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On Certain Determinantal Method of Equation and Effective Pressure Evaluation... |
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On Certain Determinantal Method of Equation and Effective Pressure Evaluation on the Basis of Laboratory Researches Author: A. Nowakowski
Analysis of rock mechanical properties usually is conducted with the assumption that this is a monophase
medium, which means that a rock is being treated as a continuous medium consisting of only
solid phase. Such an assumption leads to ignoring widely known fact that a rock is not a continuous
medium consisting of not only solid components but also of empty voids of different sizes and shapes.
The complexity level is increased when one considers the fact that such voids referred to as pore space
may be fi lled with fl uid, which may interact with a rock in various ways. In particular it may be a strictly
mechanical interaction based on stress changes resulting from pressure of porous fl uid, but it could be
also physical-chemical or even chemical interaction that changes properties of rock due to interactions
between fl uid and rock skeleton (e.g. sorptive processes and chemical reactions). In the most complex
model one deals with a mixture of all above factors.
Intuition suggests that mathematical description of processes going on in stressed porous medium
would require mathematical description of pore space itself. Such description should contain information
on the size of such space, sizes and shapes of pores, their distribution in analyzed space but it also should
take into consideration the fact that they may form a network or may be isolated voids. Determination
of pore space properties is such complex that the problem of phenomena occurring in stressed porous
medium turned out to be of vital signifi cance and researchers started to look for methods of avoidance
of this problem.
Approach which is discussed herein was worked out on the basis of laboratory investigations of rocks
in a classical triaxial state of stress („individual test” – cf. Kovári et al., 1983). In such a test a cylindrical
rock sample is placed in a Kármán type chamber and is stressed with axially symmetric stresses which
comply with the condition σ1 ≥ σ2 = σ3 = p. Confi ning pressure σ2 = σ3 = p is placed with fl uid on a side
surface of a sample whereas axial stress (σ1) is placed with press piston that loads the front section of a
sample. A sample is divided from confi ning pressure exerting medium with deformable shield. Moreover,
porous space of assessed sample is fi lled with porous fl uid (liquid or gas) under constant pressure of pp.
In fi g. 1 stresses affecting the sample during the experiment were presented.
In case of a such stressed sample we may consider any characterizing rock quantity Q being the
function of p and pp pressures. This function creates a certain surface in a space of variables Q, p and
pp (Fig. 2). On such surface we may distinguish a curve that complies with equation (10) i.e. the curve
along which interesting for us Q quantity is constant. This curve resulting from equation (10) shall be then
projected on the (pp, p) plane. As a result of such operation we achieve relation (11). Such relation defi nes
the pairs of p and pp points, for which the analyzed Q quantity is constant and shall be referred to herein
as the effective pressure law (Robin, 1973). If we substitute in equation (11) with (12), we shall receive
relation (13). At present in such relation we shall refer to p’ pressure as the value of effective pressure
for the effective pressure law (11) and Q quantity complying with the condition (10).
The value of effective pressure p’ defi ned with equation (13) may be treated as some substitute
confi ning pressure, which when applied to the rock for pp = 0 exerts on the investigated Q quantity the
same infl uence as a pair of non-zero p and pp pressures complying with the conditions of (10) and (11).
Whereas function (11) which binds confi ning and porous pressure together shows what pairs of p and
pp pressures may represent a specifi c Q quantity or if necessary to draw reverse conclusion on what Q
quantity for specifi c p and pp pressure values one may expect.
The above formalism of the description of effective pressure was coined by Robin (1973), who based
his conclusions on Nura and Byerlee (1971) and Brace’a (1972). The way to apply such formalism in
a laboratory results analysis was described by Gustkiewicz (1990) and further developed by Gustkiewicz
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et al. (2003, 2004) and Nowakowski (2005). The last papers are devoted to phenomena occurring in rock
samples where stress reached differential strength limit in particular. From the above considerations it may
be derived that effective pressure (11) shall depend on e.g.:
− analyzed quantity Q of a rock,
− level of stress in a rock sample,
− properties of pore space in a rock.
The subject of our investigations shall be sandstone from lower Triassic period from Tumlin (referred
hereto as Tumlin rock) that can be found in the northern part of Świętokrzyskie Mountains in the region
of Suchedniów. It will be shown on an example how to determine the effective pressure law in practice
and its relation to the type of used porous fl uid. The authors will analyze an example of porous fl uid
that is neutral physically and chemically for rocks (kerosene) and an example of strongly sorptive gas
(carbon dioxide). The analysis consisted of series of “individual tests” and determination on such basis
the differential strength limit (Rσ1–σ3) as the function of (pp) porous fl uid and (p) confi ning pressure and
looking for effective pressure laws and effective pressure values accordingly to methodology described
in chapter 2 of this paper.
The present authors started their investigations with „Tumlin” sandstone saturated with kerosene. The
results of this experiment in respect of Rσ1–σ3 values are shown in table 2. On the basis of shown in this
table results the authors made charts showing relations between differential strength limit (Rσ1–σ3) and
porous pressure (pp) at confi ning pressure (p) as a parameter shown in fi g. 9. The effective pressure law
and value of effective pressure for two values of differential strength limit was sought for: 60 MPa and
263 MPa as shown in fi g. 9. According to the procedure described in chapter 2 parameters of intersection
of straight lines were found with equations and bell-shaped curves of the type (15). Those points were
indicated pp ~ p coordinate system and necessary approximations were made. Results were shown in fi g.
10. Finally for selected values of differential strength limit the authors achieved equations and effective
pressure values (p’) given with relations (25).
Next step consisted of using the same procedure for the Tumlin sandstone whose porous space was
fi lled with carbon dioxide. In table 3 Rσ1–σ3 values achieved in the experiments and in fi g. 11 respective
curves are shown. Effective pressure laws and values of effective pressure were obtained for Rσ1–σ3 equal
to 195 MPa and 277 MPa. In case of selected values of differential strength limit the full procedure of
determination of effective pressure laws and values of effective pressure was achieved and showed as
a relation (28). Their geometrical interpretation is shown in fi g. 12.
Described considerations showed certain method of determination of effective pressure law and
effective pressure value for rocks which is based on a laboratory triaxial compression test. This method
allows for determination of relation between analyzed property of a rock and confi ning and porous pressures
values. Examples presented in chapter 4 show the application of this method in reference to the so
called differential strength limit.
The achieved results show that if a relation (3) is used as the effective pressure law then α coeffi cient
serves the role of a certain “balance” which determines the infl uence of porous pressure on the fi nal value
of effective pressure. Such coeffi cient may be equal to a value from three different ranges:
1) α < 1; equation (3) is the effective pressure law in the form proposed by Biot and those who further
developed this concept (e.g.: Nur & Byerlee, 1971; Rice & Cleary, 1976; Zienkiewicz & Shiomi,
1984; Detournay & Cheng, 1993). The range of applicability of the equation (3) is in this form automatically
limited to the range of applicability of Biot’s consolidation theory and as follows the range
of Hooke’s law applicability.
2) α = 1; the equation (3) has classical form coined by von Terzaghi (Terzaghi, 1923) shown in (1).
The possibility of its application in case of rocks is limited by – mentioned in the introductory part
– Handin’s conditions (cf. Handin et al., 1963). An important summary of the application of Terzaghi’s
formula may be found in Bluhm and de Boer work (1996).
3) Equation 1 < α ≤ ppp
–1; in which the value of α > 1 coeffi cient is allowed. At present there is no possibility
to check real maximum values of this coeffi cient. Given value of „upper” limit is derived only
from the conditions of the experiment (porous pressure cannot be higher that confi ning pressure – cf.
chapter 1). The fact that this coeffi cient may be higher than unit was demonstrated experimentally for
the fi rst time by Gustkiewicz et al. (2004) in reference to differential strength limit. The reasons for
increased signifi cance of porous pressure for evaluated material property must be sought for in the
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appearance of not only mechanical interactions but also physical and chemical processes – sorption in
particular between rock and porous fl uid. For more refer to e.g. Gustkiewicz et al. (2004), Gustkiewicz
and Nowakowski (2005) and Nowakowski (2005).
It must be noted also that – in contrast to Biot’s theory considerations – the procedure demonstrated
in chapters 1 i 2 of this paper that determines effective pressure laws and effective pressure values is not
conclusive in respect of a mathematical form of such equation. In particular there are no reasons not to
consider Q(p, pp) surface in chapter 1 as the straight-drawn surface. In such case a curve (10) marked
on Q(p, pp) surface does not have to be a straight line and as a consequence its projection on the (p, pp)
plane does not have to be a straight line either in the equation (11). It means that the equation of effective
pressure (11) may be non-linear because of porous pp pressure. |
2011