) 5 (0.0) 6 (+0.089)

7 (−0.088) 8 (0.0) 9 (+0.088)

5. Review of reservoir-induced stabilityof normal faults

Net earthquake-related influence on a fault isquantified using the concept of fault stabilityS(ωt). An earthquake will occur at a point on thefault near a reservoir at phase ωt if

S(ωt) = ST (ωt) + SR(ωt) = 0.

Here, ST (ωt) and SR(ωt) are contributions to S(ωt)due to ambient tectonic and reservoir influencesrespectively. S(ωt) is positive at all times butdecreases to zero at the time of the earthquake.SR(ωt) for a point on a pure dip-slip normal

fault in the ONHCEM is defined by the followingrelations (e.g., after Roeloffs 1988),

SR(ωt) = [σ(ωt)− αp(ωt)] tanφ+ τd(ωt)

= σ′(ωt) tanφ+ τd(ωt),


at a point in the P layer;

SR(ωt) = σ(ωt) tanφ+ τd(ωt),

at a point in E or H layer.The following expressions have been used by us

for σ and τd.

σ = σzz cos2 δ + σrr sin

2 δ + 2τrz cos δ sin δ

τd = (σrr − σzz) sin δ cos δ ± τrz(cos2 δ − sin2 δ).

The second expression is written for a normal faultdipping down from right to left in cross-section.The ‘+’ or ‘–’ sign is to be used according as thecoordinates of the point on the fault are (r, 0◦, z)or (r, 180◦, z).All these expressions are based on the assumptions

that compressive stresses are positive and posi-tive shear stress opposes slip on the normal fault.We assume a nominal value of tanφ = 0.6 forfault friction coefficient. A positive value of SR(ωt)implies stabilizing reservoir influence on normalfaults considered here and vice versa.

6. SR(ωt) for two locationsof a 60◦ normal fault

We consider reservoir influence on a 60◦ normalfault passing through the point (3000 m, 180◦,5000 m) in such a way that the reservoir is mostlyon its footwall (e.g., Roeloffs 1988; Talwani 1997).Fault 1 in figure 1(b) is a schematic representationof this situation. The graph in figure 3(a) showsvariation of SR(ωt) at this point on this faultover one complete cycle of water level fluctuation.SR(ωt) is positive and this normal fault is under astabilizing reservoir influence throughout the year.In other words, the reservoir actually opposes theoccurrence of an earthquake on the fault continu-ously. The earthquake is most likely at the pointat LRL because the stabilizing reservoir influenceis the weakest at that epoch. The earthquake willoccur at a given epoch of LRL only if the destabi-lizing tectonic influence equals the stabilizing reser-voir influence in magnitude at that epoch. Welearn from figure 3(a) keeping table 2 in mind thatfault stability increases and decreases in phase withreservoir level.We note here, for comparison in section 8.5,

that the computed numerical values of contribu-tions to fault stability due to steady reservoir depthH ′ = 110 m is +3.21 × 10−2 MPa, stabilizing.The corresponding value at LRL for h = 3 m is−8.60× 10−4 MPa, destabilizing. The net stabiliz-ing contribution at LRL is +3.12× 10−2 MPa.Figure 3(b) pertains to a 60◦ normal fault

passing through the point (3000 m, 0◦, 5000 m).Schematically, this fault would lie to the right of

0.0605

0.061

0.0615

0.062

0.0625

0.063

0.0635

0.064

0.0645

0 60 120 180 240 300 360

FA

ULT

ST

AB

ILIT

Y

(MP

a)

RESERVOIR PHASE (DEGREES)

−0.089

−0.088

−0.087

−0.086

−0.085

−0.0840 60 120 180 240 300 360

FA

ULT

ST

AB

ILIT

Y (

MP

a)

RESERVOIR PHASE (DEGREES)

(a)

(b)

Figure 3. (a) Variation of reservoir induced stability SR(ωt)of a 60◦ normal fault for one annual harmonic reservoircycle. Fault 1 of figure 1(b) represents this fault schemat-ically in orientation and general location. The results arefor the point (3000 m, 180◦, 5000 m) on the fault in the Player. The reservoir influence is stabilizing throughout theyear. See section 6 of text. (b) SR(ωt) for a fault describedin third paragraph of section 6. The destabilizing reservoirinfluence is maximum when ωt is 0◦ and 360◦ (i.e., HRL)and minimum when ωt is 180◦ (i.e., LRL).

and parallel to Fault 2 of figure 1(b), so that theentire reservoir is in its hanging wall. Since SR(ωt)is negative over the entire reservoir cycle, the faultis under a destabilizing reservoir influence through-out the year. The influence is most destabilizing atepochs of HRL (ωt = 0◦, 360◦, etc.) and an earth-quake is most likely at such epochs. The earth-quake will occur if the magnitudes of destabilizingreservoir influence at an epoch of HRL and stabiliz-ing tectonic influence at that epoch are equal. Thereservoir influence actively promotes the occur-rence of the earthquake in this case. We emphasizethat in this case the destabilizing reservoir influ-ence increases as the water level rises in thereservoir and vice versa.

7. Further results on variations in SR(ωt)for normal faults

Figure 4(a) shows magnitude of SR at LRL atseveral points on the Fault 1 of figure 1(b). The


−5.55E-17

0.04

0.08

0.12

0.16

0 1000 2000 3000 4000 5000

FA

ULT

ST

AB

ILIT

Y (

MP

a)

−0.2

−0.15

−0.1

−0.05

0

0.050 1000 2000 3000 4000 5000

FA

ULT

ST

AB

ILIT

Y

(MP

a)

DISTANCE ALONG DIP LINE OF FAULT (m)

(a)

(b)

Figure 4. (a) Variation of SR at LRL with distance alonga fault, which is similar to Fault 1 of figure 1(b). The dis-tance is measured up dip from the point with coordinates(3500 m, 180◦, 5000 m). The reservoir influence is stabilizingat all points considered. (b) Variation of SR at LRL withdistance along a fault which is similar to Fault 2 of figure1(b). The distance is measured up dip from the point withcoordinates (500 m, 0◦, 5000 m). The reservoir influence isdestabilizing at most points considered, but stabilizing atthe right most point. Both faults dip at 60◦.

origin of the horizontal axis corresponds to point(3500 m, 180◦, 5000 m) and the dip of the fault is60◦. We learn that the reservoir influence is stabili-zing at all points considered. Figure 4(b) is similarbut for Fault 2 of figure 1(b). The origin of thehorizontal axis corresponds to point (500 m, 0◦,5000 m) and the dip of the fault is 60◦ again. Welearn that the reservoir influence is destabilizing atmost but not all points are considered.Figure 5(a) is the visualization of a 60◦ normal

fault dipping to the left at different horizontalpositions in the P layer under the reservoir. Thephrase ‘reservoir mostly on the footwall of thenormal fault’ applies fairly for fault positions atthe extreme left of the figure. Figure 5(b) depictscomputed values of reservoir induced stabilizingor destabilizing influence on the fault. All calcu-lations are for LRL. Points marked −4000 m and4000 m on the horizontal axis have coordinates(4000 m, 180◦, 5000 m) and (4000 m, 0◦, 5000 m)respectively. Earthquakes at LRL on 60◦ normal

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

−4000 −3000 −2000 −1000 0 1000 2000 3000 4000

FA

ULT

ST

AB

ILIT

Y (

MP

a)

DISTANCE FROM Z-AXIS (m)

60 FAULT (in degree) 45 FAULT (in degree)

(a)

(b)

Figure 5. (a) Schematic representation of the same 60◦ nor-mal fault at several possible horizontal positions in the Player. (b) Variation of SR at LRL with horizontal positionof the fault. See sections 7 and 8.2 for details. The pointmarked −4000 m on the horizontal axis has coordinates(4000 m, 180◦, 5000 m). A similar curve for 45◦ normal faultis included (see section 8.2).

faults will be favoured for positions approximatelyin the range (4000 m > r > 2200 m, 180◦, 5000 m).Results for a fault dipping at 45◦ to the left are

included in figure 5(b) (see section 8.2). It emergesthat earthquakes at LRL will be favoured on thisfault for positions between (2300 m < r < 4000 m,0◦, 5000 m) also. The reservoir is on the hangingwall of the fault at these positions. We trace thisresult to the specific nature of the dependence of σand τd on fault dip δ (see sections 5 and 8.2).

8. Discussion

8.1 About NHCEM

The theory of Appendix A deals with a three-layeredNHCEM. Additional horizontal elastic layers maybe included in the earth model without conceptualchange in the theory. Similarly, more poroelasticlayers may be included also. But each poroelasticlayer must be sandwiched between two imperme-able elastic layers. This ensures that water does notflow in or out of the poroelastic layer perpendicu-larly across its upper and lower boundaries.

8.2 Possibility of normal fault earthquakesin ONHCEM at low reservoir levels

Having adopted the value of 4000 m for the radiusof the uniform circular model of Lake Mead, we


have confined our attention in the reported resultsto points whose r-coordinates have values less than4000 m because we were concerned about ratio-nalization of ‘normal fault earthquakes under LakeMead’.Figures 3(a), 5(b) and the discussion of section

6 indicate that, during steady state reservoir oper-ation, at least at some points under the reservoir,earthquakes can occur at LRL on a 60◦ normalfault in the P layer of the ONHCEM. The reser-voir should be located mostly on the footwall of thenormal fault passing through such a point. For thesequence of 60◦ normal faults shown schematicallyin figure 5(a), the results shown in figure 5(b) indi-cate that faults near the left end of the depictedhorizontal axis would be suitable hosts for suchearthquakes.Results for normal faults dipping at 45◦ are

included in figure 5(b) because Rogers and Lee(1976) noted from the analysis of seismological datathat LMEQs could have occurred on such faults.We learn that earthquakes can occur at epochs ofLRL on 45◦ normal faults in ONHCEM, even whenthe reservoir is on the hanging wall of the fault.Finally, it is expected from the theory of

section 5, and may be seen from figure 5(b) that thenature of reservoir influence on normal faults pass-ing through the same subsurface point and hav-ing the same dip direction varies significantly withthe amount of fault dip. For example, at the pointwith coordinates (3000 m, 0◦, 5000 m), the influenceis destabilizing when δ = 60◦ and stabilizing whenδ = 45◦.The influence of a circular reservoir on other

types of faults, such as reverse/thrust dip slip, ver-tical strike slip, or general oblique slip on dippingfaults, in the P layer of the ONHCEM may bededuced similarly.

8.3 Comparison of normal fault-related results forONHCEM with published results for LMEQs

LMEQs started occurring several years after theimpoundment of Lake Mead was initiated. Butthey were observed for more than two decadesthereafter (Roeloffs 1988). Roeloffs (1988) con-sidered the long history of LMEQs as a steadystate effect of sustained reservoir operation. Sheanalyzed these observations with a FHCEM, com-prising a single poroelastic layer of semi-infinitethickness. She inferred that at the focus of such anearthquake in FHCEM, the reservoir influence dueto the mean water depth should be stabilizing andthat due to water oscillations should be destabiliz-ing. The numerical values of fault stability quotedin section 6 for ONHCEM display these featuresprecisely. In other words, the occurrence of LMEQsunder sustained, periodic operation of Lake Mead

can be explained using the ONHCEM also. Thisconforms to the general geophysical experience thatmore than one earth model may explain all, or afraction, of a given set of geophysical observations.Talwani and Acree (1985) estimated hydraulic

diffusivity c ‘within an order of magnitude of5 m2 s−1’ under reservoirs exhibiting induced seis-micity. According to Roeloffs (1988), explanationof LMEQs using the FHCEM requires that itsmaterial should have low value of c. Her estimatefor the Lake Mead region is between 0.01 and0.26 m2 s−1. The value of c = 0.04 m2 s−1 used forONHCEM was fixed when we arbitrarily adoptedOhio sandstone as the material of the P layer.

9. Conclusion

Investigations of post-impoundment earthquakesnear a new man-made reservoir should be initi-ated keeping in mind that, depending on local geo-logical conditions, water from the reservoir mayor may not seep to earthquake depths. Chancesof the second possibility increase with increase infocal depths of earthquakes. We may assess thesituation through site investigations, simulations,or a combination of both. The expressions derivedhere for stresses and pore pressure may be usedwith the simulation approach. On the basis of ouranalysis of some Lake Mead earthquakes, we con-clude that results from the proposed three-layered,no-hydraulic-connection earth model may be com-bined with those from the widely-used, single-layered, full-hydraulic-connection earth model forcomprehensive interpretation of observations.

Acknowledgements

We appreciate the constructive comments of thetwo reviewers. We acknowledge, in particular, thesuggestion of one reviewer to look at Talwani(1997) also. This simplified the analysis of LMEQsusing ONHCEM significantly.

Appendix A

Response of NHCEM to a uniform circularload on its top surface

We solve the axially symmetric problem ofdetermining the influence of a uniform circular loadon a NHCEM. The basic theoretical framework foraxial symmetry in a poroelastic medium is pro-vided by Wang (2000, sections 9.1 and 9.6). It isadopted here with minor adjustments for cos(ωt)time variations. The theoretical framework for theE and H layers is obtained by suitable simpli-fication. The dependent variables are treated as


complex-valued quantities in sections A1 to A5dealing with h cos(ωt) component of water depth.The response of NHCEM to the H ′ component isconsidered in section A6.

A1. Mathematical statement of the problem

A1.1 Constitutive relations

The following constitutive equations hold for the Player (Wang 2000, equations 2.50, 2.53 and 2.68)in the axially symmetric case.

σzzP = 2GP

∂uzP

∂z+

2GPνuP1− 2νuP

εP −BPKuP ζP , (1)

τrzP = GP

(∂uzP

∂r+

∂urP

∂z

), (2)

pP = −BPKuP εP +BPKuP

αP

ζP , (3)

σ′zzP = σzzP + αPpP . (4)

The constitutive equations for the E and H layersare, with j = E, H:

σzzj = 2Gj

∂uzj

∂z+

2Gjνj1− 2νj

εj, (5)

τrzj = Gj

(∂uzj

∂r+

∂urj

∂z

). (6)

A1.2 Field equations

The following field equations hold for the P layer(Wang 2000, equations 4.25, 4.67, and 9.8).

∇2ζP =1

cP

∂ζP∂t

, (7)

∇2εP = γP∇2ζP , (8)

GP

(∂2urP

∂r2+

1

r

∂urP

∂r− urP

r2+

∂2urP

∂z2

)

+Gp

1− 2νuP

∂εp∂r

−BPKuP

∂ζP∂r

= 0, (9)

GP

(∂2uzP

∂r2+

1

r

∂uzP

∂r+

∂2uzP

∂z2

)

+Gp

1− 2νuP

∂εp∂z

−BPKuP

∂ζP∂z

=0. (10)

The simpler field equations for the E and H layersare, again with j = E, H:

∇2εj = 0, (11)

(∂2urj

∂r2+1

r

∂urj

∂r−urj

r2+∂2urj

∂z2

)+

1

1− 2νj

∂εj∂r

= 0,

(12)

(∂2uzj

∂r2+

1

r

∂uzj

∂r+

∂2uzj

∂z2

)+

1

1− 2νj

∂εj∂z

= 0.

(13)

A1.3 Boundary conditions

The following four sets of boundary conditions holdfor all r and t.

(1) As z → ∞ urH , uzH , τrzH , σzzH → 0.(2) At the lower boundary of the P layer

urH = urP , uzH = uzP ,

τrzH = τrzP , σzzH = σ′zzP ,

∂pP∂z

=0.

(3) At the upper boundary of the P layer

urP = urE, uzP = uzE,

τrzP = τrzE, σ′zzP = σzzE,

∂pP∂z

=0.

(4) At the top boundary of the E layer

τrzE = 0,

and

σzzE =

{Re[−ρghest], r ≤ rR,0, r > rR.

The Hankel transform of σzzE (Sneddon 1951;Singh et al. 2009) is:

σ0zzE =

−ρghestrRJ1(krR)

k.

The inverse Hankel transform is:

σzzE = −ρghest∫ ∞

0

(rRk

)J1(rRk)J0(rk)kdk.

A1.4 Problem solved

We discuss in sections A2–A4 evaluation of dis-placement components ur and uz, and stress com-ponents σzz and τrz, in E, P and H layers andpore pressure p in the P layer. We evaluate stresscomponent σrr in section A5.

A2. Displacements, stresses and porepressure in E, P and H layers

A2.1 The P layer

The following equations involving six unknownconstants AP–FP are based on Wang (2000,


equations 9.42, 9.46, 9.47, 9.48, 9.50 and2.68).

urP (r, z, t, s) = est∫ ∞

0

[FP e

−kz+EP ekz

+ b1uP (−DP e−kz+CP e

kz)

− ckγPs

(AP emz+BP e

−mz)

]

× kJ1(kr)dk, (14a)

uzP (r, z, t, s) = est∫ ∞

0

[FP e

−kz − EP ekz

−(b1uP z +

b2uPk

)DP e

−kz

−(b1uP z −

b2uPk

)CP e

kz

+cmγPs

(AP e

mz −BP e−mz

) ]

× kJ0(kr)dk, (14b)

τrzP (r, z, t, s) = 2GP est

∫ ∞

0

[−FP e

−kz + EP ekz

+ (b1uPkz + 0.5)DP e−kz

+ (b1uPkz − 0.5)CP ekz

− ckmγPs

(AP emz −BP e

−mz)

]

× kJ1(kr)dk, (14c)

σ′zzP (r, z, t, s) = σzzP (r, z, t, s) + αPpP (r, z, t, s)

= 2GP est

∫ ∞

0

[−FP e

−kz − EP ekz

+ (b1uPkz + b4dP )DP e−kz

− (b1uPkz − b4dP )CP ekz

+

(ck2

s+ b4dP

)

× γP (AP emz +BP e

−mz)

]

× kJ0(kr)dk, (14d)

σzz(r, z, t, s) = 2GP est

∫ ∞

0

[−FP e

−kz − EP ekz

+(b1uPkz + b4uP )DP e−kz

−(b1uPkz − b4uP )CP ekz

+ck2γP

s(AP e

mz +BP e−mz)

]

× kJ0(kr)dk, (14e)

pP (r,z, t, s) = 2GP est

∫ ∞

0

[−γP b4uP (DP e

−kz+CP ekz)

+ γP b5P (AP emz +BP e

−mz)]kJ0(kr)dk. (14f)

Equation (14d) based on (14e–f) is written aftersimplification.

A2.1.1 Estimate of Ap and Bp

We use the two boundary conditions relating tothe pore pressure gradient (see equations 2 and 3)to evaluate AP and BP in terms of CP and DP .Thus, the term γP (AP e

mz ± BP e−mz) appearing

repeatedly in equation (14a–f) take the followingforms.

γP (AP emz ±BP e

−mz) = αD±DP + αC±CP . (15)

Here,

αD±=q(emdP+mz−ekdP+mz)±q(e−mdP−mz−ekdP−mz),

αC± = q(e−kdP+mz − emdP+mz)

± q(e−kdP−mz − e−mdP−mz),

q =γP b4uPk

b5P (e−mdP − emdP ).

A2.1.2 Revised expressions for P layer

Equations (14a–f) may be rewritten now in termsof four unknown constants.

urP (r, z, t, s) = est∫ ∞

0

[FP e

−kz + EP ekz

+

(−b1uP z −

ck

sαD+

)DP e

−kz

+

(b1uP z −

ck

sαC+

)CP e

kz

]

× kJ1(kr)dk, (16a)

uzP (r, z, t, s) = est∫ ∞

0

[FP e

−kz − EP ekz

+

(−b1uP z −

b2uPk

+cm

sαD−

)DP e

−kz

−(−b1uP z +

b2uPk

+cm

sαC−

)CP e

kz

]

× kJ0(kr)dk, (16b)


τrzP (r, z, t, s) = 2GP est

∫ ∞

0

[−FP e

−kz + EP ekz

+

(b1uPkz + 0.5− ckm

sαD−

)DP e

−kz

+

(b1uPkz − 0.5− ckm

sαC−

)CP e

kz

]

× kJ1(kr)dk, (16c)

σ′zzP (r, z, t, s) = 2GP e

st

∫ ∞

0

[−FP e

−kz − EP ekz

+

{b1uPkz + b4dP +

(ck2

s+ b4dP

)αD+

}DP e

−kz

+

{− b1uPkz + b4dP +

(ck2

s+ b4dP

)αC+

}CP e

kz

]

× kJ0(kr)dk, (16d)

σzz(r, z, t, s) = 2GP est

∫ ∞

0

[−FP e

−kz − EP ekz

+

(b1uPkz + b4uP +

ck2

sαD+

)DP e

−kz

+

(−b1uPkz + b4uP +

ck2

sαC+

)CP e

kz

]

× kJ0(kr)dk, (16e)

pP (r, z, t, s) = 2GP est

∫ ∞

0

×[(−γP b4uP + b5PαD+)DP e

−kz

+(−γP b4uP + b5PαC+)CP ekz]

× kJ0(kr)dk. (16f)

A2.2 The E layer

The corresponding expressions for the E layer are:

urE(r, z, t, s) = est∫ ∞

0

[FEe

−kz + EEekz

+ b1E(−DEe−kz+CEe

kz]

× kJ1(kr)dk, (17a)

uzE(r, z, t, s) = est∫ ∞

0

[FEe

−kz − EEekz

−(b1Ez +

b2Ek

)DEe

−kz

−(b1Ez −

b2Ek

)CEe

kz

]

× kJ0(kr)dk, (17b)

τrzE(r, z, t, s) = 2GEest

∫ ∞

0

[−FEe

−kz + EEekz

+ (b1Ekz + 0.5)DEe−kz

+(b1Ekz − 0.5)CEekz]

× kJ1(kr)dk, (17c)

σzzE(r, z, t, s) = 2GEest

∫ ∞

0

[−FEe

−kz−EEekz

+ (b1Ekz + b4E)DEe−kz

− (b1Ekz − b4E)CEekz]

× kJ0(kr)dk. (17d)

A2.3 The H layer

Finally, similar expressions for the H layer:

urH(r, z, t, s) = est∫ ∞

0

[FHe

−kz + EHekz

+ b1H(−DHe−kz + CHe

kz)]

× kJ1(kr)dk, (18a)

uzH(r, z, t, s) = est∫ ∞

0

[FHe

−kz − EHekz

−(b1Hz +

b2Hk

)DBe

−kz

−(b1Hz −

b2Hk

)CHe

kz

]

× kJ0(kr)dk, (18b)

τrzH(r, z, t, s) = 2GHest

∫ ∞

0

[−FHe

−kz + EHekz

+ (b1Hkz + 0.5)DHe−kz

+(b1Hkz − 0.5)CHekz]

× kJ1(kr)dk, (18c)

σzzH(r, z, t, s) = 2GHest

∫ ∞

0

[−FHe

−kz − EHekz

+ (b1Hkz + b4H)DHe−kz

− (b1Hkz − b4H)CHekz]

× kJ0(kr)dk. (18d)

The boundary conditions in equation (1) of A1.3imply that CH = EH = 0. Equations (18a–d) thusinvolve only DH and FH .


A3. Determination of the remainingunknown constants

A3.1 Expressions in matrix form

We switch to the Hankel transform domain. Equations(16a–d), (17a–d) and (18a–d) are pertinent. Let

[u1rj, u

0zj, τ

1rzj, σ

0zzj]

T (k, z, t, s)

=[u1rj(k, z, t, s), u

0zj(k, z, t, s), τ

1rzj(k, z, t, s),

σ0zzj(k, z, t, s)

]T.

Then, with subscript j = E,P and H in turn,we have

[u1rE, u

0zE, τ

1rzE, σ

0zzE]

T (k, z, t, s)

= estZE(k, z, s) · [FE, EE, DE, CE]T , (19a)

[u1rP , u

0zP , τ

1rzP , σ

′0zzP ]

T (k, z, t, s)

= estZP (k, z, s) · [FP , EP , DP , CP ]T , (19b)

[u1rH , u

0zH , τ

1rzH , σ

0zzH ]

T (k, z, t, s)

= estZH(k, z, s) · [FH , EH , DH , CH ]T . (19c)

ZE(k, z, s),ZP (k, z, s), and ZH(k, z, s) are 4 × 4matrices of coefficients of the constants in equa-tions (17a–d), (16a–d) and (18a–d) respectively.

A3.2 Evaluation of unknown constants

We follow Singh and Rani (2006) at this stage,and note that the values of variables at the topof E layer can be expressed in terms of arbitraryconstants related to H layer through the followingmatrix relations by using the boundary conditionsin equations (1), (2) and (3) of A1.3.

[u1rET , u

0zET , τ

1rzET , σ

0zzET ]

T (k, 0, t, s)

= estJ(k, s) · [FH , 0, DH , 0]T , (20)

where

J(k, s) = AE(dE) ·Ap(dp) · ZHT ,

AE(dE) = ZET · Z−1Eb , and Ap(dp) = ZPT · Z−1

Pb .

Subscripts b and T stand for base and top of theconcerned layer respectively. The J and A matri-ces are also 4 × 4 matrices. We use a numericalprocedure to evaluate the inverse matrices.The third and fourth equations contained in the

matrix relation (20) can be combined with equation(4) of A1.3 to yield

J31FH + J33DH = 0,

J41FH + J43DH =−ρghestrRJ1(krR)

k.

These equations are solved for DH and FH . Hence,with Δ = J31J43 − J33J41,

[FH , 0, DH , 0]T =

−ρghestrRJ1(krR)

k

×[J33

Δ, 0,−J31

Δ, 0

]T

, (21a)

[FP , EP , DP , CP ]T = Z−1

Pb · ZHT [FB, 0, DB, 0]T ,(21b)

[FE, EE, DE, CE]T =Z−1

EbAP (dP)·ZHT [FB, 0, DB, 0]T .

(21c)

The values of constants so obtained may be substi-tuted into equations (18a–d), (16a–f) and (17a–d).

A4. Results after evaluation of inverseHankel transform integrals

The inverse Hankel transform integrals in equations(17a–d), (16a–f) and (18a–d) are evaluated numer-ically. The result in each case is complex-valued.It may be written in polar form whose real partcan be retained for further use. For example, theresult from evaluation of equation (16e) will be ofthe form

σzzP (ωt) =�[AσzzP

e(st+ ιφσzzP)]

=AσzzPcos(ωt+ φσzzP

).

A5. Evaluation of normal stressesσrrP and σθθP

We follow for σrrP , the same procedure as usedby Wang (2000, section 9.6) to obtain expressionsfor σzzP and τrzP from those for urP and uzP withthe help of the constitutive equations. Thus, aftersome simplification, we have in continuation withequations (14a–f),

σrrP (r, z, t, s) = 2GP

∫ ∞

0

[k(EP e

kz + FP e−kz)

+ b1uP (−kz + 2νP )DP e−kz

+ b1uP (kz + 2νuP )CP ekz

+

(ck2

s− 1

)γP (AP e

mz +BP e−mz)

]J0(kr)kdk

− 2GP

1

r

∫ ∞

0

[(EP e

kz + FP e−kz)

+b1uP z(CP ekz −DP e

−kz)

+ck

sγP (AP e

mz +BP e−mz)

]J1(kr)kdk. (22)


Alternately, in conformity with equations (16a–f),

σrrP (r, z, t, s) = 2GP

∫ ∞

0

[k(EP e

kz + FP e−kz)

+

{b1uP (kz + 2νuP ) +

(ck2

s− 1

)αC+

}

×CP ekz +

{b1uP (−kz + 2νuP )

+

(ck2

s− 1

)αD+

}DP e

−kz

]J0(kr)kdk

− 2GP

1

r

∫ ∞

0

[{(b1uP z +

ck

sαC+

)CP e

kz

+

(−b1uP z +

ck

sαD+

)DP e

−kz

}

+(EP ekz + FP e

−kz)

]J1(kr)kdk. (23)

Equation (23) can be evaluated using equation(21b).For evaluating σθθP , using the definition of mean

normal stress σMP ,

σθθP (r, z, t, s) = 3σMP (r, z, t, s)− σzzP (r, z, t, s)

− σrrP (r, z, t, s).

From equation (2.24) of Wang (2000),

σMP (r, z, t, s) = KdP εP (r, z, t, s)− αPpP (r, z, t, s).

Also, from equation (9.38) of Wang (2000),

εP (r, z, t, s) =

∫ ∞

0

[(DP e

−kz + CP ekz)

+ γ(AP emz +BP e

−mz)]J0(kr)kdk.

Expressions for σzzP , σrrP and pP have beenderived already. The constants in the integral forεP have also been evaluated above.

A6. Evaluation of the response of NHCEMto mean water load

The above theory for h cos(ωt) may be used toevaluate the response to H ′ by setting ω = 0 andνuP = νdP . The second step removes all the termsrepresenting the porous-elastic effects in the P layerfrom the above expressions. The remaining termsrepresent only the elastic effects due to a constantload on NHCEM. The results in table 1 of the maintext are obtained in this way from our computerprogram for evaluating the response at focus dueto h cos(ωt).

A7. Sign convention

The above derivations are in conformity with thesign convention adopted by Wang (2000), where

compressive normal stress is considered numericallynegative. The results quoted in the main textconform to the sign convention of Jaeger andCook (1969), where compressive normal stress isconsidered positive.

Appendix B

Symbols

b1j =1

2(1− 2νj), j = E, H

b2j =3− 4νj

2(1− 2νj), j = E, H

b4j =1− νj

(1− 2νj), j = E, H

b1jP =1

(2(1− 2νjP ), j = u (undrained) or d

(drained)

b2jP =3− 4νjP

2(1− 2νjP ), j = u or d

b4jP =1− νjP

(1− 2νjP ), j = u or d

b5P =1− νdP

αP (1− 2νdP )

cP : hydraulic diffusivity in P layer;dj : thickness of j = E, P, and H layersh: amplitude of annual oscillation in water depthk: wave number along r direction

m =

√k2 +

s

cP

p: pore pressurepP : pore pressure at a point in P layerr: radial coordinates = ιωt: timeur, uz: displacement components along r and zdirectionsz: coordinate along z-axisAj , Bj, Cj , Dj, Ej, Fj : arbitrary constants in thegeneral solution of field equations for j = E, P, HlayersBP : Skempton’s coefficient in P layer;

BP =3(νuP − νdP )

α(1− 2νdP )(1 + νuP )(Wang 2000)

E, P or H: the elastic, porous-elastic or half-spacelayerGj : shear modulus of j = E, P and H layersH ′: annual mean depth of water in the reservoirJ0, J1: Bessel Functions of order 0, 1KuP : undrained bulk modulus in P layer;

KuP =2GP (1 + νuP )

3(1− 2νuP )


KdP : drained bulk modulus in P layer;

KdP =2GP (1 + νdP )

3(1− 2νdP )S(ωt): fault stability at reservoir phase ωtSR(ωt): fault stability under reservoir influence atphase ωtST (ωt): fault stability under tectonic influence atphase ωtαP : Biot–Willis coefficient in P layer

γP =νuP − νdP

α(1− 2νdP )(1− νuP )(Wang 2000)

εj : volumetric strain in j = E, P and H layersζP : increment of fluid content in P layerνj: Poisson’s ratio of j = E and H layersνjP , j = u or d: undrained or drained Poisson’sratio in P layerφ: angle of rock frictionσ: normal stress componentσMP : mean normal stress in P layerσrrj: radial normal stress component in j = E, P,H layersσzzj: vertical normal stress component in j = E, P,H layersσ′: effective normal stress (= σ ± αp, where σ is anormal stress component; + sign for Appendix Aand − sign for main text. For use of factor α, seeWang (2000)τrzj: shear stress component in j = E, P, H layerτs: reservoir-induced resolved shear stress on faultplane along strikeτd: reservoir-induced resolved shear stress on faultplane along dipω: angular frequency of reservoir operation

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MS received 12 January 2016; revised 27 July 2016; accepted 1 August 2016

Corresponding editor: Pawan Dewangan

ona model simulatinglack of hydraulicconnection betweena

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