INTRODUCTION

Understanding abnormally high pore pressures is important for studying primary petroleum migration and drilling, as well as for analyzing sedimentary basins (Magara, 1978; Gretener, 1979; Ungerer et al., 1984; Bethke, 1986). Although many factors have been proposed to explain the causes of overpressuring, the two factors we focus on in this paper are sediment compaction and the aquathermal pressuring. The effects of these two factors are considered in most hydrodynamic evolution models.

Barker (1972) suggested that thermal expansion of pore water plays a considerable role in abnormal pore pressure and proposed the term "aquathermal pressuring." Studies on the effect of aquathermal pressuring on generating, increasing, and maintaining overpressure has been controversial (Dickinson, 1953; Barker, 1972; Bradley, 1975; Magara, 1975; Chapman, 1980; Barker and Horsfield, 1982; Chapman, 1982; Daines, 1982; Shi and Wang, 1986; Bethke, 1986; Chen and Luo, 1988).

The works of Sharp (1983), Shi and Wang (1986), and Bethke (1986), among others, proved that compaction is the principal mechanism of abnormal pressure. Compaction occurs from deposition until metamorphism and results in two effects favorable to abnormal pressure occurrence: (1) reduction of the hydraulic conductivity by decreasing porosity and permeability, and (2) steady expulsion of excess pore fluid to maintain the pore pressure equilibrium. Other factors, including aquathermal pressuring, may act only on this foundation (Magara, 1975; Chapman, 1982; Sharp, 1983; Bethke, 1986; Shi and Wang, 1986; Chen and Luo, 1988).

However, the effect of aquathermal pressuring on overpressure has not been resolved. Some authors emphasized that this factor would be very important under some "effective sealing" conditions (Magara, 1978; Barker and Horsfield, 1982; Sharp, 1983). Such sealing conditions could be ascribed to the environment, which is known to affect the occurrence of overpressure (Magara, 1978; Bethke, 1986).

In this paper, we analyze the influence of the environment on the occurrence of overpressure, emphasizing the effect of aquathermal pressuring, within a thick fine-grained detrital sediment bed, an environment considered favorable for aquathermal pressuring to influence overpressuring.

BASIC HYDRODYNAMIC EQUATION AND PARAMETERS

As shown in Appendix 1, the basic hydrodynamic equation for pore water pressure can be written as:

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(Equation 1; SEE PAGE IMAGE)

where d/dt is the derivative following the solid movement (see Appendix 1).

In this equation, the second term in the brackets on the left side:

(Equation; SEE PAGE IMAGE)

and the second term on the right side:
(Equation; SEE PAGE IMAGE)

represent the effect of compaction; the third term on the right side:
(Equation; SEE PAGE IMAGE)

represents the aquathermal effect; the last term, q, represents the discharge of fluid volume within the pores. We do not discuss these latter pressure-augmenting mechanisms in this paper.

On the right side of equation 1, the first term:

(Equation; SEE PAGE IMAGE)

is a diffusion term representing the capacity of fluid expulsion of sediments. In this one-dimensional study, we neglect horizontal components of fluid flow.

In fact, equation 1 reflects the equilibrium relation between source terms and the diffusion term; the diffusion term represents the capacity of fluid expulsion, and its magnitude depends on the permeability of sediments (k), viscosity of pore fluid (mu), and the gradient of pore pressure. The source terms represent the various factors causing excess fluid volume that should be drained. For a given condition, a balance will be established between the diffusion term and source terms. If conditions change, this balance would be upset and a different balance would be established through the readjustment of permeability, viscosity, and/or pore pressure.

The parameters used in equation 1 are discussed in the following section.

Porosity

For several decades, Athy's clayey sediments' porosity-depth relation (Athy, 1930) has been widely recognized:

(Equation 2; SEE PAGE IMAGE)

This relation, however, can be applied only in a normal compaction zone where pore pressure is hydrostatic. Based on arguments developed by those working in soil mechanics (Hubbert and Rubey, 1959), the compaction of sediments is more appropriately described as a function of the effective stress sigma by the relation:

(Equation 3; SEE PAGE IMAGE)

with
(Equation 4; SEE PAGE IMAGE)

Unlike equation 2, relation 3 is applicable even in an abnormal zone where pressure is greater than hydrostatic. If rock density is treated as constant (Dickinson, 1953; Sclater and Christie, 1980), the value of b may be written as:

(Equation; SEE PAGE IMAGE)
within the normal compaction zone.

Note that equation 3 applies only where there is little or no chemical diagenesis in sediments and where the compaction process is progressive. In fact, the compaction process is not reversible (Lambe and Whitman, 1969; Magara, 1976; Shi and Wang, 1986); if the current effective stress happens to decrease, the porosity does not increase as defined in equation 3.

However, besides the plasticity, the grains of fine-grained detrital sediments (the majority of which are clay mineral fractions) also possess some elasticity. Walder and Nur (1984) and Shi and Wang (1986) proposed that when the effective stress becomes smaller than the maximum stress ever experienced, some elastic rebound or decompaction may occur in sediments.

Permeability

Generally, in fine-grained sediments, the permeability is assumed to be a function of porosity (Terzaghi, 1925; Rieke and Chilingarian, 1974; Magara, 1978; Ungerer et al., 1984; Shi and Wang, 1986). According to Terzaghi (1925) and Rieke and Chilingarian (1974), the permeability at depth Z may be written as a function of porosity in the following form

(Equation 5; SEE PAGE IMAGE)

For argillaceous sediments, the magnitude of lambda may range from 10{-3} to 10{-7} d. In our calculation, lambda is taken as 10{-5}. From equation 5, if porosity equals 20%, the permeability is 3.2 x 10{-9} d or 3.2 x 10-{21}m{2}.

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Recent works (Foster, 1980; Freed and Peacor, 1989) suggested that during the transformation of clay minerals (principally from smectite to illite) chemical diagenesis occurs in the fine-grained sediments. Although the details of this kind of diagenesis are not yet clear, these processes effectively reduce the sediment permeability. In our calculation, the permeability changes related to this diagenesis are simulated by multiplying the results calculated from equation 5 by a small coefficient C[k] characterizing the sealing degree, with values of 10{-n} (n = 1, 2, 3...).

Viscosity

According to Mercer et al. (1975), in the temperature range from 0 to 300 degrees C, changes in the viscosity of pore fluid occurring with changes in temperature may be written as:

(Equation 6; SEE PAGE IMAGE)

where
A = (T - 150 degrees C)/100 degrees C
(Equation 7; SEE PAGE IMAGE)

This equation is experimental and one in which the possible influence of pressure is neglected.

In addition to the previous parameters, the parameters listed in Table 1 also are treated as constants.

Assumptions and Results

To discuss the influence of various geologic environments, especially the effect of aquathermal expansion, on the occurrence of overpressure, we consider several parameters representing geologic conditions: compaction coefficient (c), sediment burial rate, geothermal gradient, and sealing conditions. We used extreme values of each parameter because some researchers thought that under "ordinary" geologic conditions, permeability is not small enough to form efficient sealing conditions for the aquathermal pressuring to contribute sufficiently to the total excess pressure (Magara, 1975; Barker and Horsfield, 1982; Sharp, 1983; Bethke, 1986).

In our calculation, a fine-grained argillaceous sediment bed with a present thickness of 5900 m was deposited upon a sand layer (Figure 1) that has a much larger permeability.

In most geologic environments, the effect of aquathermal pressuring is much smaller than that of compaction in terms of excess pressure (Bethke, 1986; Shi and Wang, 1986). Thus, in each situation, we must perform the computation twice: one time with only the compaction as a source term (i.e., alpha = 0 in the equation), and again with both compaction and aquathermal pressuring as source terms.

Compaction Coefficient

Magara (1975) analyzed the geologic significance of the compaction coefficient c. Coefficient c in equation 2 is a function of the depositional environment, the deposition velocity, and the composition and properties of sediments. This coefficient changes from area to area and even from sediment section to sediment section in the same area, but for a given section, this coefficient does not change in the normal compaction zone during burial; therefore, the coefficient actually represents the characteristics of the sediment fabric, and governs the compaction of the sediments. The larger the coefficient, the greater the decrease of porosity and permeability with burial, and the greater the

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need, in terms of compaction, for expulsion of the excess pore fluid during the same period.

The range of variation of the compaction coefficient is large. Bethke (1986) declared that in nearly all porosity profiles published, the coefficients fall between Dickinson's curve, corresponding to c = 0.00014 m{-1}, and Athy's curve, corresponding to c = 0.0016 m{-1}. In our calculation, we used the following values for the compaction coefficient: 0.0001, 0.0004, 0.0008, and 0.0012 m{-1}. We used the average values for the other parameters: 100 m/m.y. sedimentation rate, 30 degrees C/km temperature gradient, and a surface porosity of 0.50 (Table 1).

Figure 2 shows four pressure profiles corresponding to different compaction coefficients c. The results in Figure 2 demonstrate that following the increase of the compaction coefficient, the excess pressure, i.e., the difference between the actual pressure and the hydrostatic pressure, increases rapidly, but the aquathermal factor contributes only very little to the total excess values (dashed lines). Moreover, the greater the coefficient, the smaller the effect of aquathermal factor. This result is inverse to that predicted by some researchers (Magara, 1975; Barker and Horsfield, 1982; Sharp, 1983), who stated that a greater compaction coefficient may more rapidly reduce the degree of sealing.

Sediment Burial Rate

In this paper, sediment burial rate refers to the rate of deposition of overlying sediments. Burial velocity mainly reflects the influence of time: during a period of time, the larger the burial rate, the greater the fluid volume that should be expelled. As a result, the potential of abnormal pressure occurrence increases.

To test the effect of the sediment burial rate over the geopressure evolution, we used the following average burial rates in our calculations: 50, 100, 500, and 1000 m/m.y., respectively, whereas we used a compaction coefficient c of 0.0004 m{-1} and a temperature gradient of 30 degrees C/km.

Figure 3 illustrates several pressure profiles calculated with different burial rates. The curves in Figure 3 show that when burial rate increases, the pore pressure also increases, but under none of these conditions is the excess pressure significantly affected by aquathermal factor. Even when the rate of deposition equals 1000 m/m.y., the maximal excess pressure value caused by aquathermal factor is only 8 bars (10{5}Pa).

Geothermal Gradient

A high temperature gradient contributes to the role played by the aquathermal factor in the development of overpressure (Barker, 1972; Magara, 1976; Sharp,

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1983; Bethke, 1986; Shi and Wang, 1986).

In our calculations, we used thermal gradient values of 10, 30, and 50 degrees C/km, respectively, whereas we used a sediment burial rate of 100 m/m.y. and a compaction coefficient of 0.0004 m{-1}.

Figure 4 illustrates the calculated results with different temperature gradients. As the temperature gradient increases, the total pressure decreases. As discussed in a following section, this unexpected effect is due to the decrease of pore-fluid viscosity. In fact, the results obtained with and without aquathermal pressuring shows that this effect increases very little with the temperature gradient (Figure 4).

Sealing Conditions

The results we obtained indicate that under general conditions, including some extreme conditions that seem favorable to the influence of aquathermal pressuring, aquathermal pressuring cannot play an important role in total excess pressure development. This means that, in a general geologic environment, the porosity decrease due to mechanical compaction and the corresponding closing situation are not enough to allow aquathermal pressuring to influence abnormal overpressuring.

Considering the possibility of permeability reduction by chemical diagenesis in clayey sediments (Foster, 1980; Freed and Peacor, 1989), we multiply the assumed permeability by 10{-n} (n = 1, 2...) when the sediment burial depth becomes greater that 1500 m, i.e., the average depth where smectite begins to transform to illite.

Figure 5 illustrates the results of using different sealing coefficients. We observe that if the permeability is reduced by one magnitude (n = 1), the excess pressure increases significantly, but further increase of n does not result in comparable pressure increase. For n greater than 4, the calculated pressure and the effect of the aquathermal factor are almost similar whatever the value of n, meaning that the sediments may be considered as completely sealed.

Along with the reduction of permeability, the excess pressure contributed by aquathermal expansion does not increase much, even under almost impermeable conditions (Figure 5). Therefore, in a nearly impermeable argillaceous sediment bed, the effect of aquathermal pressuring represents only a small part of the total excess pressure, and to a degree that is rather lower than expected by some researchers (Barker, 1972; Magara, 1975; Sharp, 1983; Bethke, 1986).

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DISCUSSION

The previous calculations of the pressure occurrence in a continuous sedimentary profile show that in general geologic environments, the aquathermal factor does not play an important role in pore pressure, even using extreme values for one or two parameters.

When Barker (1972) proposed this mechanism, he emphasized that a thermal gradient of at least 15 degrees C/km is necessary for aquathermal pressuring to contribute significantly to the pore pressure, and that this contribution would increase with an increasing thermal gradient. Our results show that as the temperature gradient increases, the effect of aquathermal expansion indeed increases slightly, but, at the same time, the total pore pressure decreases. This latter effect is due to the reduction of viscosity with temperature that affects the diffusion term of equation 1. In this case, the excess pressure caused by aquathermal expansion is so small that it cannot compensate for the pressure decrease due to the viscosity effect, even with a thermal gradient of 50 degrees C/km.

The sediment deposition rate does affect the pore pressure, but under given geologic conditions, even for a rate of 1000 m/m.y., its influence on the effect of aquathermal factor is not important (Figure 3).

The compaction coefficient c influences sediment porosity and permeability, and therefore, was considered as the most important parameter in abnormal pressure evolution. Magara (1976) emphasized that if the value of this parameter becomes large enough, it will create low permeability and produce higher excess pressure; in turn, because of these conditions, a large fraction of the total pore pressure would be due to aquathermal pressuring.

The calculated results demonstrate that the increase of this parameter indeed affects pore pressure, but its influence is not easily detected (Figure 2), and the higher the compaction coefficient c, the lower the effect of aquathermal expansion (Figure 2).

Further calculations with sediments that possess a very low permeability (Figure 5) indicate that, although the pore pressure increases rapidly with the lithostatic pressure gradient, the effect of aquathermal pressuring on the total fluid pressure remains very weak. This means that permeability reduction is not the most important factor for the influence of the aquathermal factor in overpressure occurrence.

In the following section, we study the aquathermal pressuring effect, neglecting the term of permeability in equation 1. As shown in Appendix 2, a nondimensional quantity may be defined:

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(Equation 8; SEE PAGE IMAGE)

This quantity represents the maximum potential of pore pressure caused by aquathermal pressuring scaled to the pressure effect due to overlying loading. In this expression, G[T] is the temperature gradient and alpha[phi] is the porosity compressibility, a function of the compaction coefficient c (Appendix 1). The variations of P[aqua], the excess pressure potential due to aquathermal factor relative to the overload and corresponding to various values of these two parameters, are illustrated in Figure 6, in which P[aqua] is plotted as a function of depth logarithmically on the left side of the figure and arithmetically on the right side of the figure. Note that P[aqua] does not vary much with depth.

When the porosity compressibility is not equal to zero, P[aqua] increases along with the temperature gradient and decreases with the compaction coefficient. From Figure 6, one can find that the effect of the compaction coefficient c becomes more important when its value becomes smaller. When the temperature gradient is 50 degrees C/km, a compaction coefficient of 0.0001 m{-1} may cause P[aqua] to reach 0.1, but P[aqua] decreases rapidly when the compaction coefficient increases. When c values become larger than 0.0003 m{-1}, the aquathermal pressure becomes smaller than 0.03 (Figure 6).

When the compaction coefficient c is 0, P[aqua] depends only on the ratio between the fluid thermal expansibility alpha and compressibility beta:

(Equation 9; SEE PAGE IMAGE)

In this case, P[aqua] does not vary with depth and reaches values of 0.45, 1.40, and 2.35, for temperature gradients of 10, 30, and 50 degrees C/km, respectively. This result is in agreement with Barker's work (Barker, 1972; Magara, 1978).

Assuming c = 0 means that the porosity compressibility alpha[phi] is null. In fact, actual observations, including those in the laboratory and in the field, demonstrate that porosity does decrease with increasing effective stress, even if this deformation is not reversible (Domenico and Palciauskas, 1979; Walder and Nur, 1984; Bethke, 1986; Shi and Wang, 1986). Therefore, the state equation of water (pressure, temperature, density) cannot be of direct use, as proposed by Barker (1972), for studying aquathermal mechanisms in actual geologic conditions, a fact already noted by Shi and Wang (1986).

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CONCLUSIONS

(1) Several environmental conditions, such as compaction, temperature gradient, and sediment deposition rate, can affect the relative importance of the various pressuring mechanisms, and particularly in our study, the importance of compaction and aquathermal factors.

(2) In actual geologic environments, aquathermal pressuring does not play an important role in the occurrence of abnormal overpressuring. Better sealing tends to raise the pore pressure, but it cannot offer sufficient conditions to cause significant aquathermal effect. In practical geopressure studies, the pressuring effect of aquathermal factor could be neglected.

(3) In some areas, such as the United States Gulf Coast, the excess pressure is so great that it cannot be explained by compaction alone; however, aquathermal pressuring does not significantly contribute to this excess pressure. Thus, some other, more effective mechanisms, such as reduced permeability resulting from clay mineral diagenesis or the maturation of organic materials, must be considered.

(4) Porosity compressibility seems to be the most important coefficient controlling overpressure occurrence, as well as the effect of the aquathermal pressuring. However, the mechanical aspects of the porosity compressibility of argillaceous sedimentary rocks are still poorly documented and further experimental and theoretical work on the rheology of these argillaceous rocks is needed.

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APPENDIX 1

For a porous medium, the conservation of mass of the fluid and solid phases with respect to fixed space coordinates may be expressed as follows:

(Equation 10; SEE PAGE IMAGE)

(Equation 11; SEE PAGE IMAGE)

Equation 10 may be written as:
(Equation 12; SEE PAGE IMAGE)

where
(Equation; SEE PAGE IMAGE)

is the volume flow of fluid relative to solid matrix. According to Darcy's law and considering the compressibility of the pore water, we have:

(Equation 13; SEE PAGE IMAGE)

If we assume that rho[s] is constant (that is the solid grains are incompressible), and that q[s] = 0 (during the process of diagenesis, there is no solid material obtained or lost), equation 11 becomes:

(Equation 14; SEE PAGE IMAGE)

Applying the relation
(Equation; SEE PAGE IMAGE)

equation 14 may be written as:
(Equation 15; SEE PAGE IMAGE)

Substituting equation 15 into equation 12 and arranging the result, we get:

(Equation 16; SEE PAGE IMAGE)

For a compressible fluid, the density is function of pressure and temperature:

(Equation 17; SEE PAGE IMAGE)

If the deviatoric stress is not considered, the porosity term in equation 16 may be further expressed as a function of effective stress:

(Equation 18; SEE PAGE IMAGE)

Substituting equation 3 into the previous one and differentiating the right side, we obtain:

(Equation 18; SEE PAGE IMAGE)

According to S = P + sigma, equation 19 becomes:
(Equation 20; SEE PAGE IMAGE)

where alpha[phi] = b phi is the porosity compressibility that reflects the variation of porosity under the action of the effective stress.

Substituting equations 12, 17, and 20 into equation 16, we get our final basic hydraulic equation:

(Equation 21; SEE PAGE IMAGE)

APPENDIX 2

If the sediments are placed in a completely closed environment (i.e., permeability equals 0), because the source term q is omitted in our problem, equation 1 may be written as:

(Equation 22; SEE PAGE IMAGE)

The magnitude of porosity compressibility:
(Equation; SEE PAGE IMAGE)

is about 10{-8}Pa{-1}, whereas that of fluid compressibility is about 10{-10}Pa{-1}. Equation 22 then may be approximated by:

(Equation 23; SEE PAGE IMAGE)

For a time step dt, the corresponding burial depth step is dZ. If we neglect the variation of the sediment bulk density (Dickinson, 1953; Magara, 1978), we have:

(Equation 24; SEE PAGE IMAGE)

where G[T] = dT/dZ is the temperature gradient.

Equation 24 means that the pore pressure is the sum of mechanical compaction and pore fluid aquathermal expansion. The relative importance of these two mechanisms is a function of temperature gradient G[T], porosity compressibility alpha[phi], and porosity phi. In this equation, the second term in the bracket may be defined as the aquathermal pressuring potential P[aqua]:

(Equation 25; SEE PAGE IMAGE)

APPENDIX 3:

b = compaction coefficient (Lt{2}M{-1}); c = compaction coefficient (L{-1}); g = acceleration of gravity (Lt{-2}); G[T] = temperature gradient (TL{-1}); k = intrinsic permeability (L{2}); L = length; M = mass; P = pore fluid pressure (ML{-1}t{-2}); q = fluid specific discharge (t{-1}); q[s] = solid specific discharge (t{-1});

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S = lithological pressure (ML{-1}t{-2}); T = temperature; t = time; SEE PAGE IMAGE FOR SYMBOL = velocity of fluid with respect to fixed coordinates (Lt{-1}); SEE PAGE IMAGE FOR SYMBOL = velocity of solid grains with respect to fixed coordinates (Lt{-1}); Z = observing depth (L); alpha = isobaric coefficient of thermal expansion for pore fluid (T{-1}); alpha[phi] = porosity compaction coefficient (Lt{2}M{-1}); beta = isobaric coefficient of compressibility for pore fluid (Lt{2}M{-1}); lambda = coefficient in relation of porosity and permeability (L{2}); mu = fluid dynamic viscosity (ML{-1}t{-1}); rho = fluid density (ML{-3}); rho[s] = solid grain density (ML{-3}); rho[b] = mean density of sediments (ML{-3}); sigma = effective stress (ML{-1}t{-2}); phi = porosity of sediments; phi[0] = round surface porosity of sediments