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A Quantitative Comparison of some Mechanisms Generating Overpressure in Sedimentary Basins
Institutt for energiteknikk Institute for Energy Technology A quantitative comparison of some mechanisms generating overpressure sedimentary basins
Magnus Wangen Institute for Energy Technology Box 40 N-2007 Kjeller NORWAY
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Submitted October 30, 2000 Revised Mars 7, 2001 DISCLAIMER
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Title and subtitle A Quantitative Comparison of some Mechanisms Generating Overpressure in Sedimentary Basins
Authors) l\/j. 1 \ Reviewed >0 Approved
Magnus Wan^cnj (j Tom Pedersen Tor Bjdmstad
Abstract Expulsion of fluids in low permeable rock generate overpressure. Several mechanisms are suggested for fluid expulsion and overpressure build-up, and some of them have here been studied and compared. These are mechanical compaction, aquathermal pressuring, dehydration of clays, hydrocarbon generation and cementation of the pore space. A single pressure equation for these fluid expulsion processes has been studied. In particular, the source term in this pressure equation is studied carefully, because the source term consists of separate terms representing each mechanism for pressure build-up. The amount of fluid expelled from each mechanism is obtained from these individual contributions to the source term. It is shown that the rate of change of porosity can be expressed in at least two different ways for dehydration reactions and oil generation. One way in which the reduction in the solid volume enhances the porosity, and another where the porosity remains constant. A gravity number is defined by the permeability and the Darcy flux of the expelled fluid. The gravity number is shown to be a useful indicator for overpressure build-up. The gravity number is also used to estimate the thickness a seal needs for overpressure to rise from almost hydrostatic conditions abovethe seal to almost lithostatic conditions below the seal. Simple expressions for the Darcy fluxes caused by mechanical compaction, dehydration of clays, hydrocarbon generation and cementation of the pore space are derived. It is shown how these expressions can be combined with the gravity number to obtain an upper bound for the permeability for overpressure buiol-up to take place. Mechanical compaction and cementation of the pore space are shown to be the strongest expulsion mechanisms. Although most mechanisms can generate overpressure alone given a sufficiently low permeability, it is concluded that cementation of pore space is the most likely mechanism for overpressure generation in deeply buried reservoirs.
Key Words Pressure build-up, mechanical compaction, aquathermal pressure, dehydration, hydrocarbon generation, cementation
ISSN ISBN Numbers of Pages 0333-2039 82-7017-322-3 57 Supplementary Data Abstract
Expulsion of fluids in low permeable rock generate overpressure. Several mechanisms are suggested for fluid expulsion and overpressure build-up, and some of them have here been studied and compared. These are mechanical compaction, aquathermal pressuring, dehydration of clays, hydrocarbon gen eration, and cementation of the pore space. A single pressure equation for these fluid expulsion processes has been studied. In particular, the source term in this pressure equation is studied carefully, because the source term consists of separate terms representing each mechanism for pressure build-up. The amount of fluid expelled from each mechanism is obtained from these in dividual contributions to the source term. It is shown that the rate of change of porosity can be expressed in at least two different ways for dehydration reactions and oil generation. One way in which the reduction in the solid volume enhances the porosity, and another where the porosity remains con stant. A gravity number is defined by the permeability and the Darcy flux of the expelled fluid. The gravity number is shown to be a useful indicator for overpressure build-up. The gravity number is also used to estimated the thickness a seal needs for overpressure to rise from almost hydrostatic condi tions above the seal to almost lithostatic conditions below the seal. Simple expressions for the Darcy fluxes caused by mechanical compaction, dehydra tion of clays, hydrocarbon generation and cementation of the pore space are derived. It is shown how these expressions can be combined with the gravity number to obtain an upper bound for the permeability for overpressure build up to take place. Mechanical compaction and cementation of the pore space are shown to be the strongest expulsion mechanisms. Although most mech anisms can generate overpressure alone given a sufficiently low permeability, it is concluded that cementation of pore space is the most likely mechanism for overpressure generation in deeply buried reservoirs.
Key words: pressure build-up, mechanical compaction, aquathermal pres sure, dehydration, hydrocarbon generation, cementation
0 1 Introduction
Overpressure is generated when fluids are expelled from low permeable rocks. Expulsion of fluids and the associated overpressure build-up have been suc cessfully modeled as mechanical compaction during deposition of sediments, (Lerche 1990, Ungerer et al. 1990, Audet 1996). Mechanical compaction is here understood to be compaction where the porosity is controlled by the effective vertical stress. However, several other mechanisms are proposed for overpressure build-up. Among these are thermal expansion of the pore fluid, dehydration of clays, hydrocarbon generation, and cementation of the pore space, (Osborn and Swarbrick, 1997). It is therefore a need for means to compare the different mechanisms. The study aims at this with simple ex pressions which make comparisons possible. This study is based on a pressure equation which account for the above mentioned mechanisms for overpressure build-up. This equation is derived from conservation laws for each species (or mineral) in a multi-component medium and a conservation law for the fluid. These conservation laws have source terms which accounts for the minerals and the fluid being produced or consumed. Modeling some of the pressure generation mechanisms therefore ends up in selecting proper source terms to represent the different processes involved. Another important issue in over pressure modeling is how the rate of change of porosity is modeled. The choices made here imply that the equation for the overpressure becomes a parabolic differential equation. The different source terms for each pressure generating mechanism are seen to be added together as one effective source term for overpressure build-up. A study of each single source term allows the relative importance of the different mechanisms to be studied individually.
The source terms in the equations for overpressure build-up have been identi fied and studied earlier in the case of aquathermal pressuring. Both Shi and Wang (1986) and Luo and Vasseur (1992) discuss aquathermal pressuring based on source terms. This analysis considers a wider range of mechanisms than aquathermal pressuring. The expulsion rates of the different overpres sure generating mechanisms are integrated to Darcy fluxes. These Darcy fluxes allow the relative contribution of the different mechanisms to the total Darcy velocity to be studied. These Darcy fluxes are combined with a gravity number which indicates when there will be high or low overpressures. The gravity number also allows for a simple estimate of how thick a seal ought to be for high overpressure to develop across the seal.
1 This study is based on a 1-D pressure equation with only one lithology, which is a common setting for a large number of models studying compaction and fluid flow processes in sedimentary basins, (Gibson 1958, Bredehoeft & Han- shaw 1968, Smith 1971, Sharp and Domenico 1976, Bethke & Corbet 1988, Autdet & Fowler 1992, Wangen 1992, Birchwood & Turcotte 1994 Schneider 1996, Fowler & Yang 1999, L’Heureux & Fowler 2000, Yang 2000). However, the pressure equation is derived in 3-D and then simplified to the verti cal direction. Particular concern is paid to the definition of excess pressure, boundary conditions and how Darcy’s law is expressed in the excess pressure.
The current study does not consider all possible models for compaction of sed iments and the associated pressure build-up. In particular, viscious rehologi- cal models of the sediments are not considered, see for instance Birchwood & Turcotte (1994), Schneider et al. (1996), or Connolly & Podladchikov (1998, 2000), Suetnova & Vasseur (2000). Although such models lead to a different type of pressure equation than the one studied here, the same methods to compare mechanisms for overpressure generation could be applied.
Other mechanisms which are not accounted for, but can generate overpres sures in the subsurface, are cracking of oil to gas and meteoric fluid flow. Oil to gas cracking expels fluids because of the associated volume expansion. Meteoric fluid flow can lead to high overpressure in formations with good lateral permeability.
This paper is organized as follows: The basic assumptions like the conser vation laws are presented first. Then overpressure is defined and the main variable in the pressure equation introduced. The full pressure equation is given before a gravity number is defined. It is explained how the gravity number can be used to as an indicator for high overpressure. The contribu tion to the Darcy velocity from the different mechanisms for overpressure are then compared.
2 The model
2.1 Conservation laws
The pressure equation is derived from conservation laws for solid and fluid. The solid (the sediment matrix) is considered as a composition of N minerals.
2 Conservation of each mineral phase i is written as
+ V ■ (&&va) = Sj, (1) where fa is the bulk volume fraction of mineral i, Qi is the density of mineral i, vs is the velocity of the solid phase and s, is a source term for mineral i. The source term gives the rate at which the mineral is produced or consumed in units mass per bulk volume and time. The velocity vs of the solid is the same for all mineral phases i. Conservation of fluid is written similarly as ^^ + V-(^/Vy) = gy, (2) where VD = 4>{-vf - Vs) = - — {Vpf + Qfgnz) (3) d The permeability tensor is denoted K and the viscosity of the fluid //. The fluid pressure is denoted by py and n 2 is the unit normal in the vertical direction (^-direction) . The volume fractions of each mineral phase and the porosity add up to one + Y, The conservation laws for each mineral species, equation (1), can be used to obtain an expression for the divergence of the solid velocity 1 A /Si Dfa faDQi\ (5) dt Qi dt J The operator D/dt is the material derivative D/dt — d/dt + vs • V with respect to the movement of the solid (the sediment matrix). Expression (5) can be used to eliminate V • vs from equation (2), which then leads to the following equation 3 This equation is the starting point for making pressure equations. The deriva tives of the densities can be related to pressure and temperature by means of compressibilites and thermal expansibilities. The time derivative of any density g can be written 1 Dp Dp DT (7) g dt dt dt where a is the compressibility and is the thermal expansibility, given by the following definitions 1 dg a — — — and = (8 ) g The question now is which pressures should be applied in the definition of the compressibilites of the different densities. The fluid compressibility a/ is related to the fluid pressure pf, but the solids response to pressure changes in a porouse medium is less trivial. According to Berryman (1992), the linear elastic density change can written 1 Dgi _ 1 (Dp h Dpf\ g; dt (1 — Porosity is a function of temperature, effective vertical stress, and also time, because of processes like dehydration, oil generation and cementation changes of the porosity. The rate of change of the porosity is therefore Di = a±DT + m +m +m +d± dt dT dt dt elast dt mech dt dehyd dt oil dt cem The rate of change of porosity is the sum of all active process. Notice that D 4 term and the mechanical compaction term, therefore do not apply at the same time for the same type of sediments. The linear elastic response of the porosity can be written, (Berryman, 1992) (11) The compressibility is the inverse of the bulk modulus of the drained porous frame, and the coefficients 71 and 72 are numbers close to or equal to 1, (Berryman, 1992). The temperature effect on the porosity can be written as d(p/dT = (1 — (b) (/% — fig) where and /3 S is a bulk and average solid thermal expansibilities respectively, (see Shi and Wang 1986). 2.2 Fluid potential, overpressure and boundary condi tions The main variable in the pressure equation will be the fluid flow pseudo potential, denoted (12) where g is the constant of gravity. The z-axis is positive upwards and the depths below z = 0 therefore have negative z-values. The pseudo-potential Pe = Pf Ph (14) where the hydrostatic pressure is the pressure from the weight of the entire water column above a point. A natural choice for the reference level z = 0 is the (sea level) or a water surface. The pseudo-potential $ and the excess pressure p e are then identical for those parts of the basin where hydrostatic pressure and the reference hydrostatic pressure are the same. These two quantities are not necessarily the same where the basin surface is above the reference level (sea level) as shown below. However, the quantity $ turns out to be a convenient variable for measuring overpressure build-up. 5 Darcy’s law, equation (3), can be rewritten in terms of $ K ,3PMn V£> = — (15) , ox I+ a„ where n z and n y are the unit vectors in the x- and ^-direction respectively. It is seen from equation (15) that vy is proportional to V4>, when Qf is constant, because both dph,o/dx = 0 and dph^/dy = 0. The pseudo-potential $ then becomes a true fluid flow potential. Potential flow is understood to be any type of flow driven by the gradient of a potential. The derivatives of Ph,o added to VT in equation (15) can be responsible for thermal convection, (Wangen 1994). These derivatives may play a role in two types of thermal convection. There is a weak type of convection where the convective flow has negligible feed back on the temperature equation. Such convection could be caused by non-horizontal isotherms because of differences in the thermal conductivity or differences in the heat flux from the basement. This type of weak thermal convection is often termed non-Rayleigh convec tion to distinguish it from Rayleigh convection which has a strong feed back on the temperature field, (Wangen 1994). The driving force for convection, originating from the derivatives of ph y0 in Darcy’s law (15), will normally be too weak to compete with the gradients of $ caused by overpressure build up. That is because the fluid density Qf is not very sensitive to pressure and temperature, due to the low compressibility and thermal expansibility of the fluid phase. Since thermal convection cannot compete with the often large excess pressure gradients found in overpressured basins, it is not discussed further here. The boundary conditions for the pressure equation will be either a closed boundary or an open boundary where the potential is known. A closed boundary is a boundary where v# • n = 0, and where n denotes the outward unit vector normal to the boundary. An open boundary will allow fluid to either leave or enter the basin through the boundary, and it is given by providing the potential at the boundary. The basin surface below sea level has a zero potential. The surface above sea level has a potential defined by the height of the water table above the zero level, (z = 0). If the water table is z = h(x, y), the potential along the surface becomes rh(x,y) ^surf = Qf9 dz- (16) The function h — h(x, y) is zero for all parts of the basin surface which are 6 below sea level. The pseudo-potential can easily be converted to the fluid pressure by use of relation (13). The pseudo-potential can also be converted to the excess pressure, pe, by subtracting the fluid potential at the surface p e = $ - $ sur/ (17) Equation (17) shows that the excess pressure is equal to the fluid pseudo potential for those parts of a basin which are below sea level, because $ surf = 0 there. The hydrostatic pressure, ph, is pressure of the water column all the way up to the water table, given by h — h(x, y) rh(x,y) Ph = Jz Qfgdz- (18) The hydrostatic pressure can also be written as the sum of the reference hydrostatic pressure and the pressure of the weight of the fluid column above the reference level fh(x,y) Ph = Ph, o + Qf9 dz = phfi + $ sur/ (19) Jo As a consistency check, it is seen that the excess pressure given by equa tion (17) added the hydrostatic pressure (19) yields the fluid pressure given by equation (13). The effective vertical stress is defined as the overburden minus the fluid pressure, and becomes Ps=Pb~ Phfi ~ $ (20) when it is related to the pseudo-potential $. 2.3 The pressure equation A pressure equation in $ can now be written from equation (6) using compressibilites and the thermal expansibilities given by definition (7) and Darcy’s law, equation (15) D§ 1 „ g/K dph,o , dp hfi V$ + -Hz + n„ = S,eff (21) P dx #2/ where aeg is given by 1 T \ ^ ri V(7i /I - 7 and where the right hand side is given by 1 D(j) (23) (1 — (j>) dt S f (f) Si (24) (25) (26) + - JffZJ) E " (A " A>) ir (27) This is a parabolic differential equation where the behavior of the solution $ is given by the three quantities: aeg, K//i and Seg. The first of these three quantities, aeff, corresponds to an effective compressibility of the system. It is therefore important for how fast overpressures will decay when there is no overpressure generation going on, (Seff = 0). The factor aeg can be simplified by setting as = E* 1 - ), and as~f3 = E» - 4>) where as is the average compressibility of the mineral phases. Notice that 3 Thermal expansion and compression The different mechanisms for pressure build-up manifest themselves as sep arate terms, which added together yield Seff. Each term has its own inter pretation. The last term in Seg, equation (27), which involves DT/dt, is responsible for aquathermal pressuring. This term, denoted Stherm, can be written (28) 8 where j3s = Y^i — The third and forth terms in Seg, given by equations (25) and (26) result from linear elastic compression of the fluid, matrix and porosity, and are written Dp b Dp ht0 \ ~T. 73— 77 — ) (29) The second term, equation (24) is the source term in the conservation laws for fluid and solid, respectively. The first term, equation (23), which is proportional to the rate of change of porosity, is still unspecified but clearly tells that decreasing porosity is a direct cause for fluid expulsion. The relative importance of expulsion by elastic compression and thermal ex pansion, represented by the two terms Seiast and Stherm > respectively, can be investigated by looking at the dimensionless ratio Seiast/S therm• The fluid compressibility for water is taken to be ctf — 4 • 10~10 Pa-1. The coefficients 71, 72 and 73 are all approximated by 1, and the bulk compressibility is of or der 10-10 Pa_1for certain sandstones (Berryman, 1992). The compressibility as is usually less than ab, (Shi and Wang, 1986). The source term Seiast can then be approximated by Seiast « — Selast ~ —(jxXfQf 9 + (1 — A similar expression has been derived by Luo and Vasseur 1992 for the case of zero mechanical compaction, when only the linear elastic effects of the fluid was taken into account. It is seen from the above data that Seiast/Stherm ~ 0.12 when 0 = 0.15 and dT/dz = 0.033 °C/m. Aquathermal pressuring is thus 8 times stronger than the effect of linear elastic compression. Notice that expulsion due to linear elasticity is treated separately from the mechanical compaction. 4 The gravity number; an indicator for pres sure build-up l-D models can be useful tools in studying overpressure build-up. The pres sure equation (21) simplifies when expressed in l-D, where it becomes -D$ 1 d (Qfk d«3>\ a’B~df ~ {—glj- s‘ff (31) This pressure equation can be simplified even more by setting aeg = 0, in order to study the effect of the source term Seff. The equation then be comes a stationary pressure equation, which models the pressure build-up in a medium with zero (effective) compressibility. The stationary pressure equation can also be expressed in the Lagrangian coordinate defined by the height of each sediment grain from the base of the basin as net (porosity free) sediments. (See appendix A.) This Lagrangian coordinate is denoted ( and the stationary pressure equation can be written dv D 1 k d where it is assumed that there is no fluid flow into the base of the sedimentary basin. It is seen that the stationary Darcy velocity (33) is greater than or 10 equal to zero as long as Seg > 0, with other words as long as Seg does not act as a sink term. A Darcy velocity vd > 0 leads to d$/dz < 0. The stationary fluid flow potential is therefore never decreasing with depth as long as there are no sink terms. The integral (33) giving the Darcy velocity can be calculated exactly for certain simple models. Assuming that the rock and fluid densities are constant and that there are no reactions between the solid phases and/or the fluid phases, (s* = 0 and Sf = 0), the only term left in Seg is Seff = —(D vD=co-(e- ebot) (34) where w is the deposition rate, e is the void ratio at position ( and ebot is the void ratio at the base of the basin. This Darcy velocity is also normally a good estimate when the porosity is not a function of the depth from the basin surface measured as (net) porosity free rock. The Darcy velocity towards the basin surface, where e to 1, is seen to be given by the (net) deposition rate vd ~ u> when the basin is deep enough for ebot to 0. The gradient d^/dC, is scaled with (gs — Qf)g in order to easily distinguish between the situations with high and low pressure build-up. The maximum fluid pressure occurs when the fluid pressure equals the lithostatic pressure. The corresponding excess pressure is then the lithostatic pressure minus the hydrostatic pressure. This pressure, which is termed the excess lithostatic pressure, is Pex,lith =Pb~Ph = (Q» ~ 0f) 9 (C* - C) (35) when expressed in with the (-coordinate for just one lithology with an average matrix density gs. (The (-coordinate for the surface of the basin is (*, see appendix A). The depth intervals along a well where the overpressure is increasing much, like seals, has a pressure gradient <9$/<9( which is at least of the order (gs — Q;) g. The sections with little overpressure build-up have d 11 where number Ng is given by n9 = (37 ) 11VD and where the Darcy velocity, which is not an independent variable, can be obtained from equation (33). In cases where the Darcy velocity is either given exactly by or well approximated by equation (34), the gravity number becomes = k(e.-Q,)9 (38 ) At w (e - ebot) The gravity number Ng is a useful indicator to characterize overpressure build-up. Notice that Ng as it is defined by equation (38) is a function of depth, because the void ratio is a function of depth, and the permeability is assumed to be a function of the void ratio. For depth intervals along a well where Ng 1, there will be small excess pressure gradients, which are characteristic for hydrostatic intervals. Depth intervals with JV9 < 1, on the other hand, will behave as seals and the basin below will be overpressured. A necessary condition for high overpressure is therefore that Ng < 1 somewhere along a 1-D well. This situation is plotted in figure la where the line Ng = 1 divides the plane given by log 10(u£>) and log 10(/c) diagonally into a high pressure half-plane and a low pressure half-plane. Figure la is used later to see how low the permeability has to be for a given Darcy velocity to imply overpressure. Figure lb shows the excess pressure gradient dQ/dz for three different cases of the gravity number. A gravity number equal to one, Ng = 1, is seen from equation (36) to correspond to a excess pressure gradient d$/dz being parallel to the gradient of the excess lithostatic pressure. The overpressure gradient d$/dz has to be larger than the gradient of the excess lithostatic pressure for the fluid pressure to be able to reach a level as high as the lithostatic pressure. The other regime Ng> 1 is seen, from equation (36), to imply an increasing fluid pressure with depth, but the increase is less than the increase in the lithostatic pressure. In the case Ng < 1, it is possible to estimate the minimum thickness a seal needs for the fluid pressure to go from hydrostatic pressure at top of the seal to lithostatic pressure at the base of the seal. This pressure increase across such a seal can be written —Az = (g, - g/) g ((* - () (39) 12 where Az is the seal thickness. The seal thickness can be rewritten in terms of the gravity number as Az = Nt- (C - 0- (40) For example, a seal with a thickness of 1/10 the depth from the basin surface, given as net (porosity free) rock, therefore requires a gravity number Ng = 0.1. A gravity numbers based on vp = uj and k equal to the surface permeability is shown to appear as the only explicit parameter in 1-D models for over pressure build-up when the pressure equation is made dimensionless, (Audet and Fowler 1992, Wangen 1992, Wangen 1997, Fowler and Yang 1999). This gravity number N'g is just one single number independent of depth. The num ber N'g is simply the number Ng at the basin surface, assuming that e = 1 for the shallowest sediments. The dimensionless pressure equations, controlled by the parameter N'g, yield pressure solutions consistent with the interpre tation of Ng given above. Solutions for N'g >> 1 have low overpressures and solutions for iV' < 1 have high overpressures; overpressures corresponding to fluid pressures close to the lithostatic pressure. Beware that this inter pretation of the number N'g applies for moderately thick accumulations of sediments, because Ng is based on the Darcy flux and the permeability at the basin surface. The number Ng does not represent very well the overpres sure conditions below a certain depth in thick accummulations of sediments. It is, for instance, possible to construct cases with high overpressures at large depths although N'g » 1. The gravity number based on a Darcy velocity obtained by integrating the source term Sejf is thus a simple way to decide if there is overpressure. Notice that the gravity number is proportional to the permeability k. This implies that a sufficiently low permeability leads to overpressure build-up regardless of which overpressure mechanism Seg is representing. The only condition is that Seff is positive, which implies expulsion of fluids, and therefore a positive vd- However, the relative importance of the different mechanisms generating overpressure can be assessed by comparing the different contributions which are added together in Seff. The separate terms can be compared directly, and their contribution to the Darcy velocity can also be compared by integrating them as shown in equation (33). 13 5 Different compaction models Little has yet been said about the term D(f>/dt. Different models for the porosity evolution are now studied separately. These are mechanical com paction, dehydration reactions in the sediments, oil generation and pore space cementation. These models yield different expressions for D(j)/dt, which will appear as different contributions on the right hand side, the effective source term Seff. Mechanical compaction is different from the other models of poros ity reduction by having the porosity reduction being controlled by the effec tive vertical stress. Dehydration, oil generation and pore space cementation have in common that the porosity is controlled by temperature. It is possi ble to estimate D(j>/dt for these mechanisms under the assumption of burial along a constant thermal gradient, and thereby obtain the contribution to the Darcy velocities from these mechanisms. 5.1 Mechanical compaction A common model for the porosity is to assume that it is given as a function of the effect vertical stress, p s = Pb — Pf- Such relationships originate from Terzaghi ’s model for compaction of soils, (Terzaghi 1943). Terzaghi modeled successfully soil compaction as a function of effective vertical stress. This type of compaction, where the porosity is controlled by the effective vertical stress, is called mechanical compaction, (Schneider et al. 1996). Experiments on unconsolidated clays show that the porosity can be related to the effective stress by the function (Skempton 1970) e = e0 - Cv \og{ps/p sfi) (41) where e is the void ratio, 6q is a reference void ratio, Cv is the compres sion index, and where p St0 is a reference pressure corresponding to Cq . The void ratio is related to the porosity by e = 4>/{l — 14 vertical stress instead of depth as 0 - exp(-a #ps) (42) The surface porosity is This expression for D(f)/dt is split into two parts, one part involving D$/dt, which becomes a part of aeg 1 d(j) QCmech — (44) (1 - 4>) dp s' The remaining part becomes the source term for mechanical compaction which adds to Sejj. The contribution of mechanical compaction to the source term Seff, denoted Smec^, is proportional to the factor Dp b/dt — Dph,o/dt. To simplify the discussion, it is assumed that the system is in a state of constant deposition, and that the basin surface is below z = 0. The factor Dp b/dt — Dphfi/dt can then be written (gs — Qf)ug, where gs is the average density of the sediment matrix, and uj is the deposition rate measured as net (porosity) free sediments. See Appendix A. The contribution from Seff from mechanical compaction is then (45) The source term Smech is, when computed for the Athy porosity function (42), (46) The term Smech for mechanical compaction can be immediately compared to the overpressure build-up from thermal expansion, Stherm ~ 15 be measured relative to expulsion from mechanical compaction based on the Athy porosity-function by the ratio Smech, ^ a(f>(Qs ~ Qf) 9 , The (dimensionless) number Smech/Stherm is constant independent of depth in the case for the Athy porosity function. The ratio is Smech/Stherm = 7.2 for the following choice of fluid and sediment properties: gs = 2000 kgm-3, Qf — 1000 kgm”3, pf = 4 • 10-4 K-1, a^ — 1-10"8 Pa_1and dT/dz = 0.035 °C/m. Aquathermal pressuring is seen to be an order of magnitude weaker than mechanical compaction based on an Athy type of porosity function. Notice that the porosity drops out of the ratio between the source terms in equa tion (47). This implies that this result holds for any porosity and therefore also any overpressure in this case of an Athy-type of porosity function. A similar calculation can be carried out for the porosity function for soils, equation (41), or for other porosity functions. 5.2 Reference compaction An example of a porosity-depth curve obtained from an Athy-type poros ity function is shown in figure 2a. The porosity is plotted for vertical ef fective stress at hydrostatic fluid pressure when = 6 • 10-8 Pa-1 and Qs = 2000 kgm-3. Figure 2b shows a plot of the the source term (46) for the porosity shown in figure 2a and for the deposition rates w = 50, 100 and 250 mMa"1. Figure 2c shows Darcy velocities calculated using the expres sion (33) applied to the source terms shown in figure 2b. It is seen that the Darcy velocities at the basin surface are almost the same as the deposition rates. This is no coincidence because the exact expression (34) applies with a surface void ratio e = 1 and a void ratio ebot % 0 at the base of the basin. Hydrostatic fluid pressure implies that the vertical effective stress is only a function of the (-depth from the basin surface. This condition makes the simple and exact expression (34) for the Darcy velocity applicable to hydro static compaction, when the porosity is a function of the effective vertical stress. The hydrostatic compaction, as show in figure 2a, will be used later as a ref erence porosity-depth curve, because it yields the fluid expulsion of a typical porosity-depth tend. The corresponding source term and the Darcy veloc 16 ities shown in figure 2b and figure 2c respectively, will be compared with source terms and Darcy velocities for the other expulsion processes which are dehydration of clays, oil generation and pore space cementation. 5.3 Dehydration of sediments Dehydration reactions in clay are a mechanism which has been proposed as a cause for overpressure build-up, (Powers 1967, Burst 1969). Smectite rich clays release interlayer water from its crystal structure when it is de hydrated. Although the dehydration process takes place in several stages (Colton-Bradley 1987), only one stage is modeled here. Actually, the same dehydration model as the one proposed by Audet is used here, (Audet 1995). A number of n moles of water is formed when one mole of hydrate is converted to one mole of dehydrate. Mass conservation for this reaction is Mh = Md + nM w (48) where Mh, Md and Mw are the molar masses of hydrate, dehydrate and water, respectively. The conversion of hydrate to dehydrate implies that there are source terms Sj in the conservation laws for hydrate and dehydrate, and the release of water from the reaction means that there is also a source term in the conservation law for fluid. The volume fractions of hydrate and dehydrate are denoted fa and fa respectively, and the corresponding source terms are Sh and sd . The remaining part of the solid is inert. The source term (or more precisely the sink term) for hydrate, Sh/Qh , is the rate of volume loss of hydrate per bulk volume of rock. This term is simply Sh/Qh = Dfajdt. The dehydration reaction is assumed to follow first order Arrhenius kinetics, and the source (sink) term for dehydrate can therefore be written — = —kh4>h (49) Qh where the rate fa is given by an Arrhenius law fa = Ah exp (—Eh/RT), where Ah is the Arrhenius prefactor and Eh is the activation energy. The source term for dehydrate production becomes Sd sh (50) Qd, P/i Qh where 14 and Vd are the molar volumes of hydrate and dehydrate respectively. The source term for fluid, production, s/, is similar to sd £/ _ nVf sh (51) Qf Vh Qh 17 where V/ is the molar volume of fluid. The conversion of hydrate to dehydrate leads to a volume change of the solid which implies a porosity change too, assuming that the bulk volume is conserved in the process. The rate of change of porosity in the dehydration processes is then given by (52) This change in the porosity comes in addition to eventually other porosity changing processes; for instance mechanical compaction or pore space cemen tation. This contribution to the rate of change of porosity, equation (52), was not included in Audets study (Audet, 1995) of pressure build-up from a combined process of mechanical compaction and dehydration of clay. This effect does not follow from the conservation laws for solid and fluid, and it has to be explicitly imposed on the model. As already mentioned, the poros ity reduction expressed by equation (52) assumes that the bulk volume is conserved under the dehydration reaction. If the bulk volume is not con served, but instead the porosity is conserved, there is no contribution to the rate of change of porosity from the dehydration process. This is what was assumed in Audets study, (Audet, 1995). These two situations, one where the dehydration reaction creates porosity and the other where the porosity is unchanged by the dehydration reaction, are now studied separately. The one which changes the porosity is studied first. Adding together the contri bution from the terms (23) and (24) in Seg yields the following source term for dehydration (53) The factor in parenthesis represents the net volume change in the dehydra tion process. If this factor becomes negative, then dehydration becomes a sink term rather than a source term. Audet used numbers in the numerical examples where this factor was zero. The alternative, where the dehydration process does not change the porosity, leads to (54) when the terms in equation (24) are added together. It is seen that this second source term for dehydration is stronger than the first source term given by equation (53). In a dehydration process where the total volume change is zero, and where the source term Sdehyd,i is zero, is seen that Sdehyd ,2 represents the volume of fluid generated. 18 The volume fraction hydrate is plotted in figure 3 for the deposition rates w = 50, 100 and 250 m Ma-1 using the Arrhenius data from Audet (Audet, 1995). These data are an activation energy Eh = 60 kJ mole-1 and an Arrhenius prefactor Ah = 4.2 • 10~5 s-1. The initial volume fraction of hydrate is set to Figure 4a shows the source term (53) given the volume fraction hydrate in figure 3. The numbers — nVf/Vh and a2 — Vd/Vh are difficult to give accurately. Audet (Audet, 1995) assumed that there was no total volume change in the dehydration process, thus oq + a2 — 1 = 0. It is here assumed that the total volume change is Oi + a2 — 1 = 0.1 (10%), which can be considered an upper bound. Osborne and Swarbrick (1997) estimated the total volume increase for dehydration of smectite to be 4%. The source term Sdehyd,i is seen to peak in the middle of the window of dehydration. The Darcy velocities resulting from the source terms in figure 4a are shown in 4b. These Darcy velocities are calculated by the integral (33). These velocities are compared with the Darcy velocities from hydrostatic compaction shown in figure 2c and it can be seen from figure 4c that hydrostatic compaction contribute approximately an order of magnitude more to the Darcy velocity than clay dehydration. The Darcy velocity resulting from the source term Seff— Sdehyd.i, obtained by the integral (33), can also be approximated by The source term (54) can be studied the same way. It is plotted in figure 5a 19 for the same data as in figure 4a. The number used for % is 0.14, which corresponds to n = 2.5 in Audets study (Audet 1995). The source term (54) is similar to the source term (53), with peak values at the same depths. The difference is that the source term (54) is stronger by almost a factor 2. This factor would have been much more if the overall volume change in the de hydrate reaction, + 02 — 1 had been close to zero. The Darcy velocities resulting from the source term (54) are shown in figure 5b. These Darcy ve locities are compared with the Darcy velocities for hydrostatic compaction in figure 5c and it can be seen that the Darcy velocities in this case are approx imately 2 times larger, just as the source terms. These results are consistent with the results of the Audet 1995 which showed only a small contribu tion from dehydration on overpressure build-up in a mechanical compaction model. 5.4 Oil generation Overpressure build-up caused by oil generation can be modeled by the same approach as dehydration of clay. After a sufficient impact of temperature and time, organic matter (kerogen) will generate oil and possibly an inert residue, coke. Kerogen, residue and oil corresponds to hydrate, dehydrate and water, respectively. The same two alternative source terms for pressure build-up are possible for oil generation as with dehydration of clay. The first of these source terms, equation (53), changes the porosity of the rock, while the second source term, equation (54), leaves the pore space unchanged. The question is therefore to what extent solid kerogen is replaced by the oil generated. The discussion is now simplified by considering only kerogen which is completely transformed to oil without leaving behind any residue. Furthermore, the two-phase oil-water system is analysed as a one-phase system, where only the total amount of fluid is accounted for, not the individual fluid phases. The source term where the porosity is changed represents therefore a process where the kerogen is replaced by the oil generated. Pressure generation is then possible only if there is a net volume increase in the process. It is uncertain whether this source term is more appropriate than the alternative source term which conserve the pore space. These two alternative source terms represent end members where the first source term yields minimum oil expulsion and the second yields maximum expulsion. Other source terms for oil expulsion could be realized by other assumptions about the rate of 20 change of porosity. However, the first source term is studied now. The source term (53) can then be rewritten for oil generation as (56) where k* is the reaction rate for oil generation, (in a first order Arrhenius reaction), is the volume fraction of kerogen, and and o0 are the densities of kerogen and oil, respectively. It is important to notice that a source term like S0u involves generation of porosity. The rate of change of porosity caused by oil generation from kerogen is (57) This is exactly the opposite of the rate of change of the volume fraction of kerogen, which is D(f>k/dt = —for a first order Arrhenius law. This porosity enhancement will also enhance the absolute permeability of source rock, assuming that permeability can be related to the porosity in the source rock. The source term (56) is now used to estimate how effective oil expulsion is from a rich source rock. It is assumed that the source rock has an initial volume fraction of 5% procent kerogen. The volume expansion from trans forming one unit mass of kerogen to oil is Qk/Q0 = 125. The Arrhenius data for oil generation are A& = 1015 s-1 and Ek = 221 kJ mole-1. These Arrhe nius data are based on Tissot et al. 1987, and they represent the mean in a distribution of activation energies. The volume fraction of kerogen is shown in figure 6a for the three deposition rates to = 50, 100 and 250 m Ma-1 for burial along a constant thermal gradient OT/dC, = 0.035 °C/m. Oil genera tion peaks at about 5 km depth which is about 143 °C. This is also seen from figure 6b where the source term is plotted. The Darcy velocities obtained by integrating the source term are shown in figure 6c. These Darcy velocities are compared with the reference Darcy velocities of hydrostatic compaction. It is seen from figure 6d that the amount of oil expelled is about an order of magnitude less than than the amount of fluid expelled due to hydrostatic compaction. These results are consistent with the computational results of Lou and Vasseur (1996) which showed a weak contribution from oil genera tion on the overpressure build-up. They looked at overpressure build-up from both oil generation and gas generation with a two-phase model. However, the source terms in their conservation laws for water, hydrocarbons and solid matter were not explicitly given. 21 The Darcy velocity due to oil generation can be approximated in the same way as the Darcy term for clay dehydration Ok _ \ 4>kU ^ oil (58) Bo /! — ’ where 4> is an average porosity in the window of oil generation. This approx imation is easily checked against the numerically computed Darcy velocities v0u in figure 6c. These numerical Darcy velocities in the range from 1 m Ma-1 to 3 m Ma-1 require a permeability k ~ 10-21 or lower, according to figure la, to generate high overpressures. 5.5 Pore space cementation Mechanical compaction assumes that the porosity is completely controlled by the effective vertical stress. It is uncertain as to what extent this is a good description for the porosity loss in lithified sediments. There are several au thors who argue that the porosity loss is controlled by chemical processes rather than by mechanical compaction in lithified rocks, (Bjprlykke &c Hpeg 1997, Bjprlykke 1999a, 1999b). Chemical compaction is a difficult field be cause several aspects of geochemical processes are pooly understood. Rather than trying to apply a detailed model for chemical compaction, it is simply assumed that the rate of change of porosity follows an n ’th order Arrhenius law, (Wangen 2000). This is a simple approach which is motivated by quartz cementation of sandstones with local sources for silica. There is petrographic evidence for stylolites and other quartz-mica interfaces to be the source for sil ica in many cemented sandstones, (Bjprkum 1994, 1996, Walderhaug 1994a, 1994b, 1996). The mica acts as a catalyst for dissolution of quartz. This dissolution process creates a supersaturation in the pore space in between stylolites and the quartz-mica interfaces, because precipitation of cement is the slowest step in the overall process of dissolution and reprecipitation, (Bjprkum 1996, Oelkers et al. 1996, Walderhaug 1994a, Walderhaug 1994b). The sandstones therefore dissolve and compact along the stylolites and other quartz-mica interfaces, and the dissolved silica is transported by diffusion into the pore space where it precipitates. The rate of change of porosity fol lows directly from the rate of quartz precipitation when the supersaturation is given, and it can be written (Wangen 1999), -rf = -kcemVcCS{4>) (59) dX cem 22 where kcem is the reaction rate for the precipitation of cement, vc is molar volume of the mineral cement, c is the degree of supersaturation and S(4>) is the specific surface area of the rock. The degree of supersaturation is related to the supersaturation of dissolved silica, m, and the equilibrium saturation, meg, by c = (m-meg)/meq. Petrographic work by Aase and co-workers (Aase et al. 1996) have estimated the degree of silica supersaturation created by dissolution of quartz at mica-quartz interfaces to be c % 1%. The specific surface area of the rock is assumed related to the porosity by 4> — 4>m S{4>) = S0 (60) (f>0 — 0. The specific surface area approach zero as the porosity approaches a minimum porosity 4>min , which is the porosity at which the pore space becomes disconnected. The exponent n controls how fast the specific surface decreases with decreas ing porosity. The reaction rate kcem for cement precitation is given by an Arrhenius law, kcem = Acemexp (61) with an Arrhenius prefactor Acem, an activation energy Ecem and where R is the gas constant. The burial is now assumed to take place along a constant thermal gradient when measured as (net) porosity free sedi ments. The temperature is then related to the (-coordinate and time by T = dT/dC ■ (cot — () + T0, where T0 is the surface temperature. This approach, which was originally suggested for quartz cementation of sandstones, is now used as an empirical approach to the cementation of a wider range of lithologies than quartzose sandstones under the assumption that the sources for mineral cement are local. The Arrhenius parameters are then assumed to represent the average of all sorts of grain sizes dominated by siliclastic minerals. The source term responsible for pressure build-up by the cementation process becomes D(j) ( The porosity as a function of depth due to cementation is shown in figure 7a with data for quartz cementation (Wangen 1999), A^m = 24.0 molem_2s_1, 23 Ecem = 90.0 kJ mole-1, c = 0.01, vq = 2.4 • 10~5 m3mole_1, So 1000 m2/m3and 4>min = 0. The porosity is calculated as shown in appendix B. The plots in figure show the porosity for the three deposition rates ui = 50, 100 and 250 mMa'1, and a thermal gradient dT/dC, = 0.035 °C/m. It is seen that most of the cementation takes place in the depth interval from 2 km to 4 km. This is also reflected by the source term Scem which is plotted in figure 7b. This figure shows that rapid deposition leads not only to a deeper interval of cementation but also a faster cementation and therefore a larger source term Scem. The resulting Darcy velocities are shown in fig ure 7c. It is seen that Darcy velocities towards the surface are exactly the same as the deposition rates. That is because equation (34) for the Darcy velocities applies in this case. D(f)/dt is a function of the depth from the basin surface as net (porosity free) sediments because burial is assumed to take place along a constant thermal gradient dTjdC,. The Darcy velocities are therefore vp = u ■ (e — e^t)- The expression (38) for the gravity number can therefore be used to estimate an upper bound for the permeability for high overpressure to occur. A Darcy velocity vn — 20 m Ma-1 is seen from figure la to imply that the permeability should be of the order 10-19 or less for high overpressure to build up. These Darcy velocities due to the cementation are then compared to the Darcy velocities caused by mechanical compaction in figure 7d. The Darcy velocities from the cementation are larger in most of the depth interval above the window for cementation, and they are considerably larger in the window for cementation. It is therefore likely that the cementation is an important cause for overpressures at depth. The cementation process leads to expul sion of fluids in the depth interval where overpressure is normally observed. Furthermore, the cementation of the pore space reduces the permeability. 6 Conclusion A pressure equation has been derived which models pressure build-up from several alternative mechanisms. These are mechanical compaction, thermal expansion of the pore fluid and the rock matrix, dehydration of clays, hy drocarbon generation, and cementation of the pore space. The pressure equation is derived from conservation laws for each species (mineral) in a multi-component solid and for a conservation law for the fluid phase. The 24 conservation laws are equipped with source terms which account for produc tion or consumption of the mineral species and the fluid. The pressure equa tion is formulated in the pseudo-potential defined by the fluid pressure minus the hydrostatic pressure relative to a datum line. This pseudo-potential be comes a true potential when the fluid density is constant. It is a convenient unknown in a pressure equation because the pseudo-potential becomes the excess pressure for those parts of the basin which are below the datum line. The overpressure generating mechanisms are formulated as a parabolic differ ential equation for the pseudo-potential. The different mechanisms appears in the equation for the overpressure as different source terms. These source terms are compared in order to measure the relative strength of each mech anism as a function of depth. The integral of the source terms are shown to yield the contribution to the Darcy velocity (volume flux) from each pressure generating mechanism. These contributions to the Darcy velocities are com pared. The Darcy velocity given by mechanical compaction at hydrostatic conditions is used as a reference fluid flux. A gravity number, Ng, based on the actual Darcy velocity is shown to be a useful indicator for high and low overpressure. Generation of high over pressure requires Ng < 1, while Ng 3> 1 implies negligible generation of overpressure. How low the permeability has to be for a specific mechanism to generate overpressure is obtained, in straight forward manner, by com bining Ng = 1 with the Darcy velocity for the specific mechanism. It is concluded that all mechanisms studied here can generate high overpressures given that enough fluid expulsion takes place in sufficiently low permeable sediments. For instance, a Darcy velocity ~lm Ma-1 from dehydration of clays requires that the permeability has to be order k — 10-21 m2or less. It is shown that the elastic compression of the porous rock and the fluid at hydrostatic conditions is about 12% of thermal expansion of the fluid at the same depths. Thermal expansion of the fluid is found to contribute an order of magnitude less than mechanical compaction to the Darcy velocity for an Athy type of porosity function at any overpressure. Pressure build-up from dehydration of clays and oil generation is modeled under two different assumptions about the possible changes to the pore space. The fluids generated can either replace the solid volume lost in the process, leading to a porosity enhancement, or alternatively, the porosity can be con served in the process. The last alternative to model the pressure build-up will 25 always contribute to the overpressure build-up, while the first alternative re quires a net volume gain in the fluid generating process. Darcy velocities due to dehydration of clay and oil generation from kerogen are both compared with Darcy velocities of hydrostatic mechanical compaction. Both processes are studied with parameters that can be considered an upper bound. Despite that, it seen that hydrostatic mechanical compaction still expell almost an order of magnitude more fluids than each of these two processes. The last process studied is the cementation of the pore space. The cementation pro cess is modeled by assuming that the rate of change of porosity follows an n ’th order Arrhenius law. It is seen that the cementation process can expel more fluid than mechanical compaction in the depth window of cementation. If these processes are assumed to operate simultanously in a basin during burial and deposition, then mechanical compaction will be the most impor tant expulsion mechanism for fluids in the uppermost, youngest and un- lithified sediments, while cementation of the pore space may be the most important mechanism for expulsion of fluids in lithified sediments. Both me chanical compaction and cementation of the pore space also leads to reduced permeability. The fluid expulsion mechanisms such as dehydration of clays and oil generation do not necessarily lead to reduced pore space and thereby reduced absolute permeability. 7 Acknowledgements The author would like to thank Xiaorong Luo and an anonymous referee for their comments to the manuscript. 8 Appendix A: Overburden and Lagrange co ordinates The lithostatic pressure (or the overburden) is the pressure of the weight of both the sediment matrix and the pore fluid, and can be written in the real coordinate z as (63) 26 when there is no water above the basin. The z-axis is pointing upwards and the basin surface is assumed to be z = 0. The excess lithostatic pressure, Pex,uthi is the lithostatic pressure subtracted the hydrostatic pressure rO P ex,lith =Pb-Ph = J (6a- Qf) (1 - The vertical coordinate ( measures the height of a sediment grain from the base of the sedimentary column as porosity free sediments. This implies that dQ = (1 — d>)dz. The excess lithostatic pressure is therefore, K* P ex,lith = / (Qs-6f)gdC (65) where (* is the ("-coordinate of the basin surface. The definition of the (-coordinate implies that each sediment grain has a unique (-coordinate through time. The sediments do not move along the (-axis. The ("-coordinate is therefore what is called a Lagrangian coordinate, and the material deriva tive becomes a partial derivative. The integral (33) can be done exactly for Seff = —(D fc 1 d(j> _ rC de d( = co f ‘(C) de vd = — du — uj-(e — ebot) (66) Jo (1 — 4>)2 dt Jo dt Ju(o) du by a change of integration variable from (" to u. It should be noted that the use of the (-coordinate becomes approximate for processes like dehydration and oil generation where solid is converted to fluid. However, this approxi mation is justified by the fact that only a few percent of the solid is converted to fluid. 9 Appendix B: The volume fraction of hy drate, kerogen and cement during constant deposition The volume fraction of hydrated clay or kerogen, denoted /, is given by a first order kinetics i[ = -k} (67) 27 where the reaction rate k is given by an Arrhenius law k = A exp( —TE/T). The temperature T is in Kelvin, and the temperature Te is related to the activation energy by Te = E/R, where R is the gas constant. The time derivation in equation (67) is normally a material derivative, unless it is car ried out along the (-axis. We are interested in a solution of equation (67) during constant deposition of sediments at a rate to, given as net (porosity free) sediments. The burial is assumed to take place along a constant ther mal gradient dT/d( which is measured along the (-axis. The temperature can then be written as T — dT/d( ■ (cot — () + T0 where T0 is the surface temperature in Kelvin. Integration of equation (67) can be written [ ^-=—A [ exp( —Te/T)(U = — N [ exp( —1 /x)dx, (68) Jfo J J0 Jx o where the integration variable is changed from t to x defined by x = T/TE, where x0 is T0/TE. (Notice that t, T and x are linearly related through the burial rate and the thermal gradient.) The number N is given by E N — (69) RcodT/dC The integral /* 0 exp( —l/x)dx is difficult to do exact, but there exists ap proximations. A simple and good approximation is F(x) — F(xo) where F(x) = x2 exp(—l/x). 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[46] Y ang , X.-S., 2000, Modeling mineral reactions in compacting sedimen tary basins, Geophysical Research Letters, 279, p. 1307-1310. 34 Tables Symbol Value Units Comment Table 1: Parameters 35 Figure captions Figure 1 (a) The line Ng = 1 divides the plane given by log 10(i;£>) and log 10(&) into a low pressure half-plane and a high pressure half-plane. The line Ng = 1 becomes log 10(fc) = log 10(up ) - log 10((p s - gf) g/g). (b) The figure shows the overpressure gradient d^/dz for the three cases Ng < 1, Ng = 1 and Ng > 1. The thickness, A( of a seal is plotted in the case of Ng < 1. Figure 2 (a) The porosity is plotted as a function of depth for an Athy type of porosity function of effective stress when the fluid pressure is hydrostatic. The data used are a# = 6 • 10~8 Pa-1 and gs = 2000 kgm-3. (b) The source term (46) is plotted for the porosity shown in figure 2a and for the following deposition rates: u = 50, 100 and 250 mMa’1. (c) The figure shows the Darcy velocities obtained from the source term in figure 2b. Notice that these Darcy velocities are almost the same as the deposition rates w. It is because the exact expression (34) applies in this case. Figure 3 The volume fraction hydrate Figure 4 (a) The source term (53) which accounts for the porosity change in the dehydration process is plotted for the three volume fractions of hydrate in figure 3. The total volume change in the process is ax + a2 — 1 = 0.1. (b) The plots shows the Darcy velocities obtained from the source term in figure (a). (c) The Darcy velocities resulting from dehydration described by the source term (53) are compared with the Darcy velocities of hydrostatic compaction shown in figure 2c. It is seen that hydrostatic compaction expel more than an order of magnitude more fluid than the dehydration process. 36 Figure 5 (a) The source term for dehydration, equation (54), is plotted for the volume fraction of hydrate in figure 3. This source term generates fluid in a pore space remaining unchanged by the dehydration process. The total volume change of the process is Oi + a2 — 1 = 0.1, with Oi = 0.14. (b) The figure shows the Darcy velocities obtained from the source term in figure (a). (c) The Darcy velocities of figure (b) are compared with the Darcy velocities of hydrostatic compaction from figure 2c. The Darcy velocities are at least an order of magnitude smaller than the Darcy velocities of the hydrostatic compaction. Figure 6 (a) The volume fraction kerogen is plotted as a function of depth for the three deposition rates uj = 10, 50, 250 mMa-1. (b) The source term S0u is plotted for the same deposition rates and volume fractions as in figure 6a. (c) The Darcy velocities resulting from the source term in figure (b). (d) The Darcy velocities of figure 6c, for the deposition rates uj = 10, 50, 250 mMa-1, are divided by the respective reference Darcy velocities shown in figure 2c. The Darcy flux due to expulsion of oil is considerable less than the Darcy flux due to hydrostatic compaction, even for a rich source rock at ~ 5 km depth. Figure 7 (a) The porosity as a function of depth when the porosity is reduced by cementation of the pore space. Porosity-depth curves are shown for three deposition rates, co = 10, 50, 250 mMa-1. (b) The source term Scem for the three porosity-depth curves in figure 7a. (c) The figure shows the Darcy velocities obtained from the source terms in figure 7b. (d) The Darcy velocities in figure 7b are divided by the reference Darcy velocities shown in figure 2c for the same deposition rates. 37 C\2 i --- lo g io k E 1 figure null Iqw _ I I I la Ill'll oyerp logman high ressure I I mill [m/Ma] overpressure I 38 i mini c excess pressure hydrostatic pressure - excess lithostatic pressure figure lb 39 d e p th -4000.0 -8000.0 -6000.0 -2000.0 figure 2a 9 [-] 40 d e p th -4000.0 -6000.0 -2000.0 -8000.0 0.0e+00 figure 1.0e-15 2b 2.0e-15 S„ech 3.0e-15 [1/8] 41 [m/Ma] 4.0e-15 5.0e-15 d e p th -4000.0 -6000.0 - -8000.0 2000.0 figure 2c V mcch 100.0 [m/Ma] 42 150.0 [m/Ma] 200.0 250.0 d e p th -4000.0 -8000.0 -6000.0 2000.0 figure 3 250 [m/ 9h — [-] 50 [m/Ma] 43 [m/Ma] d e p th -8000.0 -4000.0 -6000.0 - 2000.0 0.0e+00 figure 4a 5.0e s — dehydl 17 44 [1/s] 1.0e-16 1.5e — [m/Ma] 16 d e p th -4000.0 -8000.0 -6000.0 -2000.0 figure 4b Vdehydi [m/Ma] d e p t h -4000.0 -8000.0 -6000.0 - 2000.0 0.000 figure 0.020 4c Vdehydl/V 0.040 mech 46 0.060 [m/Ma] [m/Ma]_ 0.080 - d e p th -4000.0 -8000.0 -6000.0 -2000.0 0.0e+00 figure 5.0e — 5a 17 1.0e-16 s dehyd2 1.5e-16 47 [1/s] 2.0e-16 2.5e-16 d e p th -4000.0 -8000.0 -6000.0 -2000.0 figure 5b Vdehyd 2 [m/Ma] 48 d e p t h -4000.0 -6000.0 -8000.0 - 2000.0 figure 5c [m/Ma] V dehydg/ [m/Ma] ^ mech 49 d e p th -6000.0 -8000.0 -2000.0 4000.0 0.000 figure 0.010 6a [m/Ma] 0.020 % [-] 0.030 50 0.040 0.050 d e p th -4000.0 -8000.0 -6000.0 -2000.0 0.0e+00 figure 5.0e — 6b 17 50 1.0e-16 [m/Ma] S.i, =100 [l/s] 1.5e-16 [m 51 2.0e-16 2.5e-16 d e p th -4000.0 -8000.0 -6000.0 -2000.0 figure 6c v„ti [rn/Ma] 52 d e p th -4000.0 -8000.0 -6000.0 -2000.0 figure 6d [m/Ma] ^oil/^mech 53 [ ] d e p t h -5000.0 -3000.0 - - 4000.0 6000.0 2000.0 1000.0 figure 7a [m/Ma] V [-] 54 d e p t h -4000.0 -6000.0 -5000.0 -3000.0 -2000.0 - 1000.0 0.0e+00 figure 4.0e =50 7b — 16 8.0e-16 [m/Ma] Seem [l/s] 1.2e-15 55 1.6e-15 2.0e-15 d e p t h -4000.0 -6000.0 -5000.0 -3000.0 - - 2000.0 1000.0 figure 7c Vcem [m/Ma] [m/Ma] 56 d e p th -4000.0 -6000.0 -5000.0 -3000.0 - - 2000.0 1000.0 figure 7d ^cem/^mech 57 [m/Ma] [m/Ma]