Chapter 6

TWO-DIMENSIONAL FINITE DIFFERENCE CALCU- LATIONS OF DYNAMIC IN-SITU RESPONSE OF

LAYERED GEOLOGIC MEDIA TO A LARGE EXPLOSIVE LOAD

by Jimmie L. Bratton and Howard R. Pratt

The Air Force Weapons Laboratory (AFWL) has been carrying out a series of experiments whose principal purpose is to measure the dynamic response of rock media to large pressure pulses. Coupled with the field experiments have been theoretical calculations simulating this response. These two approaches allow an appraisal of the current ability to measure in-situ rock response under extreme conditions and also give a fair esti- mate of the current state-of-the-art in the area of computer calculation of the response of "real" geologic media to a large load.

The area chosen for the test series is composed of a sequence of sedi- mentary units (shale, sandstone, and limestone) located in the Estancia Valley 45 miles east of Albuquerque, N.M. (Fig. 1). The units, which dip gently off the uplifted Manzano Mountains to the west, are essentially horizontal a t the test site. An estensive geologic and geophysical investi- gation of the area was carried out in the form of surface mapping and seismic exploration. Numerous holes were cored and material properties were determined from tests of the cores. In-situ geophysical measure- ments were also carried out.

The experiment consisted of loading a 6 0 x 4 0 f t test area with a 4000-psi transient exponentially decaying pressure pulse. The pulse was initiated a t one end and traveled the 60-ft length of the test bed. The test area was instrumented with acceleration, velocity, and strain measuring devices placed in drill holes to depths of 30 ft .

Theoretical computer calculations of the response of the layered geo- logic media have been carried out on the A F W L CDC 6600 using one and

Jimmie L. Bratton and Howard R. Pratt are 1st Lieutenants, U.S. Air Force Weapons Laboratory, Civil Engineering Branch, Kirtland Air Force Base, N.M.

two-dimensional finite difference codes. The two-dimensional computer code, AFTON 2P, is a plane symmetric code used to solve transient con- tinuous motion problems. The measured material properties of the geo- logic media at the site are input into the code in the form of a subroutine.

The results from the experiment can be directly compared to those obtained using the computer code. The magnitude of results and geom-

etry of wave form are compared. Eventually, i t is hoped that a complete description of the in-situ response of rock media to a pressure pulse in various structural and stratigraphic configurations can be accomplished using the computer. A minimum number of field tests will then be re- quired to verify the calculated results.

SITE GEOLOGY

Geology

The bedrock a t the test site is part of the Pennsylvanian and Permian Magdalena Group. Only the Pennsylvanian Madera limestone formation was encountered during drilling operations. The Madera formation con- formably overlies the elastic Sandia formation. The Madera is divided into two members: a lower gray limestone member and an overlying arkosic limestone member. The gray limestone member consists mainly of a sequence of gray cherty limestones interbedded with gray and black calcareous shales. This member was encountered at a depth of approxi- mately 440 f t during drilling. The arkosic limestone member is the unit encountered at the test site and comprises the upper 440 ft. The unit gen- erally consists of interbedded arkosic sandstones and siltstones, red shales, quartz pebble conglomerates, gray limestones, and gray calcareous shales and siltstones.

A geologic map of the area (Fig. 2) shows that the beds are generally horizontal (dip <3") throughout the area of interest. Nine holes have been drilled to depths greater than 35 ft,. Drill-hole data is listed in Table 1. All holes, ranging in diameter from NX to 42 in., were cored continu- ously for their entire depth. The maximum depth drilled was 710 ft. Location of the core holes is also given in Fig. 2 and a correlation chart of the units indicating the local stratigraphy is given in Fig. 3.

Seismic Characteristics

A detailed seismic investigation was also carried out. Surface refrac- tion, uphole, and cross-hole measurements were made. Electrical re- sistivity, density, .3D velocity, continuous velocity, nuclear, and caliper logs were run. A map plan of the surface refraction surveys is shown in Fig. 4. Typical results of the seismic survey (Fig. 5) and the in-situ geophysical measurements indicate that three zones are present to a depth of 1200 ft, the maximum depth penetrated by the survey. An upper zone, with a dilatational wave velocity (Vp) of 9500-10,000 fps extends to a depth of approximately 80 f t in the Hole 1 area (elevation 6417 f t ) . At a depth of 80 f t in Hole 1, the stratigraphy changes from a predomi-

LEGEND

Gray Limestone f i

Grayish brown . Siltstone and Shale

Gray Limestone and Shale

Red Shale, Arkose, and Sandstone

Location of Drill Holes

Scale 0 1000'

Pig. 2-Geologic map o f test area.

Table 1. Drill Hole Data

Size Depth No. (In.) ( F t ) Core Core Size Location

1 76 400 x 6 300 f t north of small quarry

2 79 100 x 6 Large quarry

3 3 710 x N X Large quarry, 3 f t from Hole 2

4 79 30 Y 6 Small quarry

5 79 40 x 6 Planewave I site

6 3 400 x NX SW corner of large quarry

7 3 50 x N X SW corner of large quarry, 10 f t from Hole 6. This is a 20" slant hole

8 42 37 part 41 N E corner of large quarry

9 - 7 1 x 35 x 6 Small quarry, 60 f t west of

Hole 4

Elev.

Hole 6 I I

Hole 3 Hole 1 Hole 9

+ 250'-500' - 300' -1

F i g . 3-Drill-hole correlation chart.

UTLINE OF LARGE QUARRY AREA

F i g . 4-Seismic survey l a y w t .

S P ~ S SPI S k 4 0 0 f 1 d

(OFFSET 1250 ft) HORIZONTAL DISTANCE

Fig . 5-Seismic uelocities.

nately red shale sequence to one of limestones and competent gray cal- careous shales. This middle zone with an average seismic velocity of 12,000 fps extended to a depth of 440 ft. Below 440 ft, a competent lime- stone section with minor gray, black, and red shales, and having a seismic velocity of approximately 15,000 fps, was encountered.

The results of an uphole survey near drill hole No. 8 (Fig. 6) indicated a consistent seismic velocity of 9700 fps for the upper 25 f t of material.

A detailed cross-hole survey was carried out to determine the compres- sional and shear wave velocities of the individual layers to a depth of 40 f t . The survey was carried out in the large quarry (Fig. 4 ) . The five- hole configuration shown in Fig. 7a was used. The three-dimensional downhole seismometer was placed in the central 400-ft-deep central NX hole. Small explosive charges were placed sequentially a t depths of 40, 30, 25, 20, 12.5, and 5 f t in each of the four holes. The depths given cor- respond to depths of 32, 22, 25, 20, 12, and 5 ft , respectively, in the area in which the test was carried out. The borehole seismometer was raised to the appropriate level (the depths indicated previously) for each sequence of shots. The total number of shots was 24. Three sequences were fired in the upper red shale, two in the conglomerate, and one in lower red shale.

The detailed lithology of the test area is shown in Fig. 7b. The average velocity of the three layers is as follows : (1) upper red shale, Vp= 8400 fps, shear wave velocity (Vs) = 3900 fps ; (2) conglomerate, Vp=7850,

UPHOLE HOLE NO. I TIME (rnsec)

UPHOLE HOLE NO. 4

Fig. 6-Uphole seismic velocities:

Vs=3600 ; (3) lower red shale, V p = 10,000, Vs=4800. Data from the survey is summarized in Fig. 7b.

Mechanical Properties of the Rock

In order to provide constitutive relations for the fin'ite difference cal- culations, a number of laboratory tests were conducted. These tests may be categorized into four types: (1) sonic wave velocity measurements, (2) static one-dimensional compression tests, (3) static unconfined com- pression tests, and (4) high-pressure one-dimensional wave propagation tests. The testing was conducted by the U.S. Army Engineers Waterway Experiment Station (WES), the E.H. Wang Civil Engineering Research Facility (CERF) , and Physics International ( P I ) under the direction of the AFWL.

The sonic tests involved the measurement of dilatational (P) and shear (S) wave velocities. These measurements were made on intact samples obtained from the drill holes shown in Figs. 2 and 3 and are summarized in Table 2. Although the samples were taken from different holes, they can be readily correlated with a specific depth in Hole 9. The depths indi- cated in Table 2 reference to elevation datum for Hole 9 (elevation= 6386).

Recording Bole

65' 6x: A Shot Bole

Lithology Survey Depth Vp Vs

Elevation 6386' ( f t ) ( f p s ) ( f p s )

Depth 0 8710 3850

Bed shale w/ siltstone lenses 4.5 7665 3850

12.0 8710 4090

Gray sandstone grading to quartz 17.0 7370 3150 pebble conglomerate at 17.5 feet - - - - - - . 22.0 8330 4000

siltstone lense at 27 feet

32.0 10060 4800

B Pig. 7-Cross-hole seismic survey configuration ( A ) and detailed

lithology at test site and cross-hole seismic results ( B ) .

The sonic tests were conducted in the standard manner on NX samples. They provided the basis for the dynamic elastic properties used in the calculation. The tabulated elastic constants were calculated from the measured propagation velocities and the known densities.

The one-dimensional compression tests were performed to provide infor- mation for the one-dimensional calculation. This is a static test in which the radial stress is increased in order to keep the radial strain zero during axial compression. The ratio of radial to axial stress is defined as K and is shown along with the constrained secant modulus (axial stress divided by axial strain for a condition of zero radial strain) a t an axial stress of 3000 psi in Table 3. From these data, the elastic constants a t 3000 psi

Table 2. Summary of Sonic Tests

Sample Depth

7

Sample Description

Shale Conglomerate Conglolnerate Limestone Limestone Limestone Limestone Sandstone Sandstone

Testing Agency

PI WES WES WES WES WES WES WES WES

Dilatational Velocity,

Fvs

Shear Young 's Velocity, Modulus,

FDS Psi

* All depths referenced from the elevation datum of the test pit, 6386 f t

Shear Bulk Modulus, Modulus, Poisson 's C3

Psi Psi R'atio 8

Table 3. Values of Elastic Constants Calculated from WES 1-D Tests at 3000 Psi

Dila- 0 - -.-~

Young's t Shear Bulk tational * Young 's Depth, Sample Me,.* Poisson's Modulus, Modulus, Modulus, Velocity, Modulus,

F - Ft Description PSI K Ratio Psi Psi Ps i

a FPS Psi ti

B 17 Conglomerate 6.98 X 1 0 V . 7 5 0.43 2.45 X 10" 0.87 X lo0 5.84 X 10" 15,550 8.0 X 10"

150 Sandstone 4.65 X 10' 0.15 0.13 4.40 X 10" 1.45 X 10' 1.98 X 10' 14,850 7.2 X 10' 8 60 Limestone 15.6 X 10" 0.375 0.27 12.50 X 10" 4.92 X 10' 9.60 X 10' 21,300 16.5 X 10' z

rn

* Secant modulus a t 3000 psi. t At 3000 psi.

were calculated and are also shown. The axial stress-axial strain curves are shown in Fig. 8. Since these curves were relatively linear after the initial seating, it was decided that a linear stress-strain curve or a con- stant modulus (Young's modulus) could be used for the one-dimensional calculation. Fig. 9 shows the axial vs. radial stress curves for the same tests shown in Fig. 8. Since these curves are also relatively linear, a con- stant I< (and consequently a constant Poisson's ratio) was used in the calculations.

Dynamic one-dimensional tests were also conducted at WES to a pres- sure of about 9 kilobars (kb) on a limestone and a sandstone sample. The "air-gun" procedure was used. This technique involves measuring the pressure and wave propagation velocity resulting from the impact of a flat projectile on one end of the sample. One-dimensional wave theory is then used to calculate the stress and strain. Fig. 10 shows the stress- strain curves. These samples were taken from below 120 f t and conse. quently were not used in these calculations. They are included here only to give an indication of the high pressure stress-strain properties.

Pig. 8-One-dimensional stress-strain curve.

Pig. 9-Axial us. radial stress.

The unconfined compression tests served to: (1) provide a minimum yield strength in compression (neglecting the-effect of joints and other natural discontinuities), (2) establish the shape of the stress-strain curve under a zero lateral restraint condition, and (3) provide data on the rela- tionship between axial and radial strains under a two-dimensional condi- tion. A summary of this data is contained in Table 4. Based upon these unconfined compressive strengths, it is evident that no yielding would occur under the anticipated pressure of 3000 psi. This is especially true since the rock will be in a partially confined condition a t the time of the test. As a result, it was not necessary to define the yield criteria and fail- ure envelope. Since the calculation of plastic flow would not be required, a plastic flow rule did not have to be defined.

Combined with the one-dimensional stress-strain curves in Fig. 8, the curves in Fig. 11 represent the extremes of the possible conditions of lateral restraint. The in-situ stress-strain relationships should lie some- where between these bounds. The curves shown were chosen to represent

Table 4. Summary of Unconfined Compression Tests

Sample Ultimate Young's * Depth, Sample Testing Strength, Modulus, Poisson's t Poisson's $

F t Description Agency Psi Psi Ratio Ratio

1 Shale & S S CERF 11,658 - - E? - P

1 Shale & SS CERF 10,500 0.77 X 10' 0.104-0.254 0.130 0

4.5 Sandstone CERF 15,765 - - - k 4.5 Sandstone CERF 14,000 7.30 X 10' 0.154 0.27 G Shale CERF 9,264 - - - 2 6 Shale CERF 8,607 0.90 X 10' 0.05 -0.26 0.125

10.5 Shale & SS CERF 9,636 - - - k a 10.5 Shale & SS CERF 10,050 0.61 X 10' 0.35 -0.4 0.39 s 17 Conglomerate CERF 9,742 4.6 X 10' - - 17 Conglomerate CERF 9,150 4.8 X 10' - -

-5 Sandstone CERF 12,220 1.7 X 10' 0.1 -0.5 0.12 -5 Sandstone CERF 9,728 1.3 X 10' 0.03 -0.23 0.12 g -5 Sandstone CERF 9,330 - - - E

-10 Limestone CERF 18,721 12.5 X 10' 0.41 -0.33 0.48 -10 Limestone CERF 20,243 30.0 X 10' 0.02 -0.37 0.11 z -10 Limestone CERF 16,242 - - - -20 Limestone CERF 15,300 - - - -20 Limestone CERF 20,900 - - - F -20 Limestone CERF 21,438 16.7 X 10' 0.28 -0.25 0.32 5

C) U)

Scant modulus a t 3000 psi. t Secant values a t 1 ksi and failure. $ Secant value a t 3000 psi.

STRAIN, percent

Fig . 10-Presszcre vs. strain, dynamic one- dimensional tests.

Fig. 12-Unconfined stress-strain curves.

the four major rock typs of interest although many more were available for establishing the values to be used in the calculations. Because of the large particle sizes, tests of the conglomerate were made on 6 ~ 1 2 - i n . samples.

Typical plots of axial vs. radial strain are shown in Fig. 12. Note that these represent neither a linear nor a constant value of Poisson's ratio. If one assumes a linear elastic material model, then these curves will have to be approximated by straight lines. It is most common to take secant values to the maximum stress of interest or an average secant value. It is more correct, however, if one is using an incremental elastic theory to use tangent values from the curves plotted as Poisson's ratio vs. mean stress.

I t is significant to point out that the laboratory testing program is con- tinuing (1968), but the material presented here is that which was avail- able a t the time of the experiment and was all that was available for establishing a material model for the calculations that will be discussed in the following sections.

I EmnTE DIFFERENCE CALCULATIONS

I Background of the AFTON Code The AFTON codes have been developed over a period of about eight

years. Work was begun at the Lawrence Radiation Laboratory (LRL)

Fig. 1.2-Axial ws. radial strain from .unconfined compression tests.

.25

.20

.I5

3 4 a + m

.I0 0 4 a

0.5

I ~ I ~ I ~ I ~ I ~ I ~ I

- - LIMESTONE - - .- SANDSTONE I / - SHALE i /

/'

/' /' - i - /'

i /' - i /'

/' I' - /" - /' - i 0.

/. - ,.

/' 0' - .OJO -

I I I I I , I l l , 0 .I .2 .3 4 .5 .6 .7 .8

AXIAL STRAIN, %

by John G. Trulio, and has been continued by him while at Nortronics and most recently at Applied Theory Inc. under the sponsorship of the Air Force Weapons Laboratory. There are now three AFTON codes employing the same philosophy of development, but specialized to solve different transient continuum motion problems: (1) one spatial dimen- sion (AFTON I ) , (2) plane symmetric systems of two spatial dimensions (AFTON 2P), and (3) axisymmetric systems of two spatial dimensions (AFTON 2 8 ) . The version used for the two-dimensional calculations discussed in this chapter is the AFTON 2P code, which has been made operational on the AFWL CDC 6600 computer. The code has been modi- fied somewhat by AFWL personnel.

Description of the AFTON Code Numerical solution to continuum mechanics problems begins by replac-

ing the continuous variables of space and time by a set of discrete points. As the spacing between these points approaches zero, the set approximates a continuum. Finite difference equations are then written which approxi- mate the principles of continuum mechanics. Basically, this means that principles of the conservation of mass, energy, and momentum, and the First Law of thermodynamics are rigorously followed.

The conservation of mass simply states that what flows out of a given zone representing an area of the continuum must flow into an adjacent one. Total energy is conserved by stating the kinetic and internal energy analogs in such a way that the finite difference equations explicitly con- serve energy. AFTON has been constructed so that momentum analogs are conserved exactly in the difference equations.

The AFTON code is unique in the manner in which it zones a problem. Most procedures for the solution of continuum mechanics problems as- sume either an Eulerian or Lagrangian mesh. Each of these techniques present inherent problems in various calculations, particularly if a calcu- lation is to be carried from a point source to very large distances and long times. This "rezoning" problem has been solved in the AFTON code by writing equations in a form in which the space coordinates can be moved in an arbitrary manner; the problem is, therefore, continually being rezoned. This zoning can be defined by a subroutine which identifies areas of greater activity so that the computer storage is more effectively used by placing a. greater concentration (smaller zones) of zones in the active area (space) of the problem and much larger zones in the relatively dormant areas. This procedure presumably results in an optimum use of the zones (therefore the computer storage) a t all times.

The AFTON finite differencing technique is of the "time marching" kind. The variables of motion are divided into two classes: (1) those associated with the zone boundaries and (2) those associated with the zone interiors. The first type includes the mesh-point positions and their time derivatives (displacement and velocity), and the second type consists

of the thermodynamic variables (strain, st,ress, and internal energy). The code was constructed so that the finite difference equations are as self- consistent a,s possible and constitute the most direct possible statement of the principles of continuum motion on finite regions. This has led to the very satisfying property that the equatioils each can be given a precise meaning in elementary physical and geometric terms. The code has been constructed so that the constitutive relations for the material comprising the continuum are handled in a subroutine and consequently may be varied at will.

More detail on the theory and operation of the AFTON codes is avail- able (Trulio 2 - 5 ) and is beyond the scope of this chapter. Because of the length of the codes, a listing is not given here, but is available (Trulio 3 ) .

Nature of the One-Dimensional Calculation

The primary purpose of this chapter is to present calculations of the response of a layered geologic sequence and to compare these calculations with the measured response. In order to make the two-dimensional calcu- lation (which is rather expensive in terms of computer time) as efficient and as meaningful as possible, a preliminary one-dimensional plane strain calculation was run by using a Lagrangian code programmed at the AFWL. This code was patterned after the techniques developed for the AFTON codes.

Based on the laboratory t,est data and the geologic section discussed previously, the one-dimensional constitutive relations shown in Table 5 were.selected. The four layers were considered to be linearly elastic. I t should be pointed out that this was not a restriction of the code, but a decision based on the laboratory test data. The anticipated pressure pulse (3000-psi peak) and 6-in. zoning were used for this calculation. Since the rock contained nearly horizontal bedding planes and a number of hori- zontal shale partings which were not cemented, i t was assumed that the effective tensile strength was zero. This was the only correction which was made for the inherent structural discontinuities. I t is believed that joints

Table 5. Properties of the One-Dimensional Model

Depth, Layer F t

1 0-13.5 2 13.5-22 3 2244 4 44-200

Unit Young 'S Weight, Constrained Modulus, Poisson 's Lb per Modulus,

Psi Ratio Cu F t Psi

may have a pronounced effect on the energy absorption of propagating waves, but a t the present (1968) no rational method of treating this in continuum calculations has been developed. As a part of the current study, an investigation of this phenomenum is being undertaken.

Results of One-Dimensional Calculation

The one-dimensional calculations included stress, strain, particle velocity, and displacement-time histories. Because the calculation was run with an elastic model in one dimension, the displacement calculations have little meaning and will not be discussed here. They will be discussed in more detail in conjunction with the two-dimensional calculations. The calculated velocity-time histories are shown in Fig. 13. I n all figures dis- cussed in this section, R refers to the distance below the ground surface. The effect of the geologic layering is best seen in these velocity-time histories. Referring to the time history a t the ground surface (Fig. 13), first reflections from each layer may be identified. By using the con- strained moduli listed in Table 5, the wave propagation velocity, Cp, the travel times through each layer may be calculated. The first reflection from the conglomerate layer should arrive a t the ground surface a t a time equal to twice the travel time through the shale layer or 5.2 m-see. (Note that the abscissa of the figure is lo-?). Since the conglomerate is stiffer than the shale, this reflection would be a compression wave with a n asso- ciated particle velocity in the upward direction. This is quite evident in the time-history plot. Note that the impedance mismatch between the layers is large enough to cause a significant drop in downward particle velocity, but not large enough to turn the material flow upward. A t about 7 m-see, the tensile reflection caused by the red shale underlying the conglomerate arrives and greatly increases the downward velocity. The compressive reflection from the very hard limestone layer arrives a t the ground surface a t 14.4 m-sec and is strong enough to turn the particle flow upward. This same reflection sequence can be seen in the strain- time histories (Fig. 14) and to a lesser extent in the stress (u)-time histories (Fig. 15).

The differences in the wave forms in the various layers may be assessed by comparing time histories on either side of the interface. For example, compare the time-histories from 120 in. (10 f t ) and 180 in. (15 f t ) which are near the red shale-conglomerate interface a t 13.5 f t .

By definition the strain a t the initial zone boundary is zero ; therefore, the first strain-time history shown is for 6 in. below the surface. The layering effect can be seen quite distinctly in the strain plots as large changes in the amplitude of the strain pulses.

The stress shown in Fig. 15 a t the ground surface is the input forcing function. Only a t depths of 20 f t and below were net tensile stresses de-

Fig. 18-Velocity-time hiatories.

veloped during the times shown. The seemingly cutoff portions of the deep stress plots are a result of not allowing a net tensile stress to develop. If a tensile stress was calculated, the stress was set to zero. Because the rock was assumed elastic, there was no energy dissipation or attenuation associated with the rock response and since the calculation was one- dimensional, there was no spatial atte~iuation of energy. In reality, both of these factors undoubtedly exist in a "real world'' situation. The cal- culation, however, always shows an increase of pressure as a consequence of the reflections from the various rock layers.

Nature of Two-Dimensional Calculation

This version of the AFTON 2P code is limited to a case involving only three layers. This is undoubtedly a limitation when dealing with sedi-

T IME I N 10 SEL

Pig . IS (continued)-Velocity-time histories.

mentary sequences similar to one like the Estancia test site, which has a number of relatively thin layers. In order to obtain the most realistic calculation in the upper 20 f t a t early times, i t was necessary to sacrifice some of the late time response. The three layers used in this calculation were the top three layers used in the one-dimensional calculation. This means that the limestone a t 44 f t was omitted, and the red shale which began at 22 f t was assumed to extend to a depth of 200 ft. The very stiff limestone causes a rather strong reflection (refer to the one-dimensional results) which, in the one-dimensional calculation reversed the particle velocity direction and therefore reduced the displacement. This probably would have occurred in the two-dimensional calculation if the layer could have been included.

The mechanical properties used in the two-dimensional calculation

d

4 .ca

3 ," 3.m - rn

2 .m

1 .m = a IN

0 . 0. . I.W 1.91 e.m e.m 3.m 3.m r.w r.m s.m

TIME IN SEC Fig. I4a-Strain-time histories.

were altered from those used in the one-dimensional calculation because spatial effects of the two-dimensional calculation result in an earlier un- loading than in the one-dimensional case, and to take into account, at least in part, the inherent joint pattern in the rock. The 'first rational method of treating joints in a finite element calculation was presented by Good- man et a1.= after our calculations were completed.

To account for the joints, the modulus was reduced in this calculation. Since bounded strain gages on laboratory specimens have been shown to

1 .M R = am I N

0 . 0 . .SD I-m 1.m e.m e.sn 3.m 3 . c u.m u.s, s.m

TIME \N I O - ~ SEC Fig. 14b-Strain-time histories.

yield moduli that are higher than those determined from large gage- length measurement^,^ the lower values presented previously were selected for correction. The reduction of moduli was based on the relationship between the laboratory sonic velocity and the field cross-hole velocity measurements tempered with engineering judgment. The stress-strain curves for the three layers are shown in Fig. 16. These have the same shape as the unconfined compression curves, but lower slopes. The un- loading moduli were taken to be 20% larger than the loading moduli.

Fig. 14c-Strain-time histories.

Since no unloading data were available, this value was assumed in order to introduce some energy dissipation by the rock. This correction ac- counts for irreversible deformation or compression of the joints.

The Poisson's ratios were determined from Fig. 12 using average tangent values for stresses of 4000 psi and below. These values are about half those used in the one-dimensional calculation. A yield strength of 10,000 psi was used in all three layers. This was slightly higher than the values given in Table 4 for the unconfined compressive strengths, but both

Fig. 15a--Stress ( u ) -time histo~.ies.

are well above the maximum stresses attained. The tensile strength was assumed to be 300 psi for all layers. This information is summarized in Table 6.

The problem was zoned with 4-ft-wide by 3-ft-deep zones in the upper 21 ft. Below that depth the zones were increased in depth a t a rate of 3% per zone. Since the center line of the test area is a line of symmetry, only one side was calculated.

The horizontal plane was normal to the direction of travel of the pres-

0. .m 1.m I - = e.m e.m ~ . a o Y r-m 4.a s.m

TIME IN lo-' SEC

Pig. 15b-Stress (u)-t ime histories.

sure pulse. This is the same as saying the wave traveled an infinite dis- tance. The pressure pulse was applied-to four zones (20 f t ) on one side of the half space beginning at the axis of symmetry. The time step was 1 x sec.

The output was as follows: time histories of vertical and horizontal particle velocity; vertical and horizontal displacement; vertical and horizontal stress; and shear stress all a t depths of 0.5, 2, 4, 7, 10, 15, 20, 25, and 40 f t for horizontal distances of 0, 6, 10, 15, 20, 40, and 60 ft. In addition, vector plots were made of the particle velocity every 25

Pig. 15c-Stress (0)-time histories.

cycles (2.5 m-see) for the first 50 m-see. This information was plotted on microfilm by the computer. Select time histories and vector plots were then plotted from the microfilm for presentation.

Results of Two-Dimensional Calculation

Only a portion of the results of the two-dimensional calculation will be presented here. More will be presented later as comparisons with mea- sured data and with the one-dimensional calculation.

Fig. 17 shows the calculated particle velocity time history a t 0.5 f t on

LINEAR APPROXIMATION

STRAIN, percent

Fig. 16-Stress-strain curwes for the three layers.

the axis of symmetry. Recalling the rock properties in Table 6, the layer reflections may be identified on this figure. The compressive reflection from the hard conglomerate arrived a t about 5.7 m-sec and results in a rapid decrease in the downward particle velocity. At about 4 m-sec the stress front reaches the soft shale layer and is reflected in tension back

Table 6. Rock Properties for Two-Dimensional Calculation

LAYER

Depth, f t Unit Weight, lb per cu f t Bulk Modulus, Loading, psi Bulk Modulus, Unloading, psi Constrained Modulus, Loading, psi Young's Modulus, Loading, psi Poisson's R.atio Yield Strength, psi Tensile Strength, psi

I I I I 1 l L I I I I I I 1 l l l l I .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

T I ME x sec

Fig. 17-Particle velocity-time history.

into the conglomerate. This wave reaches the 0.5 f t depth a t 8 m-sec and causes a large increase in the downward velocity. At about the same time a shear wave which was initiated at the edge of the loaded area' (to be discussed later) arrives and also increases the downward velocity. This shear wave interacts at this point with another initiated at the opposite side, therefore, propagating in the opposite direction. This results in a stress relief and reduction in downward particle velocity. Beyond this time (about 12.5 m-sec), a large number of reflections occur and calcula- tion "noise" becomes significant. Rather than follow these, the smooth

curve shown superimposed on the calculation was used for the compari- sons discussed subsequently in Summary.

Fig. 18 shows the particle velocity vectors a t various times. I n all cases the loading was over the area shown in Fig. 18a. Vectors having a very small magnitude were not plotted. This is the reason for the areas with no vectors. These represent " quiet7 areas with respect to the pri- mary motions. The zones where the arrows are vertically downward indi- cate areas which are unaffected by the finite size of the loaded area or

GROUND SURFACE

Y SYMMETRY- I

L 2 O - - - - - 2 0 t 20 40

DISTANCE, FT ( 0 ) l = 2 5 msec 4 0 . .- .- ........ -.-.

C 20 40 I b l I =- 5.0 msec

DISTANCE, FT ( c l I = 7.5 msec

DISTANCE, FT ( d l 1: 10.0 msec

. ....... . . . . 8 0 ..i . l t 20 40 60 80

DISTANCE, FT ( e l l = 12.5 msec

Fig. 18 a-e-Particle velocity vectors.

DISTANCE, FT (f 1 t = 15 m x c

. . . .

C 2 0 40 6 0 8 0 100 DISTANCE, FT

( q l 1 = 2 0 msec

Pig. 18f, g-Particle velocity vectors.

q * . . ' I - . ......... J- . ..-=.* L + *1L~f~\!.*-j!....~.-- -+-- ......... * a * A,...

j i ! I 180 ' L 20 40 6 0 8 0 100 120 140 160 180

DISTANCE, FT I h l 1 = 4 0 msec

Pig. 18h-Particle velocity vectors.

which are responding one-dimensionally. I t can be seen that at very early ' times (about 7.5 m-sec), stress relief caused by the shear disturbance is

reaching the axis symmetry. One can also see that although the wave is initially plane (Fig. Ma), it very rapidly becomes spherical in nature. Fig. 18h shows that the strong reflection from the bottom on the calculation grid has traveled up to about 135 ft. Notice also the extreme complexity of the flow field at 20 and 40 m-sec.

The changing nature of the stress pulse is shown in Figs. 19, 20, and 21. These are shown here since no measurements are available for comparison. The vertical values are on axis, the horizontal values are 6 f t from the center line and both have been smoothed in the same manner as the velocity calculations.

EXPERIMENTAL RESULTS

Instrumentation The instrumentation used to measure the response of the rock to the

imposed loading consisted of velocity and acceleration transducers and

TIME IN 1 0 - ~ s e c

Fig. 19-Calculated vertical stress-time histories.

0"' ' ' I ' 0 .5 1.0 1.5 \ 2.0'

TIME IN IU -sec

Fig. 20-Calculates vertical stress-time hGtories.

strain gages. In addition, air pressure gages were used to record the forc- ing function. However, because this chapter is primarily concerned with a comparison of calculations and measurements and since neither of the codes calculate accelerations, the acceleration data will only be used to obtain high-frequency velocity data by integration.

AIR PRESSURE GAGES-The air pressure gages (Schaevitz-Bytrex Model HFL 10,000 SP-A) were 10,000 psi range transducers with a natural frequency of 160 kHz. The gage was a half-bridge semiconductor strain-gage-type with an acceleration sensitivity of 0.01 psi per g and was temperature-compensated within the operating temperature range. The gages were mounted in canisters and placed in grout with the sensing ele- ment flush with the rock surface.

The Sandia-type velocity transducer (Spartan Model 601V256L and 601H25L) is a variable-reluctance fluid-damped pendulum gage with a frequency response of about 500 Hz and requires a 3 kHz carrier excita-

TIME IN 10-2 rec

fig. 21-Calculated horizontal stress-time histories.

tion. The viscosity of the damping fluid and therefore the gage output is temperature-sensitive. This requires subsurface temperature measure- ments prior tb the test event so that appropriate temperature corrections may be determined. The transducers were placed in canisters to protect the fluid from the (environment) pressure and placed down 9-in. drill holes and grouted into place with an impedance matching grout.

ACCELERATION TRAXXDUCERX-The acceleration transducers varied somewhat with the peak outputs anticipated. The model AV 20,000 (Endevco Corp.) is a half-bridge semiconductor strain-gage-type which will respond linearly to &20,000 g and will survive, although the response is nonlinear to k 30,000 g. It has a natural frequency of 40 kHz and a cross-axis sensitivity of less than 5%. The model 2261M6 (Endevco Corp.) is a full-bridge semiconductor strain gage with a linear accelera- tion range of &10,000 g. It will also survive up to -e30,000 g and has a cross-axis sensitivity of 5%. I ts natural frequency is 80 kHz and its fre- quency response is 10 kHz. The model 2261C (Endevco Corp.) is linear to k2500 g and can withstand a 300% overrange. It is a full-bridge

strain-gage-type with a natural frequency of 30 kHz and response frequency of 6 kHz. The cross-axis sensitivity is 3%. These transducers were placed in tlie same manner as the velocity transducers.

Strain Gages

Model SR-4 (Baldwin Lima-Hamilton) strain gages with gage lengths of 3, 4, and 3 in. were bonded directly to rock cores taken from the depth of measurement. These gages have a "nominal" gage factor of two and were applied to the split cores in order to measure both vertical and hori- zontal strains. The 6-in.-diam cores were placed in 9-in.-diam drill holes and grouted into place.

The grout used in refilling all instrumentation holes was designed so as to have the seismic impedance equal to an average of the layered sequence. The sonic velocity after 14 days airing was 10,000 fps and the unit weight 165 lb per cu f t .

DATA RECOVERY-The transducer output was recorded on mag- netic tape in a protected van about 600 f t from the test area. This analog information was converted to digital lists ; engineering units were applied based on predetermined calibration values and results were plotted by Cal Comp plotters a t the AFWL computer facility. All data analysis was done from these plots.

INSTRUMENTATION LAYOUT-The instrumentation was placed in order to record the two-dimensional effects normal to the direction of travel of the pressure pulse. A plan view of the strain, velocity, and ac- celerations instrumentation locations is shown in Fig. 22. Although acceleration was not calculated, accelerometers were installed to monitor,

A

DIRECTION OF TRAVEL

SCALE - FT - 0 5 10

LEGEND STRAIN VELOCITY

A ACCELERATION

Fig. dd-Inst~~nte9Ltati07t locations.

by integration, the higher frequency portion of the velocity pulse. This was necessary because of relatively low frequency of the velocity data. The maximum depth of instrumentation was 30 ft. The horizontal trans- ducers were oriented in order to monitor the motion in the transverse di- rection (normal to the direction of travel of the pressure pulse).

Forcing Function

The explosive source used for this experiment generated an exponen- tially decaying pressure pulse which can be represented by :

where P ( t ) is pressure at any time, Po is maximum pressure at zero time, t is time, to is total pulse duration, and A, B, a are constants controlling the pulse shape. The constants, A, B, a, are determined by performing a least-square fitting procedure using the measured airblast data. For the case discussed in this chapter, they were 0.131, 0.869, and 11.671, respec- tively.

This pressure pulse (Fig. 23) was applied to a surface area 40 x 60 f t in plan beginning at one end and traveling toward the other end (60-ft length) with a velocity of 14,400 fps. Based on a propagation velocity of 4750 fps in the top layer of the rock, this results in a shock wave which makes an anele of 19.3" with the rock surface.

u

The total impulse associated with the pressure pulse was 110.0 psi-sec and the total duration was 226 m-see. The impulse-time history is also shown. This represented a peak pressure about 1000 psi larger than the anticipated 3000 psi peak. As a result it was necessary to perform a second two-dimensional calculation after the test using the measured Dressure ~ u l s e . Since the one-dimensional calculation was intended to be only of a preliminary nature, it was not rerun.

Ground Motions

The measured particle velocity and strain measurements will be dis- cussed in detail subsequently. The acceleration time-histories presented in Fig. 24 are representative of the measurements. No calculations were made of acceleration. The acceleration data were generally rather noisy although in the cases where the signal was large, interpretation was not too dScult . Fig. 24 shows a comparison of 3 gages at 0.5 f t below the surface. The frequency is quite high, making determination of the peak value difficult. However they show good general agreement with peaks in the range of 12,000-15,000 g and similar wave forms. Fig. 25 shows data from greater depths. Note that the negative values are not nearly as large but the wave forms are quite similar. The peak amplitudes at-

tenuated very rapidly between 0.5 and 4.0 ft, but rather slowly below that depth.

The primary purpose in installing acceleration gages was to obtain velocity data from a high-frequency response transducer. This was ac- complished by integrating the accelerometer output with respect to time. The data obtained in this manner are compared with the velocity gage measuremknts a t similar depths in Figs. 26, 27, and 28. The data from 0.5 f t compare reasonably well in both amplitude and wave form. The initial upward motion indicated on both records is unexplanable a t this time. Whether this is real or was introduced by the recording system is uncertain. This is being investigated in more detail. The late time discrepancy is probably a result of tilt and the resulting zero drift of the velocity gages. I n this case the velocity gage seems to record more defini-

Fig. 23-Overpressure-time history.

Pig. 24a-A

cceleration-time

histories.

( S0 1 NQ

Iltl831333tl

Fig

. 24b-Acceleration-tim

e h

istories.

8

D

CI

=

I

4 - U

W

V

)

F.3 r W

E

I-

9

U)

IS91 NO

Iltl&I31333tl

Pig

. 25a-Acceleration

-time

histories.

C

z 3

g T

z

e l- 0 I

I I

I I

I I

I I

8 8

W q

Q

@

0

In

- Y

- 0

- 4

-R - U

W

V

)

-= E

W

x - I-

- ca

- L"

- 0

- "7 0

q ( SO 1 NO

1 ltlU31333tl

Pig

. 15

b-A

cceleratio

n-th

e h

istories.

133S/ljl A113013A

Fig. 2

5~

Accelera

tion

-time histories.

h

8-

I

I I

I I

I

8 8

H @

? 0

-R

- YI

- 0

- YI

- 0 - U

W

In

-K E W

r, I-

- 0

N

-L"

- 0

- VI

0

[SO

1 NOIlB8313JSB 4

Fig. 26-C

omparison

of m

easured velocity and in

tegrated acceleration,

D =

0.5 ft.

Fig

. 27-Co

mp

a7ism

of m

easwed

velo

city and

integ

rated

acceleration

, D

= 4

ft.

Fig. 28-C

omparison

of measured velocity and in

tegrated acceleration, D =

8 ft.

tion of the stress wave than does the accelerometer. This probably re- sulted from the very large accelerations causing the lower values to be averaged out. A t 4 f t , the accelerometer responded about 2.5 m-sec before the velocity gage and recorded a maximum value not a t the initial peak, but a t a later peak. Also the accelerometer recorded a positive pulse about 5 m-sec longer than the velocity gage. The peak value was also higher on the accelerometer. Fig. 28 shows gages a t somewhat different depths and the contrast between 8 and 10 f t . (The accelerometer a t 10 f t was dam- aged before the velocity peak was attained.) The velocity gage a t 10 f t lags in initial response (the wave travel time between 7 and 10 f t is 0.72 m sec) and shows a maximum behind the front. This gage is nearer the con- glomerate interface and is undoubtedly seeing a larger reflection.

I n general, the agreement between these two transducer types was better than expected. Since no pronounced trend was evident, i t is most likely that the differences observed were random instrumentation errors and not limitations of the gages.

The quality of the strain data was fair. The noise was quite consistent and hampered data interpretation where the signal levels were low. Fig. 29A is a typical raw strain record. I n order to expedite analysis, the strain data were smoothed by applying a five-point averaging pro- cedure. The smoothed record is also shown in Fig. 29B. Fig. 30 shows typical smoothed records drawn by hand to illustrate the nature of the wave forms measured on axis.

COMPARISON OF CALCULATIONS AND RESULTS

This section will be divided into three parts: (1) a comparison of the one and two-dimensional stress calculations, (2) a comparison of the one- dimensional calculated and the measured strains, and (3) comparisons of two-dimensional calculations and measured particle velocities and velocity flow fields.

The stress comparison (Fig. 31) demonstrates the effect of the limestone layer on the vertical stress. Based on the comparison one can deduce this effect on the two-dimensional motion predictions. Nearer the interface (Fig. 31b) the calculated effect is rather significant, but further away the influence decreases.

The comparison of measured and calculated strain shown in Fig. 32 indicates general qualitative agreement. Quantitatively, however, the measured magnitude of the reflections were less pronounced than was calculated. Also, the measured rise time was somewhat longer. This is believed to be caused by the joints. The measurements also indicate that the strain remains higher after the peak. This is believed to be caused by

-800 ] I I I I I 0 10 20 30 40 50

TIME (MSEC)

Pig. 29-Comparison of natural and smooth strain-time histories.

the spherical dispersion (two-dimensionality of the strain) and to a lesser extent by the joints. I t is felt that the strain comparisons are meaningful.

A typical vertical velocity comparison is shown in Fig. 33. This repre- sents a good comparison, all things considered. I t is desired, of course, that the comparisons will improve, but recognizing the many assumptions and simplifications that go into the calculation, this kind of comparison is considered quite encouraging. As with the one-dimensional calculation, the calculated reflections are much more pronounced than actually mea- sured. There is also a time dieerenee in reflection arrivals indicating that the moduli used in the calculation were somewhat in error. Calculated

o ~ ' ~ " ' ~ " " ' 0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

TIME XI@, sec

Fig. SO-Smoothed strain-time histories.

--- I-D

- 2-D

TIME x sec 31-Comparison of one and two-dimensional stress

calculations.

Fig. 32-Comparison of measured and calculated one-dimasional strain.

velocities typically do not cross the zero abscissa, whereas the data con- sistently show a negative phase.

Comparisons of horizontal particle velocities were not as encouraging, however. A t the 4-ft depth a t 15 f t from the center line, the comparison (Fig. 34) is fair for about the first 10 m-sec then becomes quite poor. A t the 20-ft depth under the edge of the loaded area (Fig. 34b), the compari- son is good only for the magnitude of the initial peak and grossly different for all times. These indicate that there is some basic phenomena that is affecting the horizontal motions which has not been accounted for in the calculations.

At instrumentation locations where both the horizontal and vertical gages functioned properly, vector diagrams have been drawn a t several times (Fig. 35) . The head of the arrow is a t the gage location. Although only a few vectors could be constructed, i t appears that the trend indi- cated in Fig. 18 is generally correct.

SUMMARY AND CONCLUSIONS

The AFTON two-dimensional finite difference code has been discussed and its application to the calculation of stresses and motions in a layered geologic sequence resulting from a large-amplitude pressure pulse has

Fig. $3-Comparison of measz~red and calculated vertical velocity.

been presented. The calculated results have been compared with experi- mental measurements. It has been shown that certain assumptions and simplifications are necessary in order to use laboratory tests results to describe the in-situ material properties and to represent the real situation in the calculation.

In addition, the results of geologic and geophysical surveys and labora- tory material property tests have been discussed. Engineering judgment must still be applied to interpret this data as it pertains to a dynamic calculation.

The field experiment and its instrumentation was briefly presented in order to allow an evaluation of the degree of confidence which should be placed on the data. The measured data were discussed and their limita- tion examined.

\ I - - TIME, m a w

Fig. 34-Comparison of measured and calculated horizontal velocity.

The experimental data has shown that even though the stresses were well within the supposedly elastic range of the intact rock, large residual strains did occur. This illustrates the importance of natural discontinui- ties (joints, bedding planes, faults, etc.) on the response of a rock mass and how far one might be from reality if a great deal of significance is placed on laboratory tests of intact samples. The measurements demon-

SCALE, f t 111 0 2 4 6 8 1 0

4 10 fps Pig. 35-Measured velocity vector8.

strated a rather satisfying agreement between the measured particle velocity and that obtained by integration of acceleration records.

The comparison of the one and two-dimensional calculated and mea- sured vertical wave forms showed an encouraging agreement both quanti- tatively and qualitatively. The disagreement in the horizontal wave forms indicates that the assumption of isotropy within a rock mass is not valid. This can pose a very difficult problem in terms of calculations. The over- emphasis of layer reflections by the calculation illustrates that calculations must treat not a limited number of layers, but a combination of large num- bers of layers separated by transitional zones of gradually changing properties.

I t is also concluded that a calculation assuming the rock may be repre- sented by a continuum will never be completely realistic for real geologic conditions. Rational techniques for determining the compression and frictional properties of geologic discontinuities and for including them in calculations must be developed. In addition, the anisotropy of the rock media must be measured and included in calculations.

REFERENCES

1. Ainsworth, D. L., and Sullivan, B. R., "Shock Response of Rock a t Pressures Be- low 30 Kilobars," Technical Report No. 6-802, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss., Nov. 1967.

2. Trulio, J. G., " Studies of Finite Difference Techniques for Continuum Mechanics, " WL-TDR64-72, Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., Dec. 1964.

3. Trulio, J. G., "Theory and Structure of the AFTON Codes," AFWL TR 66-19, Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., June 1966.

4. Trulio, J. G., Carr, W. F., & Germroth, J. J., "Optimum Coordinate Systems for One-Dimensional Finite Difference Calculations," AFWL TR 67-27, Vol. 11, Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., 1968.

5. Trulio, 3. G., et al., "Ground Motion Studies and AFTON Code Development," AFWL TR 67-27, Vol. 111, Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., 1968.

6. Goodman, R. E., Taylor, R. L. and Brekke, J. L., "A Model for the Mechanics of Jointed Rock," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 94, No. SM3, May 1968.

7. Fairhurst, C., "Laboratory Measurements of Some Physical Properties of Rock," Proceedings of the Fourth Sy7nposium on Rock Mechanics, College of Mineral In - dustries, The Pennsylvania State University, University Park, March 1961.