Arch. Min. Sci., Vol. 52 (2007), No 3, p. 311–330

Arch. Min. Sci., Vol. 52 (2007), No 3, p. 311–330
311
JANUSZ NURKOWSKI*
THE CORELESS INDUCTIVE SENSOR FOR STRAIN MEASUREMENT OF ROCK SAMPLES
IN A PRESSURE CELL – SOME ADVANTAGES AND DISADVANTAGES IN RELATION
TO THE ELECTRICAL RESISTANCE STRAIN GAUGES
BEZRDZENIOWY INDUKCYJNY SENSOR DO POMIARU ODKSZTAŁCEŃ PRÓBEK SKALNYCH
W KOMORZE CIŚNIENIOWEJ – ZALETY I WADY WZGLĘDEM TENSOMETRÓW
REZYSTANCYJNYCH
This paper presents a sensor for measuring strains ranging from below 0.1% to over 50%, which is
considered as the one-ply coreless inductor. This sensor is used mainly for measuring the compressibility
of rock samples in a triaxial cell under hydrostatic pressure up to 1 GPa, in the case when it is difficult or
impossible to use a resistance strain gauge. Moreover, the paper describes the electronic system working
with the sensor and discusses the influence of temperature and high pressure on the sensor properties.
To conclude, some problems referring to the application of the referential resistance strain gauge in
a pressure cell are discussed.
Keywords: displacement sensor, high pressure, inductive sensor, strain measurement
W artykule przedstawiono indukcyjny bezrdzeniowy czujnik do pomiaru odkształceń materiału
a zwłaszcza próbek skał w komorze ciśnieniowej wypełnionej nieprzewodzącą cieczą sprężaną do setek
MPa. Omówiono jego konstrukcję, sposób stosowania i przykładowe wyniki pomiarów. Na zakończenie
poruszono pewne problemy związane z zastosowaniem tensometrów rezystancyjnych (zwanych dalej
krótko tensometrami) do takich pomiarów i skonfrontowano obie metody ze sobą.
Pomiar deformacji próbek skał w trójosiowym stanie naprężenia oparty na zastosowaniu tensometrów
przyklejonych wprost na próbkę wiąże się z pewnymi problemami kiedy skała jest spękana, porowata,
nasycona wodą lub odkształcenia przekraczają kilka procent, ponieważ ciśnienie hydrostatyczne wgniata
ścieżkę rezystancyjną w szczeliny, podczas gdy obecność wody może być przyczyną zwarcia. Pewne
problemy związane z tensometrami były opisane przez Attingera i Koppela (1983), Hoque’a i in. (1997),
Lintona i in. (1988) i Wawersika (1975). Inna metoda oparta na tensometrach przyklejonych na sprężystą
taśmę, która jest zamocowana wahliwie do próbki na zaczepach (LDT) jest zalecana do stosowania w stałym
ciśnieniu hydrostatycznym (Hoque i in., 1997; Besuelle i Desrues, 2001). Ponadto dystans pomiędzy próbką
skały a ścianą komory ciśnieniowej może być zbyt mały do zainstalowania takiego przetwornika.
*
STRATA MECHANICS RESEARCH INSTITUTE OF THE POLISH ACADEMY OF SCIENCES, UL. REYMONTA 27, 30-059
KRAKÓW, POLAND
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Opracowano zatem nową metodę opartą na wykorzystaniu jednowarstwowej, bezrdzeniowej cewki
indukcyjnej. Tak wykonany czujnik jest instalowany na próbce badanego materiału; jej odkształcenie
powoduje zmianę długości cewki a zatem jej indukcyjności (rys. 1). Indukcyjność czujnika razem z pojemnością tworzy elektryczny obwód rezonansowy tranzystorowego oscylatora LC (rys. 2). Zmiana
częstotliwości oscylacji jest źródłem informacji o odkształceniu badanego materiału (zmiany indukcyjności
są zbyt małe do ich bezpośrednich pomiarów z wystarczającą rozdzielczością). Taki sposób pomiaru
jest prosty i niezawodny. Głównym problemem było zredukowanie wpływu temperatury i ciśnienia na
czujnik. Dzięki specyficznym właściwościom wysokorezystywnego drutu stalowego użytego do wykonania czujnika jest możliwe ograniczenie wpływu temperatury na częstotliwość, co umożliwiło pomiar
odkształceń od 0,1% do ponad 50%. Opracowano dwie odmiany czujnika: liniową – montowaną do
zaczepów lub pierścieni na próbce – do pomiaru małych odkształceń np. ściśliwości i drugą – toroidalną
do pomiaru odkształceń obwodowych np. w teście trójosiowego ściskania próbki walcowej. Czujnik ma
małą średnicę zwojów (kilka milimetrów), więc zajmuje niewiele przestrzeni w komorze ciśnieniowej.
Brak bezpośredniego kontaktu z badanym obiektem (z wyjątkiem punktów mocowania) w przypadku
czujnika liniowego umożliwia osiągnięcie rozdzielczości 0,001% (kilka mikrometrów). Te właściwości
czynią prezentowane czujniki konkurencyjnymi w porównaniu do tensometrów rezystancyjnych. Są one
używane do pomiaru odkształceń próbek skał umieszczonych w komorze ciśnieniowej pod ciśnieniem
nieprzewodzącej cieczy aż do 1 GPa.
Dokładność pomiaru takim czujnikiem jest limitowana głównie przez niestabilność:
1. indukcyjności i rezystancji czujnika wywołanej zmianami ciśnienia i temperatury w komorze.
2. pasożytniczych pojemności i indukcyjności przewodów łączących czujnik z generatorem – składają
się na to przewody wewnątrz i na zewnątrz komory oraz przepusty elektryczne w ścianie komory,
na które wpływa zmienna temperatura i ciśnienie.
3. pojemności układu elektronicznego generatora zależnych od temperatury otoczenia i napięcia
zasilania.
Jakkolwiek można eksperymentalnie określić wpływ ciśnienia i temperatury na charakterystykę
czujnika i korygować matematycznie uzyskane wyniki, to korekcja ta jest ograniczona przez dokładność
określania charakterystyk korekcyjnych, różnicę bezwładności cieplnej czujnika odkształcenia i temperatury oraz mnogość czynników zakłócających. Lepszym sposobem zwiększenia dokładności pomiaru
odkształceń jest użycie czujnika odniesienia. Czujnik ten ma postać identyczną jak czujnik pomiarowy
(długość, średnica, liczba zwojów, rezystancja) z tym, że jest zamocowany na próbce materiału o znanej
ściśliwości, np. na stalowym wsporniku. Umieszcza się go w komorze ciśnieniowej razem z czujnikiem
pomiarowym. Podłączając do oscylatora na zmianę raz jeden raz drugi czujnik za pomocą przełącznika
(najlepiej tranzystorowego) otrzymamy względny pomiar odkształcenia badanego materiału ze znacznie
zredukowanymi błędami spowodowanymi ciśnieniem i temperaturą. Rysunek 4 przedstawia schematycznie
zasadę działania oraz czynniki zakłócające pomiar, natomiast na rysunku 5 pokazano fotografię czujnika
odniesienia zamocowanego do stalowego wspornika. Rysunki 6, 7 i 8 przedstawiają rezultaty pomiaru
ściśliwości aluminium, soli i łupka uzyskane tą metodą przy użyciu czujnika liniowego.
Jako przykład zastosowania czujnika toroidalnego do pomiaru odkształceń poprzecznych (obwodowych) przedstawiono efekty pomiaru odkształceń próbki piaskowca w klasycznym trójosiowym stanie
naprężenia w urządzeniu GTA-10. Próbka jest zabezpieczona osłonami lateksowymi przed kontaktem
z naftą, która wypełnia komorę ciśnieniową. Zewnętrzna osłonka lateksowa pod wpływem nafty pęcznieje,
nawet jeśli obciska ją ciśnienie nafty rzędu setek MPa. Proces pęcznienia trwa przez cały czas kontaktu
osłony z naftą. Prowadzi to oczywiście do błędów pomiaru odkształcenia, bowiem czujnik jest nałożony
na osłonki. Ogólnie można stwierdzić, że pęcznienie to jest tym większe im mniejsze jest ciśnienie nafty
i dłuższy czas kontaktu z naftą. Wpływ pęcznienia osłonki na wskazania czujnika przedstawia rysunek
11. Rozwiązaniem tego problemu jest wykonanie osłonek z materiału odpornego na naftę (np. na bazie
silikonu) lub wypełnienie komory innym płynem, np. olejem silikonowym lub alkoholem. Zastosowanie
koszulek termokurczliwych daje również dobre rezultaty, choć dla odkształceń powyżej 20% istnieje obawa
ich pęknięcia. Możliwości pomiarowe czujnika toroidalnego w zakresie małych odkształceń przedstawiono
na rysunku 12, na którym widać efekty trójosiowego testu piaskowca Tumlin przy ciśnieniu okólnym
50 MPa. Dodatkowa pozioma oś przedstawia odkształcenia w mikrometrach, pozwala to oszacować
rozdzielczość pomiaru odkształceń obwodowych na kilka mikrometrów.
313
W pomiarach tensometrami rezystancyjnymi naklejonymi wprost na próbkę, aby zredukować błędy
oddziaływania zmiennego ciśnienia i temperatury stosuje się tensometr kompensacyjny, który na ogół
jest naklejony na stalową płytkę i umieszczony wewnątrz komory ciśnieniowej. Uzyskuje się wówczas
względny pomiar odkształceń skały, odniesiony do ściśliwości płytki kompensacyjnej. Wydaje się jednak,
że stal nie jest najlepszym podłożem dla tensometru kompensacyjnego ze względu na około dwukrotnie
większą rozszerzalność cieplną w stosunku do skał. W Pracowni Odkształceń Skał Instytutu Mechaniki
Górotworu PAN w Krakowie uzyskano lepsze efekty stosując do tego syntetyczny rubin (korund), który
miał rozszerzalność i przewodność cieplną zbliżoną do skał. Porównanie pomiaru ściśliwości próbki
granitu o średnicy 22 mm i długości 44 mm w przypadku zastosowania kompensacji na płytce stalowej
i korundowej wewnątrz komory, a także dla kompensacji umieszczonej poza komorą ciśnieniową pokazano
na rysunku 13. Histereza krzywej ściśliwości gdy płytka kompensacyjna jest z korundu jest kilkakrotnie
mniejsza niż dla stalowej.
Tensometry przed naklejeniem na próbkę skały powinny być wstępnie kondycjonowane ciśnieniem
hydrostatycznym. Realizuje się to przez umieszczenie ich na stalowej płytce (bez przyklejania) i nałożenie
lateksowych osłonek oraz zanurzeniu w cieczy ciśnieniowej i sprężenie jej do maksymalnej dopuszczalnej
dla komory wartości. Efekt tego zabiegu widać na makrofotografii przedstawionej na rysunku 14.
Indukcyjne bezrdzeniowe czujniki odkształceń rozwiązały problem badania skał porowatych lub przewodzących, eliminując ryzyko związane z naklejaniem tensometrów rezystancyjnych wprost na próbkę.
Mają wysoką czułość oraz bardzo szeroki zakres pomiaru od mikronów do centymetrów. Prostota wykonania, odporność na udary mechaniczne i łatwość mocowania czyni je atrakcyjnym narzędziem pomiarowym.
Ciągle udoskonalane są zarówno czujniki, jak sposób ich mocowania oraz współpracujący z nimi oscylator,
w efekcie systematycznie rośnie ich dokładność i konkurencyjność względem tensometrów.
Słowa kluczowe: czujnik indukcyjny, pomiar odkształceń, tensometr rezystancyjny, wysokie ciśnienia
1. Introduction
The simplest method of measuring deformations of a sample of rock in a triaxial state
of stress in an arbitrary direction is measurement by means of an electrical resistance
strain sensor (called a strain gauge) cemented onto a sample. However, the application of
this method has a few problems, especially when the rock is cracked, porous, saturated
with water or when the strains exceed a few per cent. Under such conditions the use
of a strain gauge is very difficult because the hydrostatic pressure forces the wire into
cracks and pores, and the presence of water in the rock can cause a shorting of the wire.
Some problems caused by strain gauges were described by Wawersik (1975), Attinger
& Koppel (1983), Hakami et al. (1987). Non-contact methods based on strain gauges
cemented onto a thin elastic metal strip and attached to the sample by hinges called an
LDT (local deformation transducer), shown e.g. in Hoque et al. (1997) are recommended
for conditions of constant confining pressure (Besuelle & Desrues, 2001). However, the
distance between the rock sample and the wall of the pressure cell may be too small for
the installation of such a transducer.
Eventually, the analysis of the already existing solutions considered with respect
to the needs and the technical and financial limitations of the laboratory has led to the
development of an original method of measuring deformations under the conditions of
a triaxial compression test. The new method is based on the use of a one-ply, coreless
inductive sensor (Nurkowski, 1998).
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The sensor is installed on a sample of the tested material. Changes in the dimensions of
the sample cause changes in the length of the sensor. This causes changes in inductance,
which together with the capacity, constitutes a resonant circuit of a transistor oscillator.
The change of oscillation frequency is the source of information about the deformation
of a sample (changes in inductance are too small to be directly measured with sufficient
resolution). The main problems were to reduce the temperature sensitivity of the sensor
and to recognize the influence of pressure. Due to the specific qualities of the wire used
for the coil, it is possible to reduce, to a considerable degree, the influence of the temperature on frequency. This makes it possible to measure strains ranging from below 0.1%
to over 50%. There are models of the sensor: linear – mounted by means of catches or
clamping rings (Fig. 1) and toroidal for measuring circumferential strain. The sensor
has a small diameter of the turn (a few millimeters), therefore, it takes little space in the
pressure cell. The lack of any direct contact with the measured object in the case of the
linear sensor enables it to achieve a resolution of 0.001%.
Fig. 1. Linear and toroidal inductive sensors fastened to rock samples
Rys. 1. Liniowy i toroidalny sensor zamocowany na próbkach skał
These features, under certain circumstances, make the sensor presented here competitive in relation to strain gauges. The inductive sensors can be used for measuring
strains of rock samples placed in a high pressure cell under the hydrostatic pressure of
a non-conducting liquid up to 1 GPa.
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2. Principle of operation of the sensor
The oscillator, located outside the high pressure cell and connected to the sensor,
operates in the Colpitts system with divided capacity (Fig. 2). The oscillation frequency
is approximately:
f where:
L, C
C1, C2
LP, CP
LS
h11, h22
—
—
—
—
—
1
2
1
h22
CC
C1C2 h11
( LS LP ) 1 2 CP
C1 C2
(1)
total inductance and capacity of the resonant circuit,
capacity of the oscillator capacitors,
total parasite inductance and capacity,
inductance of sensor,
input resistance and output conductance of transistor.
The frequency of the oscillator working with the sensor can easily be measured with sufficient accuracy and
resolution.
The linear sensor, connected to an oscillator, is calibrated by controlled stretching. Changes in length and consequent frequency changes are measured. The toroidal sensor
is scaled by means of compressing a rubber cylinder with
an attached gauge in a vice. It is important to calibrate the
sensor preserving the identical character of its connection
with the oscillator as it was during the measurement. The
actual characteristic of a sensor for strains up to 100% is
presented in Figure 3. The characteristic of a sensor can
also be calculated from the formula (1), where the coil inductance LS for an environment with the magnetic relative
permeability coefficient equal to 1 is given by:
Fig. 2. The scheme of an
oscillator
Rys. 2. Schemat oscylatora
LS 0 z 2 D2 4l ( z D)2107 l
(2)
where:
z, D, l — number of turns, diameter, length of a sensor, [m],
µ0 — abs. magnetic permeability = 4 π . 10–7, [T.m/A].
The theoretical and experimental curves are nearly consistent for strains up to 50%.
Then the formula (2) becomes incorrect due to the excessive extension of the coil.
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The calculation was made for the following parameters: z = 150, D = 3 mm, C = 650 pF,
LP = 950 nH.
The parasite inductances cause a decrease in the sensor sensitivity. For example, a parasite inductance LP of about 5% of the sensor inductance, decreases the sensitivity by
10%. But if LP is equal to the inductance of the sensor, the sensor’s sensitivity becomes
three times smaller. Moreover, parasite inductances decrease the stability of oscillations.
The values of parasite inductances can be measured by connecting a short piece of wire
to the oscillator instead of to the sensor. In this case, the oscillator will operate with the
frequency fs (short-circuit), determined by the total of all possible parasite inductances.
By measuring fs, the parasite inductance can be calculated from the formula (1).
Fig. 3. The sensor characteristic
Rys. 3. Charakterystyka sensora
The sensor sensitivity s is a derivative of frequency (given by (1)) in relation to the
inductor length l (h11, h22 is negligible):
df 107 zD s
dl 4 CR l 2
( zD ) 2 LP 7 l
10
1.5
, Hz/m
(3)
In practice, the sensitivity of a sensor is 20 ÷ 40 kHz/mm for C = 1 nF and frequency
2500 kHz.
An increase in absolute sensitivity by diminishing the number of turns, their diameters
or the capacity of the resonant circuit capacitor results in a greater impact of parasite
inductances and capacities. The relative sensitivity, defined as the relation of absolute
317
sensitivity to resonance frequency, depends inversely proportionally solely on the extension of the sensor and parasite inductance. In order to obtain high sensitivity, the turns
of the sensor should be placed close to each other. In this case the coil inductance is also
greater so the influence of parasite inductance diminishes.
3. The influence of temperature and pressure
on the measurement results
The accuracy of the measurement is limited by the instability of the resonant circuit
caused by:
1. the sensor’s inductance and resistance caused by a change in pressure and temperature inside the cell;
2. parasite capacities and inductances of the wires connecting the sensor to the oscillator (including the wires inside and outside the cell as well as the electric seal wires
in the cell wall, which are affected by changeable temperature and pressure);
3. the capacity of the electronic system as a result of changes in the environment
temperature and in the supply voltage.
To decrease the impact of parasite capacities, the capacities C1, C2 should be as
large as possible. On the other hand their values are limited by the quality factor of the
resonant circuit.
By the application of high quality electronic elements, as well as the stabilization
of the temperature and supply voltage and/or thermal compensation, the drift of the
oscillator caused by the electronic elements can be reduced to the required level (few
Hz per hour).
The most difficult problem is to reduce the influence of temperature upon the sensor
(Nurkowski, 1999). A temperature increase causes an increase of the wire length ld,
which results in an increase in the diameter of the sensor turns D:
T ldT DT
1
f TS
fTS T
1 TS T f f fT f 1
ld
D
1TST 1TS T (4)
where:
αTS — thermal expansion coefficient of the inductor wire,
ldT, DT, fT — the length of wire, diameter and frequency at temperature T.
For a copper inductor with the αT = 16.10–6/K, and for the frequency 3 MHz, and the
temperature fluctuations in the cell of about 10 K, the changes in frequency will be about
500 Hz, which is too much for measuring strains below 1% in the case where the length
of a sample is a few cm because the sensitivity of sensor is about 30 kHz/mm.
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The temperature also changes the resistance of the wire. This has a direct influence
on the resonance frequency, but this effect is of no great importance because it only
produces a change of few Hz.
The changes in the resistance will also affect the operating conditions of the oscillating
transistor but in the case of a copper inductor their impact on the oscillation frequency is
negligible. The change in resistance is small (0.002 Ω/K) in comparison with an emitter
resistor RE ≈ 1000 Ω.
A steel spring wire has a great specific resistance (≈10–6 Ωm). As a consequence, the
quality factor of the resonant circuit decreases considerably. Therefore, in order to force
oscillations and obtain the required amplitudes, the oscillator transistor has to be strongly
coupled with the resonant circuit. This causes significant changes in the transistor electric
parameters on the oscillation frequency. The high resistance of a spring steel sensor also
results in its considerable absolute changes. For a steel sensor of R = 30 Ω resistance,
when heated by 1 K, the resistance will increase by 24 mΩ, while for a copper sensor,
it would increase by 2 mΩ. Such big changes in the resistance affect the operating conditions of the oscillating transistor and compensate for the influence of temperature on
diameter of the sensor. In this way it is possible to reduce the temperature sensitivity of
the sensor by more than ten times.
Measurements of strains less than 1% using such inductive sensors tend to be difficult
as the sensor itself and the electric seals are affected by pressure changes and the associated temperature fluctuations. Though the effects of pressure and temperature on the
sensor characteristic can be determined experimentally and correction terms computed
accordingly, the range for correction is limited by the required accuracy of correction
characteristics and the difference in thermal inertia between the strain and temperature
sensors.
Measurement precision can be improved through the use of a reference sensor. Its
structure is identical to that of the measuring sensor, yet it is fixed on an object of known
compressibility, for example a steel support. It is placed inside a pressure cell together
with a measuring sensor. The reference sensor, made of the same material as the measuring sensor and having identical mechanical and electric parameters (length, diameters,
number of turns, resistance), is connected to an electric seal whose parameters are similar
to those in the measuring sensor. The two sensors are alternately and in cycles connected
to one oscillator, preferably via a transistor switch. The principle of this method and
factors disturbing the measurement are shown in Figure 4. Figure 5 presents a photo of
the reference sensor mounted on a steel support.
Compressibility of the tested material and the support can be measured whilst the
systematic error due to temperature and pressure fluctuations should be the same. Once
the compressibility of the support material is known, the compressibility of the support
can be measured and sensor readouts will be compensated accordingly. In addition,
the method allows us to compensate for the effects of ambient temperature and supply
voltage fluctuations on the system electronics.
319
Fig. 4. Method of the reference sensor usage and factors disturbing the measurement
Rys. 4. Sposób użycia sensora odniesienia i czynniki zakłócające pomiar
Fig. 5. Construction of the reference sensor
Rys. 5. Konstrukcja sensora odniesienia
The method outlined in this study allows strain measurements of a rock sample under
hydrostatic pressure of the order of several hundred MPa with the accuracy comparable
to resistance strain gauges. The measurement accuracy data are presented below.
320
4. Examples of application of the linear sensor to the measurement
of the compressibility of various materials
The measuring capabilities of an inductive sensor incorporated in a circuit comprising
a reference sensor are demonstrated by the linear compressibility measurements of an
aluminium cylinder with the same dimensions as a rock sample (22 mm in diameter,
44 mm in height) and fixed in the same manner. This enables us to evaluate sensor sensitivity and accuracy and to find out how thermal expansion of the tested material might
affect the test results. Figure 6a shows plots of predicted compressibility (tabled data
are indicated with a broken line), frequency from the reference sensor fixed on a steel
support, frequency from the measuring sensor and liquid temperature in the chamber.
b)
300
300
250
250
200
200
p, MPa
p, MPa
a)
150
150
steel
support
100
100
aluminium compressibility
50
50
1, %
0
0
0.05
0.1
0.15
2883
d)
300
300
250
250
200
200
p, MPa
p, MPa
c)
150
aluminium
sample
100
2883.5
2884
2884.5
2885
liquid
temperature
150
100
50
50
f 2, kHz
0
2895
f 1, kHz
0
2895.5
2896
2896.5
2897
T, °C
0
15
20
25
30
Fig. 6. Linear compressibility of the aluminium sample (a), signals from the sensors: referential (b)
and measuring (c), liquid temperature inside the cell (d) vs hydrostatic pressure
Rys. 6. Ściśliwość liniowa próbki aluminium (a), sygnał z sensorów: odniesienia (b) i pomiarowego (c)
oraz temperatura cieczy w komorze (d) vs ciśnienie hydrostatyczne
321
The compression and decompression rate was rather high (1.5 MPa/s) so that differences in liquid temperature in the chamber should be large.
At 150 and 300 MPa, compression was interrupted to find out how the accompanying
temperature decrease of about 5°C might affect the measurement data.
Due to a hydrostatic pressure increase the aluminium cylinder would shrink so the
length of the attached sensor would decrease. Therefore, the sensor inductivity would
increase. On the other hand, the length of sensor wire would decrease as well as the
turn diameter which means that the sensor inductivity would decrease simultaneously.
Moreover, the sensor resistance would also decrease, which would increase the resonant
frequency of the LC circuit, and – as a consequence – would change the operating point
of the transistor in the oscillator. But the effect of the decrease in sensor length would be
significantly greater than other effects so finally the frequency would decrease. As the reference sensor was fixed on a steel support with a compressibility two times lower than that
of aluminium, this factor seemed predominant in producing an increase in frequency.
The temperature increase at 150 and 300 MPa was responsible for a larger frequency
fluctuation of the compensating sensor than on the measuring one, mostly due to differences in the thermal expansion of steel and aluminium. This evidences the high sensitivity
of the sensor and this aspect should be taken into account, which might prove difficult
as the reference material and the tested specimen have different thermal inertia.
Figure 7 shows the linear compressibility of rock salt measured by an inductive sensor
and, for comparison, by a resistance gauge. Figure 8 shows the linear compressibility
300
250
saturated
with water
p, MPa
200
150
non saturated
100
50
0
, %
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Fig. 8. Linear compressibility of shale, measured using an inductive sensor
Fig. 7. Linear compressibility of rock salt
Rys. 7. Ściśliwość liniowa soli kamiennej
Rys. 8. Ściśliwość liniowa łupka zmierzona
za pomocą czujnika indukcyjnego
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of shale in two cycles of compression and decompression under water-saturated and
non-saturated conditions. Note that water in pores strengthens the rock.
5. Selected applications of a toroidal sensor
A toroidal sensor was used in measurements of transverse strains of a sandstone specimen in an axi-symmetric triaxial state of stress. Measurements were taken in a GTA-10
triaxial apparatus which allows confining pressures of up to 400 MPa and an axial load of
up to 1500 kN to be applied (Długosz et al., 1981). Specimens were covered with latex
jackets to protect them from kerosene used as a confining liquid. Axial strain ε1 of the
specimen was obtained by measuring the movement of a piston integrated with a dial
gauge (mounted on the low-pressure end). Transverse strain ε3 was measured using an
inductive sensor (Fig. 9). In order to compute such considerable strains, it is required
that the sensor characteristic be approximated by a second-degree curve.
Fig. 9. Differential stress vs strain: axial (ε1) and transverse (ε3). Transverse strain measured by inductive
sensor, axial strain determined based on the measured displacement of the piston; sandstone
Rys. 9. Naprężenie różnicowe vs odkształcenie: osiowe (ε1) i poprzeczne (ε3).
Odkształcenie poprzeczne zmierzone za pomocą czujnika toroidalnego, osiowe uzyskane
na podstawie pomiaru przemieszczenia tłoka; piaskowiec
6. Errors involved in measurements using toroidal sensors
When a rock specimen has to be insulated from the liquid filling the chamber, thin
latex covers are recommended. Normally, a specimen is protected by three latex jackets
to ensure that it remains leak-proof even if one of the jackets gets damaged by a rough
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part or if it has leaks not revealed by visual inspection. Figure 10 shows the sensor fixed
on an exposed specimen resting on the plug of the pressure chamber. There are thicker
sections made from latex directly above and below the sensor to stabilise its position
under the huge strains that cause the cylindrical sample to become barrel-shaped (Fig. 1).
Under atmospheric pressure the total thickness of these jackets is about 1 mm. Due to
the action of hydrostatic pressure in the chamber, the thickness of the latex cover is
reduced. If the chamber is filled with kerosene, the external jacket begins to swell even
when subjected to pressures of the order of hundreds of MPa. The swelling process
continues while the jacket is in contact with the kerosene, which leads to strain measurement errors. Generally, the lower the pressure of kerosene and the longer the time of
exposure, the more intense the swelling process. The effects of cover swelling on sensor
readouts are depicted in Figure 11. In the experiment, a granite specimen was protected
with three latex jackets and the sensor was fixed upon them. The jackets were subjected
only to a constant pressure of kerosene. Measurements were taken for pressures 20,
50, 100, 200, 300 MPa, for about one hour. For example, for the confining pressure of
50 MPa, the swelling of the cover during 60 minutes led to an apparent increase of the
sample diameter by about 0.3 mm. In the example quoted here, the measured transverse
deformations of sandstone approaching 40% for a sample 22 mm in diameter would
be overestimated by about 3% due to the swelling of the cover. For lower pressures the
error is greater, for pressures in excess of 200 MPa it will be many times lower.
Fig. 10. The toroidal sensor installed
on a sample covered with latex jackets
Rys. 10. Toroidalny sensor na próbce ubranej
w lateksowe osłonki
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Fig. 11. Effect of the swelling of latex jackets on the sensor signal as a result
of the influence of compressed kerosene (top graph) and the temperature of the kerosene
in the triaxial chamber (bottom graph)
Rys. 11. Wpływ pęcznienia osłonek lateksowych na sygnał z sensora w skutek oddziaływania sprężonej
nafty (wykres górny) oraz temperatura nafty w komorze trójosiowej (wykres dolny)
The bottom graph in Figure 11 shows the temperature inside a chamber once the
kerosene compression process at the rate 30 MPa/s is completed. The effects of large
temperature fluctuations on frequency are evident during the first stage, i.e. for the first
10 minutes. Temperature affects both the jackets and the sensor and electric pressure
seals. In the experiment the sample in the triaxial state of loading is squeezed by the
piston; the squeezing test begins when the temperature inside the chamber stabilises at
a level 1-2°C higher than the temperature prior to liquid compression.
Calculation of the Poisson’s ratio involves a significant relative error, approaching
10%, particularly under low pressures as the sample deformations will be minor and
cover swelling – considerable.
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To solve this problem, it is recommended that the protective jackets should be made
from kerosene-resistant materials (for example silicone-based). Alternatively, the chamber might be filled with a different liquid, such as silicone oil or alcohol, though alcohol
is hydrophilic and the compressibility of silicone oil is greater than that of kerosene.
Heat-shrinkable jackets seem a good solution, too, though they tend to break under
strains in excess of 20%.
The measuring performance of a toroidal sensor in the range of small strains is depicted in Figure 12, showing the Tumlin sandstone stress-strain characteristics under the
triaxial test conditions at the confining pressure of 50 MPa. Axial strain was determined
based on piston movement measurements. The initial Poisson’s ratio of 0.8 would decrease to about 0.3 and then would increase again beyond the elastic range. Since the
Poisson’s ratio values were calculated by dividing increments in circumferential strain
by increments in axial strain, for very small strain increments when the loading began
Fig. 12. Results of a triaxial test of a sample of Tumlin sandstone, ε1 determined based on the
measurement of the piston displacement, ε3 measured using a toroidal sensor
Rys. 12. Wyniki próby na trójosiowe ściskanie próbki piaskowca Tumlin; ε1 określone na podstawie
pomiaru przemieszczenia tłoka, ε3 zmierzone czujnikiem toroidalnym
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even relatively small measurement errors might lead to highly erroneous values of this
ratio. Even though the sample was protected with latex jackets, the liquid was kerosene
and pressures were small, the accuracy of the measurements of circumferential strain
seems satisfactory, mainly because the experiment did not last long, just 260 s. The
additional horizontal axis shows the strain in micrometers, which shows the resolution
in strain measurements to be several micrometers.
Other errors result from the inaccurate and non-repeatable mounting of the sensor on
the sample. In such a case, the plane of the torus is not perpendicular to the sample axis
and/or the torus turns do not rest on one and the same plane. The maximum resulting
error is about 0.2%.
Of course, the sensor may be also used in uniaxial compression tests. In such a case
the strain measurement accuracy may be better because no pressure error will occur.
7. Shortcomings of strain gauges in the conditions of high pressure
As was mentioned in the introduction, the main cause of any shortcomings in strain
measurements using resistance gauges cemented onto a sample is the stretching of the
resistive grid as it is squeezed into the sample unevennesses of the sample’s surface
under the action of pressure. The grid deformation depends on pressure and surface
structure, in some cases the grid might even be broken, leading to greater or smaller
measurement errors. Certain errors are considerable and render the measurement data
utterly unreliable. Yet even some erroneous measurements might be classified as correct leading to major inaccuracies in the definition of rock properties. Smoothing of
the sample surface by applying a thick layer of epoxy resin is a good solution only in
the case of hard rocks with minor roughness. When investigating porous rocks of high
compressibility such smoothing is of little use. Besides the local properties of the rock
might change, for instance rock-resin composites might be produced that display entirely
different properties.
Another source of errors, though of minor significance, is the action of variable
hydrostatic pressure and the accompanying temperature changes upon the resistive grid.
The resistive material of the gauge (constantan) is temperature-compensated in the range
of 20°C with a several degree accuracy whilst temperature fluctuations in the chamber
are several tens degrees. In order to minimize the errors due to variable pressure and
temperature action, a compensating strain gauge is employed, which is normally attached
to a steel plate and placed inside the pressure chamber. Together with the measuring
strain gauge it forms a part of the resistance bridge, enabling measurement of rock strains
relative to compressibility of the compensating plate. It appears, however, that steel is
not an optimal base for a compensating strain gauge as its thermal expansion is nearly
twice as large as that of rocks. Better results were achieved when using the synthetic
ruby (corundum) whose expansion and thermal conductivity is close to that of rocks. The
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ruby plate was several times thicker (5 mm) than the steel plate, so it was a better match
for a rock sample because of its thermal inertia. Figure 13 summarises the compressibility measurement data for a granite sample 22 mm in diameter and 44 mm in length
obtained in tests with compensating steel and corundum plates and with a compensating
element outside the pressure chamber. The continuous line indicates measurements taken
during the compression phase, the broken line – during decompression. Hysteresis of
the compressibility curve for the corundum plate is many times smaller than that for
a steel plate.
Fig. 13. Linear compressibility of granite measured using a resistance strain gauge
and various methods of compensation
Rys. 13. Ściśliwość liniowa granitu mierzona czujnikiem rezystancyjnym
przy różnych sposobach kompensacji
Before strain gauges are attached to the rock specimen, they are first pressure-conditioned. This procedure was first adopted by Prof. J. Gustkiewicz. The gauges are
placed on a steel plate (without cement), covered with latex sleeves and put inside the
chamber where the pressure is increased to the maximum admissible level. The results
are evident on photos shown in Figure 14. On the left-hand side there is a strain gauge
after pressure-conditioning at 350 MPa, showing paper stamped on beams connecting
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the wires and paper squeezed due to improper binding of the strain gauge. These effects
cause serious measurement errors and the conditioning prevents them.
Fig. 14.The strain gauge after (left) and before conditioning (right) at 350 MPa pressure
Rys. 14. Tensometr po (z lewej) i przed kondycjonowaniem (z prawej) ciśnieniem 350 MPa
8. Advantages and disadvantages of inductive and resistance sensors
in strain measurements
The major advantage of a linear inductive sensor is that the structure of the tested
material (pores and cracks) does not affect the measurement results; besides the range
of measurable strains, both positive and negative, is much larger. Other benefits include:
repeated use of the sensor, its easy and fast assembly on the rock sample, low price,
no need to use an A/D converter for digitisation. Besides, the measurements of global
strains are possible (also by toroidal sensors), which is of key importance particularly
in the case of heterogeneous rocks. The major drawback is the inferior measurement
accuracy in relation to strain gauges. This is because the sensors are more sensitive
to pressure and temperature fluctuations that affect the inductivity and capacity of the
sensor’s resonance circuit. Besides, friction produced between a toroidal sensor and the
specimen leads to additional error.
The chief benefit of strain gauges is higher accuracy and improved measurement
resolution when the sample surface is smooth. Otherwise, measurements may be rendered impossible or, even worse, wholly unreliable, which is a major shortcoming. Other
drawbacks are: relatively small measuring range of strains (up to several percent) and
the measurement base restricted to the size of the strain gauge. Measurements at and
after failure are impossible due to cracks even if the strains do not exceed the admissible
levels for the given strain gauge type. Another disadvantage is that cementing the gauge
onto the sample is time-consuming. In addition, the gauge cannot be used repeatedly
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which vastly increases the cost of the experiments, especially when tests are conducted
on a large number of samples to allow for the statistical treatment of data.
An alternative approach to transverse strain measurements utilises a strain gauge
bonded to an elastic metal strip and swing-attached on the opposite sides of the sample’s
middle section, thus forming a strain reducer. A major benefit of this approach is that
friction is eliminated whilst the drawbacks include the point-to-point strain measurements,
uncertain quality of gauge attachment to the sample, particularly when protected with
covers, and a smaller measurement range when compared to toroidal sensors. Although
the measuring range can be extended through the control of the sensor shape, it will
occupy too much space inside the chamber.
9. Summary
Coreless inductive sensors for strain measurements are a good solution in the testing
of porous and conductive rocks as the risk involved in attaching resistance gauges
directly to the sample is thus eliminated. Inductive sensors are highly sensitive and
have a wide measuring range, from micrometers to centimeters. Simple construction,
mechanical strength and easy assembly are their major assets as a measuring instrument.
In this method an A/D transducer is quite unnecessary. Sensors, sensor-fitting methods
and dedicated oscillators are still being improved and so their measuring accuracy is
enhanced. The application of a reference sensor eliminates the effects of oscillator instability associated with ambient temperature or voltage supply fluctuations, thus enabling
long-term measurements (lasting for several days) and helping to reduce the effects of
variable pressure and temperature on the measuring sensor.
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Received: 14 March 2007