Poroelasticity and tidal loading

Transcription

1 Chapter 6: Poroelasticity an tial loaing 63 Chapter 6: Poroelasticity an tial loaing 6. Introuction It is establishe above, in Chapter 4, that there are many known examples of the tial moulation of seafloor hyrothermal systems. Variations in temperature, effluent velocity an chemistry can often be correlate to the local ocean tie, an it is therefore expecte that the changing tial pressure fiel on the seafloor is responsible for a majority of the observe tial signals. In contrast, Chapter 5 is concerne with the steay-state structure of a seafloor convection cell, an in particular its epenence on the properties of water. The aim of this chapter is to establish whether the nonlinear thermoynamic properties of water (Chapter 5 are sufficient to explain the tial signals which have been observe at mi-ocean rige hyrothermal systems (Chapter 4. The effect of tial loaing on subseafloor convection cells is investigate with reference to the equations of poroelasticity, which escribe the response of a flui-fille porous meium to applie stress (Biot, 94; Rice & Cleary, 976; Van er Kamp & Gale, 983; Kümpel, 99. When a porous meium is place uner loa, the resultant stress is borne partly by the soli matrix an partly by the interstitial flui where it is manifest as a change in flui pressure known as the incremental pore pressure. The partitioning of the total stress between the flui an the soli matrix is a function of their elastic properties an the porosity. Consequently, if any one of these properties varies spatially, it is possible for a locally uniform loa, such as the ocean tie, to prouce a non-uniform incremental pore pressure fiel. The subsequent spatial graients in the incremental pore pressure can then rive interstitial flui from one region of the flui-fille meium to another. For this reason, it is of great interest to consier how the poroelastic properties of a subseafloor convection cell vary spatially. As in Chapter 5, it is assume for simplicity that the elastic an transport properties of the soli matrix are constant an homogeneous. The only source of spatial variability is the interstitial flui, whose properties are known to be strongly epenent on temperature. For example, it is shown that the bulk moulus of water changes significantly over the temperature range encountere in a subseafloor hyrothermal system. The partitioning of the total tial stress between the matrix an the interstitial flui

2 Chapter 6: Poroelasticity an tial loaing 64 iffers markely between regions where the flui is liqui-like an regions where it is gaslike. When tial loaing is applie at the seafloor, the graients in incremental pore pressure inuce tially moulate flui flow between the col, liqui-like regions an the hot, gaslike regions. The physical quantities associate with the flui in a subseafloor convection cell - such as pressure, velocity an temperature (Section 5.3 are expecte to fluctuate as a result of tial loaing. The label incremental is use to refer to the tially inuce variation of any physical quantity about its unisturbe value. For some years, there has been a wellestablishe analytical solution giving the incremental pore pressure in an infinite oneimensional halfspace subject to uniform tial loaing (Van er Kamp & Gale, 983; Wang & Davis, 996. Here, this solution is extene to cover the two physical quantities which are commonly measure on the seafloor the vertical velocity an the temperature (Chapter 4. For an infinite halfspace, it is shown that the incremental vertical velocity at the seafloor lags the ocean tie by 35. Consequently, the maximum effluent velocity at the seafloor is preicte to occur ~4.5 h after high tie for semi-iurnal components, an ~9 h after high tie for iurnal components. If the permeable crust is taken to be of finite thickness the corresponing phase lag is shown to lie between 9 an ~35, epening on the thickness of the permeable layer relative to the lengthscale associate with pore pressure iffusion. Consequently, observe phase lags between the ocean tie an effluent velocity at the seafloor coul be use to infer the vertical istance over which the crust is permeable. An approximate expression for the tially inuce incremental temperature is erive for the case of buoyant flui ascening through a linear temperature graient. It is shown that the incremental temperature lags the incremental velocity by 9. Hence, for an infinite halfspace, the incremental temperature at the seafloor lags the ocean tie by 5 an the hottest effluent is expelle ~7.5 h after high tie for semi-iurnal components, an ~5 h after high tie for iurnal components. If the permeable crust has finite thickness, however, the seafloor effluent temperature lags the ocean tie by an angle between 8 an ~5, epening on the thickness of the permeable layer. These preicte phase lags are shown to be consistent with many of the seafloor ata iscusse in Chapter 4. Analytical solutions of simple one-imensional moels allow the physical processes responsible for seafloor tial signals to be ientifie. In Section 6.3., some preliminary

3 Chapter 6: Poroelasticity an tial loaing 65 numerical simulations are presente in which the two-imensional convection cell of Section 5.3 is subjecte to uniform tial loaing at the seafloor. The results of these simulations are interprete with reference to the one-imensional analytical solutions (Section Funamental concepts of poroelasticity The purpose of this section is to review the theory of poroelasticity, with particular regar to hyrothermal systems uner tial loaing (Biot, 94; Rice & Cleary, 976; Van er Kamp & Gale, 983; Kümpel, 99. It is assume that the properties of the crust are constant an homogeneous. Consequently, the objective is to investigate the effect which temperatureepenent flui properties have on the principal poroelastic parameters. 6.. The epenence of poroelastic parameters on flui temperature When a flui-fille porous meium such as the ocean crust is place uner external loa, it experiences incremental stress ( σˆ ij an incremental strain ( ê ij. Prior to loaing, the absolute pressure of the flui in the pores (p is constant. Uner loa, however, there is an aitional, time-varying pressure known as the incremental pore pressure pressure ( pˆ c is a scalar measure of the stress, efine by: pˆ c ( pˆ. The confining = σˆ kk (6. 3 (The Einstein summation convention is use throughout, so that there is an implie summation over a repeate tensor suffix. Three istinct bulk mouli are relevant to the behaviour of the flui-fille porous meium an their physical interpretation is summarise in Figure 6.. Firstly, the grain bulk moulus (K g is the bulk moulus of the meium in the absence of any pore space. Seconly, the matrix bulk moulus (K m is the bulk moulus of the meium when the pore space exists but is empty. (For this reason, it is sometimes known as the raine bulk moulus. Values of the grain an matrix bulk mouli for several geological materials are reprouce in Table 6..

4 Chapter 6: Poroelasticity an tial loaing 66 pˆ in pores (a pˆ c soli liqui voi pˆ c (b (c pˆ c ( pˆ c K g K m K f Figure 6.: Schematic illustrations of the three bulk mouli relevant to poroelasticity. (a A flui-fille poroelastic meium subjecte to an incremental confining pressure pˆ c has incremental pore pressure pˆ. (b Situation efining the grain bulk moulus: e ˆ = p ˆ / K. kk c g (c Situation efining the matrix bulk moulus: e ˆ = pˆ / K kk c m. ( Situation efining the flui bulk moulus: e ˆ = p ˆ / K kk c f Thirly, the flui bulk moulus (K f is the bulk moulus of the flui which fills the pore space, an is a function of its pressure an temperature. If the pore flui has temperature T, specific enthalpy h, an ensity ρ, then the isenthalpic flui bulk moulus (K fh is efine (Van Wylen & Sonntag, 978 by: K fh ρ = ρ p =. h const while the isothermal flui bulk moulus (K ft is efine by: (6. K ft ρ = ρ p =. T const (6.3 For water, the values of K fh an K ft are sufficiently similar to be regare as interchangeable, an both may be consiere synonymous with the flui bulk moulus (K f. (The values quote in this issertation are erive from the steam tables embee in the

5 Chapter 6: Poroelasticity an tial loaing 67 HYDROTHERM coe an are, strictly, isenthalpic flui bulk mouli. Values of K f are plotte in Figure 6. for a range of pressures an temperatures relevant to seafloor hyrothermal systems. material K g (GPa K m (GPa ν φ k (m seafloor seiment to -5 seafloor seiment.6-5 seafloor seiment Ruhr sanstone Berea Sanstone Weber sanstone Charcoal granite Westerly granite Tennessee marble MOR basement typical Basalt 6 3. Table 6.: The transport an elastic properties of various geological materials. K g grain bulk moulus. K m matrix bulk moulus. ν Poisson s ratio. φ porosity. k permeability. References: ( Wang & Davis (996. ( Fang et al. (993. (3 Hurley & Schultheiss (99. (4 Van er Kamp & Gale (983. (5 Fisher et al. (997 MOR refers to Mi-Ocean Rige. (6 Carmichael (98, where values for several basalts are given - typical values are reprouce here. Figure 6. shows that K f epens strongly on temperature but only weakly on pressure in the region of p-t space relevant to seafloor hyrothermal systems. The temperature epenence is similar to that of the thermoynamic properties consiere in Chapter 5 (Figure 5.6, in the sense that there is a sharp change in behaviour at ~4 C as water moves from a liqui-like state to a gas-like state. At seafloor pressures, the bulk moulus of water ranges from a col, liqui-like value of ~. GPa at C to hot, gas-like values between ~. GPa an ~.5 GPa above ~4 C. (A flui obeying the laws for a perfect gas has a bulk moulus (K f equal to the flui pressure (p. At seafloor pressures, water above ~4 C approximates this perfect gas behaviour fairly closely. Comparison of Figure 6.a with Table 6. shows that the bulk moulus of water in the seafloor crust (K f is much smaller than the grain bulk moulus (K g for all of the liste geological materials. It is therefore assume that K f <<K g for all subseafloor convection cells. However, it is not always true that K f <<K m. For example,

6 Chapter 6: Poroelasticity an tial loaing 68 Table 6. shows that K f (~. GPa when col may even excee K m (~.5 GPa, when the poroelastic meium is seiment. (a (b x 9 flui bulk moulus, K f (Pa.5 a (P K f.5 p = MPa p = 3 MPa p = 4 MPa p = 5 MPa temperature ( C Figure 6.: The bulk moulus of pure water, K f. (a K f as a function of pressure an temperature. (b K f as a function of temperature at a range of typical subseafloor pressures. Data are erive from the steam tables embee in the HYDROTHERM coe. ( 7 Pa correspons to km below sea level. The three bulk mouli (K g, K m, an K f, an the porosity (φ together constitute a set of four funamental parameters escribing the elastic properties of a flui-fille porous meium. Three further parameters (α, S an β are often erive from this funamental set, an their epenence on the thermoynamic state of the pore flui is consiere below Coefficient of effective stress: α Firstly, the coefficient of effective stress (α is efine (Nur & Byerlee, 97 by: K m α = (6.4 K g Hence, α is a measure of the relative magnitue of the matrix an grain bulk mouli (Table 6.. It shoul be note that K m K g, an so α is a imensionless number in the range [,]. The value of α epens solely on the properties of the rock matrix, an not on the properties of the interstitial flui. Consequently, it is assume for simplicity that α is constant an homogeneous within any particular subseafloor convection cell. For the geological materials liste in Table 6., the values of α range from. (Tennessee marble to.99 (seafloor seiment.

7 Chapter 6: Poroelasticity an tial loaing Storage compressibility: S The secon erive parameter is the storage compressibility (S efine by: S = K m K g φ (6.5 K f K g As its name implies, the storage compressibility has units of inverse pressure, an it can be interprete as the volume of interstitial flui which is ae to the pore space, per unit volume of rock, per unit increase in pore pressure (Kümpel, 99. Since K g, K m, an φ are regare as constants for any particular problem, it follows that spatial variations in S can only be ue to spatial variations in K f. Figure 6. shows that K f changes little with pressure an is a ecreasing function of temperature. It therefore follows from equation (6.5 that S changes little with pressure an is an increasing function of temperature. Table 6. suggests that K g = 5 GPa an K m = 3 GPa might be reasonable values for the bulk mouli of the oceanic crust. Unless otherwise state, these values will be use throughout this chapter. The fact that S epens on the flui bulk moulus suggests that there will be a marke change in its value at ~4 C, when water changes from a liqui-like state to a gas-like state. Equation (6.5 shows that the relative magnitue of this change is controlle by the value of the porosity, φ. Consequently, a critical porosity, φ c (p,t, can be efine by: ( K m K g φ c p, T = (6.6 K f K g It shoul be note that this critical porosity epens on the flui bulk moulus (K f an is therefore pressure- an temperature-epenent. The variability of S as a function of flui temperature can now be classifie into one of three regimes corresponing to large, intermeiate an small values of the porosity ratio (φ / φ c. In the intermeiate porosity regime - when (φ / φ c ~ for all temperatures - the full efining equation for S must be use (equation (6.5. However, simplifications can be mae when the porosity is either large or small in relation to the critical porosity, φ c. For K g = 5 GPa an K m = 3 GPa the critical porosity (φ c is.3 for col water (K f =. GPa an.3 for hot water (K f =. GPa. The epenence of S on pressure an temperature for the three porosity regimes is illustrate in Figure 6.3.

8 Chapter 6: Poroelasticity an tial loaing 7 (a (b - a (P S 5 x -9 storage compressibility, S (Pa porosity =.5 p = MPa p = 3 MPa p = 4 MPa p = 5 MPa temperature ( C (c (e ( (f - a (P S x - storage compressibility, S (Pa - p = MPa p = 3 MPa p = 4 MPa p = 5 MPa. porosity = temperature ( C x - storage compressibility, S (Pa a p = MPa (P S porosity = 5e-5 p = 3 MPa.5 p = 4 MPa p = 5 MPa temperature ( C Figure 6.3: The epenence of the storage compressibility (S on the thermoynamic state of the interstitial flui in the large, intermeiate an small porosity regimes with K g = 5 GPa, K m = 3 GPa. The interstitial flui is assume to be pure water. Thermoynamic ata are taken from the tables embee in the HYDROTHERM coe. (a,(b φ =.5 (large. (c,( φ =. (intermeiate. (e,(f φ =.5 (small. ( 7 Pa correspons to km below sea level.

9 Chapter 6: Poroelasticity an tial loaing 7 The large porosity regime - when (φ / φ c > for all temperatures - implies that: S φ (6.7 K f Equation (6.7 shows that the storage compressibility (S is proportional to the flui compressibility (K - f in the large porosity regime (Figures 6.3a,b. In contrast, the small porosity regime - when (φ / φ c < for all temperatures - implies that S K m K g α = (6.8 K m Hence, the storage compressibility is approximately constant as a function of temperature when the porosity is small (Figures 6.3e,f. In summary, it shoul be note that S is always an increasing function of flui temperature at seafloor pressures. The greatest change in S occurs at ~4 C, as the water changes from a liqui-like state to a gas-like state. The contrast between the liqui-like an gas-like values of S is greatest when the porosity is large Skempton ratio: β Thirly amongst the erive poroelastic parameters, the Skempton ratio (β is efine by K m K g β = (6.9 φ K m K g K f K g The Skempton ratio is a imensionless parameter taking values in the range [,]. When a poroelastic meium is loae isotropically, the Skempton ratio (β can be interprete as the ratio of the incremental pore pressure ( pˆ to the applie incremental confining pressure ( pˆ (Figure 6.a. Equivalently, β is the proportion of the total applie stress which is borne by the pore flui. Equations (6.4, (6.5 an (6.9 imply that β = (α / K m / S. Since α an K m are constant properties of the rock matrix, it follows that the variation of β with the thermoynamic state of the flui can be ivie into the same three porosity regimes as the variation of S. c

10 Chapter 6: Poroelasticity an tial loaing 7 (a (b.8.6 porosity =.5 Skempton ratio, β p = MPa p = 3 MPa p = 4 MPa p = 5 MPa β temperature ( C (c ( Skempton ratio, β β p = MPa p = 3 MPa p = 4 MPa p = 5 MPa. porosity = temperature ( C (e (f Skempton ratio, β.8 β.6.4 porosity = 5e-5 p = MPa p = 3 MPa p = 4 MPa p = 5 MPa temperature ( C Figure 6.4: The epenence of the Skempton ratio (β on the thermoynamic state of the interstitial flui in the large, intermeiate an small porosity regimes with K g = 5 GPa, K m = 3 GPa. Interstitial flui is pure water. Thermoynamic ata from the tables embee in the HYDROTHERM coe. (a, (b φ =.5 (large. (c, ( φ =. (intermeiate. (e, (f φ =.5 (small. ( 7 Pa correspons to km below sea level. For intermeiate porosity - when (φ / φ c ~ - the full efining equation for β (equation (6.9 must be use. However, for large porosity - when (φ / φ c > - equation (6.9 reuces to:

11 Chapter 6: Poroelasticity an tial loaing 73 α K f K β (6. φ m Hence the Skempton ratio (β is a scale version of the flui bulk moulus (K f when the porosity is large an it is therefore highly temperature epenent (Figures 6.a,b, 6.4a,b. In contrast, for small porosity - when (φ / φ c < - the temperature epenence of the Skempton ratio is much reuce (Figures 6.4e,f an equation (6.9 reuces to β (6. In summary, β is a ecreasing function of flui temperature for all seafloor pressures, with the greatest change occurring at ~4 C, where water changes from a liqui-like state to a gas-like state. The contrast between the liqui-like an gas-like values of β is greatest when the porosity is large. Since β represents the proportion of the applie stress borne by the interstitial flui, it follows that uniform tial loaing at the seafloor can create incremental pore pressures which iffer between the hot an col parts of a hyrothermal convection cell. In general terms, when the ocean tie is high, col regions of the cell (where β is large will have greater pore pressures than hot regions (where β is small. Consequently, flui is expecte to flow from col regions to hot regions at high tie. The pressure ifference which rives the flow is particularly large if the hot flui is gas-like (>~4 C an the col flui is liqui-like (<~4 C. The magnitue of the inuce flow is etermine by the spatial graient of the incremental pore pressure. Consequently, the inuce tially moulate flow is expecte to be large where spatial graients in temperature are large. This heuristic argument is investigate with greater rigour in Section The governing equations of poroelasticity Given a value for the Poisson s ratio of the soli matrix (ν it can be shown (Rice & Cleary, 976; Van er Kamp & Gale, 983 that the stress-strain relationship for a flui-fille porous meium is: ( ν ν ˆ σ ˆ ij σ kkδ ij αpˆ δ ( ν ( ν eˆ ij = ij (6. 3K m Typically, the Poisson s ratio takes a value ν. for geological materials (Table 6., an so ν =. is assume throughout this chapter. With the convention that a repeate suffix inicates summation, equation (6. yiels: eˆ kk = [ ˆ σ kk 3αpˆ ] = [ pˆ c αpˆ ] (6.3 3K K m m

12 Chapter 6: Poroelasticity an tial loaing 74 Equation (6.3 can be use to provie a physical interpretation for the effective stress ratio (α. For a soli meium, the volume ilatation ( ê kk is proportional to the confining pressure ( pˆ c (Figure 6.b. For a poroelastic meium, however, equation (6.3 shows that the volume ilatation is proportional to the so-calle effective pressure ( ˆ αpˆ p c. This issertation is concerne with the effect of tial loaing at perios of ~ h an greater. This timescale is much greater than the time taken for seismic waves (spee > km.s -, to cover the lengthscales appropriate to hyrothermal systems (< km. Consequently, the stresses ue to tial loaing of the seafloor are, effectively, transmitte instantaneously throughout a subseafloor convection cell. The equations of compatibility for an elastic meium (Love, 97 can be applie to equation (6. to give an equation expressing elastic equilibrium: ν ij kk ij kk ij (6.4 xix j xix j [( ˆ σ νσˆ δ ] ˆ σ α( ν pˆ δ pˆ = Equation (6.4 can be contracte on the suffices to yiel: ( ν ( ν α ˆ σ ˆ kk p = (6.5 The conservation of pore flui is expresse by the equation: k µ S pˆ ˆ σ pˆ = β t t 3 kk (6.6 where k is the permeability of the rock matrix an µ is the ynamic viscosity of the interstitial flui. In general, solutions to problems of poroelastic loaing are foun by solving equations (6.4 an (6.6 for the incremental pore pressure ( σˆ ij pˆ an the incremental stress tensor (. In certain simple cases, however, the number of unknowns reuces to two - pˆ an σˆ - an kk solutions are foun by solving equations (6.5 an (6.6. Furthermore, if σˆ kk is known, equation (6.6 becomes a force conuction equation for the incremental pore pressure ( pˆ with a iffusivity (κ efine by k κ = (6.7 µ S The permeability of the seafloor (k is very poorly constraine (Table 6.. Consequently, it is helpful to efine a scale iffusivity ((µs - in orer to examine the epenence of the iffusivity (κ on the temperature (T of the interstitial flui.

13 Chapter 6: Poroelasticity an tial loaing 75 (a (b x 3 scale iffusivity, (µs - (s - porosity =.5 - p = MPa ( 8 s p = 3 MPa - 6 p = 4 MPa p = 5 MPa (µs temperature ( C (c ( 8 x 4 scale iffusivity, (µs - (s - porosity =. - s ( - (µs 6 4 p = MPa p = 3 MPa p = 4 MPa p = 5 MPa temperature ( C (e (f.5 x 5 scale iffusivity, (µs - (s - - s ( - (µs.5.5 porosity = 5e-5 p = MPa p = 3 MPa p = 4 MPa p = 5 MPa temperature ( C Figure 6.5: The epenence of the scale iffusivity ((µs - on the thermoynamic state of the interstitial flui in the large, intermeiate an small porosity regimes with K g = 5 GPa, K m = 3 GPa. (The true iffusivity, in m.s - is k (µs -. The interstitial flui is assume to be pure water. Thermoynamic ata from the tables embee in the HYDROTHERM coe. (a,(b φ =.5 (large. (c,( φ =. (intermeiate. (e,(f φ =.5 (small. ( 7 Pa correspons to km below sea level. Figure 6.5 shows how the scale iffusivity epens on pressure an temperature in each of the three porosity regimes for K g = 5 GPa an K m = 3 GPa. In the large porosity regime - when (φ / φ c > - equations (6.7 an (6.7 imply that:

14 Chapter 6: Poroelasticity an tial loaing 76 K f [ φ ] (6.8 µ S µ Consequently the iffusivity is proportional to (K f / µ (Figures 6.5a,b, an is maximise at ~ C for seafloor pressures. Conversely, in the small porosity regime - when (φ / φ c > - equations (6.8 an (6.7 imply that: K m µ S α µ (6.9 Consequently, the iffusivity is proportional to the inverse viscosity (µ - (Figures 6.5e,f, an is maximise at ~4 C for seafloor pressures. In summary, Figure 6.5 shows that the scale iffusivity is maximise somewhere between ~ C an ~4 C for all seafloor pressures, epening on the value of the porosity. Hence, assuming that crustal permeability is homogeneous, iffusive incremental pore pressures ue to tial loaing will ten to propagate furthest in regions of a subseafloor convection cell where the temperature is between ~ C an ~4 C. Equations (6.6 an (6.7 show that the propagation of incremental pore pressure is escribe by a iffusion equation with iffusivity κ. When tial loaing is applie to the seafloor at angular frequency ω, iffusion occurs over a characteristic lengthscale known as the skinepth ( efine by: κ = = k (6. ω ω µ S It has alreay been remarke that the crustal permeability (k is very poorly constraine for seafloor hyrothermal systems. Nonetheless, using equation (6. it is possible to make some general euctions concerning the magnitue of the skinepth (. Figure 6.6 shows how the skinepth for semi-iurnal loaing (ω =.4-4 s - epens on permeability for typical values of (µs - taken from Figure 6.5. In broa terms, the minimum plausible skinepth for a semi-iurnal tial signal in seafloor crust is ~ m when k = -7 m an the flui is at ~ C. Conversely, the maximum plausible skinepth is ~, m when k = - m an the flui is at ~4 C. It is important to note that this range of skinepths spans a very wie range. A skinepth of m is very much less than the typical imensions of a seafloor convection cell (H ~ m, L ~ 5 m while a skinepth of, m is consierably larger. The epenence of the skinepth on angular frequency (ω in equation

15 Chapter 6: Poroelasticity an tial loaing 77 (6. implies that the skinepths for iurnal components are greater than those for semiiurnal components by a factor of. 5 semi-iurnal skinepth, (m C 4 C (m -5 - permeability, k (m Figure 6.6: The variation of the skinepth ( for semi-iurnal tial loaing as a function of permeability, k. Using the values from Figure 6.5c, it is assume that (µs - = -3 s - at ~ C, an (µs - = -5 s - at 4 C. Note the logarithmic scales on both axes. It is now possible to escribe qualitatively the nature of solutions of equation (6.6 in a seafloor uner tial loaing, accoring to the size of the skinepth (. By consiering the case of a layere seafloor, Wang & Davis (996 emonstrate that a iffusive pressure signal is inuce wherever there is a sharp contrast in the elastic properties of the seafloor. In the case of the iealise convection cell of Chapter 5, there are two such sharp bounaries in elastic properties the seafloor itself, an the bounary separating liqui-like water from gaslike water which is approximate by the 4 C isotherm. The amplitue of the iffusive pressure signal ecays exponentially over a lengthscale away from the bounaries in elastic properties. In fact at istances greater than from these bounaries, the iffusive pressure signal is negligible an the solution is ominate by the instantaneous pressure signal : pˆ ˆ σ ˆ kk β = βpc (6. 3 It shoul be note that the approximate solution of equation (6. applies at every point of the cell which is more than a istance from a sharp bounary in elastic properties. Consequently, the magnitue of the skinepth ( is of great importance. In regions more than a few skinepths from bounaries in elastic properties, the incremental pore pressure is in phase with the tial loaing. Therefore, if the skinepth is very small ( < m the incremental pore pressure is well approximate by equation (6. over almost all of the

16 Chapter 6: Poroelasticity an tial loaing 78 subseafloor. It follows that the incremental pore pressure will be in phase with the ocean tie over most of the cell, although its amplitue varies accoring to any spatial variations in the Skempton ratio (β. In contrast, if the skinepth is comparable with the lengthscales of the cell ( > m, then the incremental pore pressure at epth is generally not in phase with the ocean tie. In Section 6.3, these qualitative remarks above concerning the incremental pore pressure response to tial loaing are illustrate with analytical solutions an numerical simulations. 6.3 Tial loaing of the seafloor This section is concerne with the effect of loaing by the ocean tie on a poroelastic seafloor. The nature of the global ocean tie is iscusse in Chapter, where it is illustrate with the use of cotial maps (Figure.8. The area of seafloor associate with an iniviual hyrothermal system typically has a lengthscale of at most km. This is much smaller than the typical wavelength of the ocean tie. Consequently, it is reasonable to suppose that at any time the tial loaing is uniform over the area of seafloor associate with an iniviual hyrothermal convection cell. (a tial loaing: p T cos(ωt z (b tial loaing: p T cos(ωt z seafloor z = x seafloor z = p ˆ ( z, t p ˆ ( z, t x permeability (k extens to z = - permeability (k extens to z = -H Figure 6.7: The loaing of a - seafloor by the ocean tie. It is suppose that there is no horizontal spatial variation. (a Infinite - halfspace. (b Finite permeable layer of epth H Tial loaing of a - seafloor Consier the uniform tial loaing of a seafloor in which the poroelastic properties are homogeneous an the incremental pore pressure is p ( z, t ˆ (Figure 6.7a,b. Symmetry implies

17 Chapter 6: Poroelasticity an tial loaing 79 that there is no horizontal strain ( e = eˆ (, t p ( iωt ˆ =. Since the governing equations are linear, the incremental pore pressure ue to a single component of the ocean tie can be consiere in isolation (Table.. The impose pressure on the seafloor is therefore taken to be a timeharmonic oscillation of magnitue p T an angular frequency ω. In complex notation, the seafloor bounary conition on the incremental pore pressure is: p ˆ = exp (6. T The ties in the open oceans generally have an amplitue of ~ m (Figure.8, an so p T typically has an amplitue of about kpa Incremental pore pressure in an infinite halfspace The simplest moel is one in which the permeable crust is of infinite extent below the seafloor (Figure 6.7a. The solution for this infinite halfspace is erive by Van er Kamp & Gale (983 an is iscusse in etail by Wang & Davis (996. It is of funamental importance, an is reprouce below. Van er Kamp & Gale (983 show that the incremental vertical stress is the same for all epths: ( i t z ˆ σ 33 = p T exp ω (6.3 Consequently, equation (6.6 can be re-arrange to rea: ˆ 33 Since ( z, t 3( ν k ( 3( ν αβ ( ν pˆ pˆ µ S = z t ( pˆ 3( ν k ( ν αβ ( ν µ S ν β 3( ν αβ ( ν t κ = (6.5 3 ( ν ( ν αβ ( ν β ( ( ˆ σ 33 (6.4 σ is known from equation (6.3, equation (6.4 constitutes a force iffusion equation for the incremental pore pressure. From equation (6.4, it is possible to efine a - iffusivity (κ accoring to: The assumption of a - seafloor leas to the introuction of the imensionless factor in square brackets, which istinguishes κ (equation (6.5 from κ (equation (6.7. Equation (6.4 also suggests the efinition of a - loaing efficiency (γ accoring to: γ = (6.6 3 The assumption a - seafloor leas to the introuction of the imensionless factor in square brackets which istinguishes γ (equation (6.6 from the Skempton ratio, β (equation (6.9.

18 Chapter 6: Poroelasticity an tial loaing 8 Referring to the - parameters of equations (6.5 an (6.6, equation (6.4 yiels an equation governing the iffusion of incremental pore pressure in a homogeneous - seafloor: κ pˆ pˆ = γ z t t ( p exp( iωt T (6.7 By analogy with the skinepth ( (equation (6., a -skinepth, representing the lengthscale of iffusion of the incremental pore pressure, can be efine by: κ = (6.8 ω Since κ an γ are not here functions of space, equations (6. an (6.7 have the (complex solution: i = T (6.9 ( z, t p ( γ exp z γ exp( iωt pˆ Equation (6.9 can be written in real notation as: z z ( z, t = p ( γ exp cos ωt γ cos( ωt pˆ T (6.3 Typical epth profiles associate with this solution are shown in Figure 6.8. It is conceptually useful to split the solution of equation (6.3 into the sum of an instantaneous signal (p T γ cos(ωt an a iffusive signal (p T ( - γ exp(z/ cos(z/ ωt. The instantaneous signal is in phase with the ocean tie an has magnitue p T γ for all epths. In contrast, the iffusive signal ecays exponentially from the seafloor over a lengthscale an is generally not in phase with the ocean tie. Figure 6.8 shows that the nature of the incremental pore pressure solution epens on the value of z/. Close to the seafloor (i.e. within one or two skinepths the incremental pore pressure contains a significant iffusive component. Further from the seafloor, however, the incremental pore pressure is ominate by the instantaneous signal (p T γ cos(ωt. Therefore the - loaing efficiency (γ represents the proportion of the tial loa which is borne by the interstitial flui at epths greater than skinepths.

19 Chapter 6: Poroelasticity an tial loaing 8 z/ ωt= ωt=π/8 ωt=π/4 ωt=3π/8 ωt=π/ ωt=3π/4 p(z ^,t ( z, t pˆ p T ( z, t pˆ p T z z = γ cos cos t ( ωt ( γ exp ω ( ωt p/ p ^ T Figure 6.8: Typical epth profiles of the incremental pore pressure ( p ˆ( z, t as γ cos a function of imensionless epth (-z/ in an infinite halfspace at various times uring the tial cycle with γ =.5. High tie occurs when ωt=. Half tie occurs when ωt=π/. Maximum vertical velocity at the seafloor occurs when ωt=3π/4. The iffusive pressure signal is limite to epths for which (-z/ < ~ Incremental velocity in an infinite halfspace Previous authors have been concerne solely with the incremental pore pressure (Van er Kamp & Gale, 983; Wang & Davis, 996. In contrast, the aim here is to investigate the flow of interstitial fluis inuce by the incremental pore pressure. In other wors, the point of interest is the spatial graient of the incremental pore pressure rather than the incremental pore pressure itself. The tially inuce incremental velocity at the seafloor (z = is of particular interest since it can be measure by instruments such as Meusa (Chapter 4. Applying Darcy s law to equation (6.9, the incremental vertical velocity (in complex notation is: k pˆ k i i wˆ (6.3 ( z, t = = p ( T γ exp z iωt µ z µ Therefore the tially inuce incremental velocity at the seafloor is: wˆ k i T µ = (6.3 (, t p ( γ exp( iωt Equation (6.3 shows that the tially inuce outflow of water at the seafloor leas the ocean tie by a raian angle arg(-(i = -3π/4 raians. In other wors, the tially inuce outflow of pore water at the seafloor lags the ocean tie by 3π/4 raians (or 35. Therefore peak outflow occurs ~4½ hours after high tie for the semi-iurnal components,

20 Chapter 6: Poroelasticity an tial loaing 8 an ~9 hours after high tie for iurnal components. In principle, this preicte phase lag of 35 coul be compare with observe effluent velocity time-series from the seafloor for each harmonic component of the tie (Table.. However, the effluent velocity time-series which have been collecte to ate are too short an too noisy to allow the extraction of iniviual harmonic components (Chapter 4. Comparison of this theory with observation will therefore have to await the collection of higher quality ata. The tially moulate effluent velocities observe on the seafloor take the form of a small ŵ incremental velocity ( impose on a steay backgroun flow (w. From equation (6.3, the amplitue of the velocity variations is: wˆ p k µ ( γ T = (6.33 Assuming that the backgroun flow consists of flui of ensity ρ ascening a col hyrostatic pressure graient (gρ, the steay velocity (w is given by: ( ρ ρ gk w = (6.34 µ From equations (6.33 an (6.34, it is possible to efine the imensionless magnitue of the incremental seafloor velocity as follows: wˆ w p g ( γ ( ρ ρ T = (6.35 Equation (6.35 expresses the magnitue of the tial oscillations in effluent velocity as a fraction of the backgroun steay velocity. Using the fact that S = α / (K m β, an equations (6.5, (6.6 an (6.8, it can be shown that: m ( ν γ ( ν µ 6K = k (6.36 ω α Equations (6.35 an (6.36 then imply that the imensionless magnitue of the incremental seafloor velocity is: ( ν ( ν ( a [ ω ] ( c ( wˆ pt α w g K { m ( b 3 k µ γ = ( ρ ρ γ ( (6.37 The terms in square brackets in equation (6.37 can be consiere in turn, to reveal how variations in parameter values affect the magnitue of the velocity perturbations observe at the seafloor. Term (a contains quantities which are either fixe constants or elastic parameters of the rock matrix, while term (b shows that the imensionless incremental velocity increases with the frequency of the loaing. Consequently, uner this theory,

21 Chapter 6: Poroelasticity an tial loaing 83 seafloor velocity signals are preicte to be high-pass filtere relative to the ocean tie. Equation (6.37 preicts that the ratio of semi-iurnal to iurnal amplitues in the outflow velocity signal is greater than that in the ocean tie by a factor of. Term (c shows that the imensionless magnitue of the incremental velocity ecreases as the permeability increases. Term ( reveals the epenence of the velocity perturbations on the thermoynamic state of the pore water an can be use to efine a tial flow magnitue parameter (M T as follows: ( γ µ M T= (6.38 γ ( ρ ρ Graphs of M T are presente in Figure 6.9 over a range of p-t conitions appropriate to hyrothermal systems. When the porosity (φ is very small, γ an so: µ M (6.39 T ( ρ ρ Figure 6.9 shows that M T is minimise at ~4 C for all seafloor pressures. Hence, the ratio of the tially inuce incremental velocity to the buoyancy riven backgroun velocity is a ecreasing function of temperature for the temperatures below ~4 C which characterise the ischarge zone of a subseafloor convection cell (Chapter 5. Therefore equations (6.37 an (6.38 preict that the tial variations in effluent velocity are most significant as a fraction of the steay flow for: ( cool effluent ( high frequency loaing (3 low permeabilities The observation that effluent velocity variations are greatest for high frequency loaing implies that the relative magnitues of semi-iurnal to iurnal components will be greater in an effluent velocity time-series than in the local ocean tie. It has alreay been remarke (Chapter that the semi-iurnal components of the ocean tie are typically larger than the iurnal components. For these two reasons, it is expecte that tial variations in effluent velocity will, in general, be ominate by semi-iurnal frequencies. In the subsequent sections it is assume for simplicity that the loa at the seafloor is a pure sine wave of angular frequency ω. By linearity, the iurnal an semi-iurnal components may be consiere separately. In the case of semi-iurnal loaing ω =.48-4 ra.s -, while for iurnal loaing ω = ra.s -.

22 Chapter 6: Poroelasticity an tial loaing (a (b / / ḳḡ 5 / ṣ- (m T M tial flow magnitue parameter, M T (m 5/.s -/.kg -/ K g = 5 GPa p = MPa K m = 3 GPa p = 3 MPa p = 4 MPa porosity =.5 p = 5 MPa temperature ( C 84 (c (e ( (f / ḳḡ 5 / ṣ- (m T M tial flow magnitue parameter, M T (m 5/.s -/.kg -/ - K g = 5 GPa p = MPa K / m = 3 GPa p = 3 MPa -3 porosity =. p = 4 MPa p = 5 MPa / / ḳḡ 5 / ṣ- (m T M temperature ( C tial flow magnitue parameter, M T (m 5/.s -/.kg -/ - K p = MPa g = 5 GPa -3 p = 3 MPa K m = 3 GPa p = 4 MPa porosity = 5e-5 p = 5 MPa temperature ( C Figure 6.9: The epenence of the tial flow magnitue parameter (M T on the thermoynamic state of the interstitial flui in the large, intermeiate an small porosity regimes with K g = 5 GPa, K m = 3 GPa. The magnitue of the velocity perturbations at the seafloor, relative to the backgroun velocity, is proportional to M T. Interstitial flui is pure water. Thermoynamic ata from the tables embee in the HYDROTHERM coe. (a, (b φ =.5 (large. (c, ( φ =. (intermeiate. (e, (f φ =.5 (small. ( 7 Pa correspons to km below sea level.

23 Chapter 6: Poroelasticity an tial loaing Incremental temperature in an infinite halfspace In the previous section, the incremental pore pressure solution of Van er Kamp & Gale (983 ( p ˆ( z, t is extene to yiel an expression for the incremental velocity ( w ˆ ( z, t. In this section, an expression for a thir quantity of interest - the incremental temperature ( T ( z, t ˆ - is erive. (a no tial loaing z (b tial loaing: p T exp(iωt z seafloor z = x seafloor z = x steay velocity: steay pressure: steay temperature: w p ρgz T Γz velocity: pressure: temperature: ( z t gz pˆ ( z t z Tˆ ( z t w wˆ, p ρ, T Γ, loaing: flow: ( z t Figure 6.: The incremental temperature ( T ˆ, ue to tial loaing, just below the seafloor. (a In the absence of tial loaing, the vertical velocity is assume to be constant while the pressure an temperature are linear functions of epth. (b Uner tial loaing, the pressure, velocity an temperature are all tially moulate. For simplicity, it is suppose that pressure an temperature are linear functions of epth in the absence of tial loaing (Figure 6.a. This is approximately true in the section of the hyrothermal ischarge zone which lies just below the seafloor (Figure 5.. If the pressure graient is col hyrostatic (Section 5.3 an the temperature graient is enote by Γ, then: ( z p g z p ρ = (6.4 ( z T z T Γ (6.4 = It is suppose for simplicity that the ensity (ρ an ynamic viscosity (µ o not vary with epth. The buoyancy-riven vertical velocity (in the absence of tial loaing therefore has the constant value: ( ρ ρ k p gk w = ρ = µ g z (6.4 µ This steay state flow (Figure 6.a satisfies mass conservation because:

24 Chapter 6: Poroelasticity an tial loaing 86 ( w = z ρ (6.43 However, the avective energy accumulation per unit volume (Q is non-zero because: h Q = ( ρ hw = ρw (6.44 z z Supposing for simplicity that enthalpy is a linear function of temperature - h(p,t = c p T - it then follows that: Q = ρ c wγ (6.45 p In a real seafloor system, the avective energy accumulation of equation (6.45 woul be balance by the lateral conuctive flow of heat away from the ischarge zone. It is not possible to incorporate lateral conuction into this moel explicitly because it is assume that there is only one spatial imension. However, energy can be balance by assuming that heat is remove at a rate Q per unit volume at all epths below the seafloor. In other wors, heat sinks must be introuce to simulate the effect of conuctive heat loss an ensure that energy is balance in the steay state. It is suppose that the steay, one-imensional solution escribe above is isturbe by tial loaing (p T exp(iωt on the seafloor (z = (Figure 6.b. The pressure, velocity an temperature are then perturbe from their steay values as follows: p ( z p g z pˆ ρ ( z, t w w wˆ ( z, t T ( z T Γz Tˆ ( z, t Assuming that the incremental pressure ( p ( z, t (6.46 ˆ an incremental velocity ( w ˆ ( z, t are given by the - solutions iscusse in previous sections (equations (6.9 an (6.3, an expression for the incremental temperature ( T ( z, t ˆ can be erive. If tially inuce changes in flui ensity are negligible, the conservation of energy uner tial loaing is expresse by the avection-iffusion equation: t ( T T ( w wˆ ( T Tˆ ˆ Q = (6.47 z ρ Equation (6.47 can be linearise by neglecting the prouct of any two incremental properties. Using equations (6.4 an (6.45, this linearisation yiels: Tˆ Tˆ w = Γwˆ (6.48 t z c p

25 Chapter 6: Poroelasticity an tial loaing 87 The steay state velocity (w an the steay state temperature graient (Γ are both constant. Therefore equation (6.48 can be use to erive an expression for the incremental ( temperature T ˆ ( z, t in terms of the incremental velocity ( w ˆ ( z, t that the tially inuce incremental velocity has the form: ( z, t = Aexp z iωt. Equation (6.3 shows i wˆ (6.49 where A is a complex number expressing the amplitue an the phase relative to the ocean tie. Equation (6.48 suggests that the incremental temperature has a similar form to equation (6.49. Accoringly, it is postulate that: i T ˆ (6.5 ( z, t = B exp z iωt where the value of the unknown complex constant (B remains to be calculate. Equations (6.48, (6.49 an (6.5 together yiel an expression linking the complex amplitues A an B: B w ω w i = Γ A ( Equation (6.5 shows that the incremental temperature T ( z, t velocity ( w ˆ ( z, t by the raian angle: arg ω ( ( = A arg B arg i w (6.5 ˆ lags the incremental (6.5 Since ω,, an w are all positive real numbers, equation (6.5 implies that the incremental temperature lags the incremental velocity by an angle in the range [π/4, π/] (i.e. between 45 an 9. The exact value of this phase lag epens on the value of the imensionless parameter: 3( v ( v αβ ( ν k ω ωµ = w g( ρ ρ S 3 (6.53 For semi-iurnal tial loaing, ω =.4-4 s - an all parameters except the permeability (k are known accurately. Assuming typical values (g = 9.8 m.s -, (ρ -ρ = 5 kg.m -3, µ = 5-5 Pa.s, S = 5-9 Pa -, ν =., α =.5, β =.5, suggests that, in SI units: ω 7 4 w k (6.54

26 Chapter 6: Poroelasticity an tial loaing 88 Using equations (6.5 an (6.54, the phase ifference between the incremental velocity an the incremental temperature can be graphe as a function of permeability (k (Figure 6.. Table 6. suggests that the permeability of the seafloor lies between -7 m an - m. Figure 6. shows that the incremental temperature lags the incremental velocity by almost exactly 9 within this range. Phase lags significantly smaller than 9 are only possible for permeabilities greater than -9 m. Such high permeabilities coul only exist if there were consierable cracking of the ischarge zone. Consequently, seafloor measurements of the phase lag between effluent velocity an temperature are iagnostic. Phase lags significantly below 9 present evience for unusually high seafloor permeability. phase lag of temperature behin velocity ( 9 8 g ( la Figure 6.: permeability, k (m The preicte phase lag of the incremental temperature behin the incremental velocity as a function of permeability (k for typical seafloor parameter values. Note the logarithmic scale on the horizontal axis. ( w ˆ( z, t ( T ˆ( z, t In the infinite halfspace moel, it has been shown that the incremental velocity at the seafloor lags the ocean tie by 35. Consequently, the infinite halfspace moel preicts that the incremental temperature at the seafloor lags the ocean tie by an angle in the range [8, 5 ]. Furthermore, for semi-iurnal loaing an permeabilities less than - m, the phase lag is preicte to be almost exactly equal to 5 (Figure Incremental pore pressure in a finite permeable layer The assumption of an infinite halfspace is a significant simplification, an it is of interest to exten the well-establishe halfspace moel of Van er Kamp & Gale (983 to the case of a

27 Chapter 6: Poroelasticity an tial loaing 89 finite permeable layer. It is therefore suppose in this section that the crust is only permeable between z = an z = -H (Figure 6.6b. The new bounary conition ( (, ˆ = t H w means that the solution of equation (6.7, which is given by equation (6.9 for an infinite halfspace, is now given by: ( ( ( ( ( t i H i H z i p t z p T ω γ γ exp cosh cosh, ˆ = ( Incremental velocity in a finite permeable layer By Darcy s law, it follows from equation (6.55 that the incremental velocity for a finite permeable layer is given (in complex notation by: ( ( ( ( ( ( t i H i H z i i p k t z w T ω γ µ exp cosh sinh, ˆ = (6.56 The incremental velocity on the seafloor is then: ( ( ( ( ( t i H i i p k t w T ω γ µ exp tanh, ˆ = (6.57 It shoul be note that the presence of the impermeable bounary at z = -H affects both the magnitue an phase of the outflow at the seafloor (equation (6.57, compare with the solution for an infinite halfspace (equation (6.3. It can be shown that: ( = H H H H H i cos sinh sin sinh tanh (6.58 an that: ( = H H H i sinh sin tan tanh arg (6.59 Therefore the tially inuce outflow at the seafloor has magnitue:

28 Chapter 6: Poroelasticity an tial loaing 9 pt k µ ( γ sinh sinh H H sin cos H H (6.6 It has alreay been shown (equation (6.3 that the effluent velocity at the seafloor lags the ocean tie by a raian angle (3π / 4 in the case of an infinite halfspace. It follows from equation (6.59 that the presence of an impermeable bounary at z = -H means that the effluent velocity at the seafloor lags the ocean tie by a raian angle: 3π tan 4 H sin H sinh (6.6 The preicte magnitue (equation (6.6 an phase lag (equation (6.6 of the incremental velocity at the seafloor are shown in Figure 6. as functions of the imensionless epth of the impermeable bounary (H /. (a.5 amplitue factor for velocity (b phase lag of velocity behin ocean tie ( 4 3 fac tor.5 g ( e la a s ph epth of impermeable bounary: H/ epth of impermeable bounary: H/ Figure 6.: The effect of an impermeable layer at epth H on the magnitue an phase of the incremental velocity at the seafloor. The horizontal axes measure the imensionless epth of the impermeable layer (H/. (a The amplitue of the outflow as a fraction of the infinite halfspace solution, which is approache as H/. (b The lag of the outflow behin the ocean tie in egrees, which approaches the infinite halfspace value of 35 as H/.

29 Chapter 6: Poroelasticity an tial loaing 9 In summary, the incremental velocity at the seafloor lags the ocean tie by an angle between 9 an 35, epening on the relative magnitue of the - skinepth ( an epth over which the seafloor crust is permeable (H. If the skinepth is much smaller than the thickness of the permeable layer (H / >> the incremental velocity follows the infinite halfspace solution an lags the ocean tie by 35. In contrast, if the skinepth is greater than H/ (so that H / <, then the incremental velocity at the seafloor is reuce in magnitue compare to the infinite halfspace solution (Figure 6.a, an its phase lag behin the ocean tie takes a value in the range [9, 35 ]. When sufficient high quality ata are available, observe values of the phase lag between the ocean tie an the effluent coul be use to constrain the ratio (H / Tial loaing of a - seafloor In Section 6.3., the investigation of tial loaing is restricte to one spatial imension for simplicity. In this section, the same principles are extene to two imensions by consiering the tial loaing of the simulate convection cell of Chapter 5. The numerical steay state solution (Figure 5.4 efines the properties of the interstitial flui as functions of space. Figure 6.3 shows the istribution of pressure (p, temperature (T, ensity (ρ an flui compressibility (K f - within the simulate convection cell. It shoul be note that the seafloor is place at z = m in these figures in common with the convention followe throughout this chapter. (In Chapter 5, however, it is more convenient to place the seafloor at z = m. The ivision between the col, ense, liqui-like region below ~4 C an the hot, buoyant, gas-like region above ~4 C is clearly shown in Figure 6.3c,. Uner tial loaing, it is expecte that the iffering incremental pore pressures in these regions will rive a tially moulate flow. In the following sections, a numerical simulation is use to examine this flow Governing equations in - In Section the equations governing the propagation of incremental pore pressure in one imension are erive (Van er Kamp & Gale, 983. In this section, the analogous ( equations governing the incremental stress ˆ ( x, z t ( p ˆ ( x, z, t σ an incremental pore pressure ij, in two spatial imensions are erive. In this case, the assumption of only two spatial imensions implies that e ˆ =. Therefore equations (6. an (6.3 imply that: ( ν ( ˆ σ ˆ σ α( pˆ ˆ kk 33 ν σ = (6.6

30 Chapter 6: Poroelasticity an tial loaing 9 (a (b (c ( Figure 6.3: The thermoynamic properties of the flui in the numerically simulate convection cell of Section 5.3. In each plot, the omain is the same as that shown in Figure 5.4a. The seafloor is at z = m an a magma chamber lies at z = - m. On all plots the isotherms at 3 C, 4 C an 5 C are shown in white. The change from liqui-like properties to gas-like properties occurs at ~4 C. (a The pore pressure, p. (b The temperature, T. (c The ensity, ρ. ( The flui compressibility, K f -. It then follows from equation (6.4 that: ( ( ( ( ˆ ˆ ˆ 33 = σ ν ν σ ν ν ν α z x z x p z x (6.63 ( ( ( ( ˆ ˆ ˆ 33 = σ ν ν σ ν ν ν α z x z x p z x (6.64 Equation (6.6 implies that: ( ( ( ( ( 33 ˆ ˆ 3 ˆ ˆ 3 3 σ σ ν αβ β ν µ ν αβ = t t p z p S k (6.65 Equation (6.65 suggests the efinition of a - iffusivity accoring to:

31 Chapter 6: Poroelasticity an tial loaing 93 3 k κ = αβ ( ν ( µ S an a - loaing efficiency accoring to: ( ν ( ν β γ = ( αβ It then follows that conservation of flui mass is expresse by: pˆ pˆ κ γ ( ˆ σ ˆ = σ 33 (6.68 z t t Equations (6.63, (6.64 an (6.68 are the governing equations for the three scalar fiels p ˆ ( x, z, t, ˆ σ ( x, z, t an ˆ σ 33 ( x, z, t. A numerical solution for these fiels can be foun when tial loaing (p T cos(ωt is applie to the seafloor (z = Numerical simulation Bounary conitions are require in orer to obtain a numerical solution to equations (6.63, (6.64 an (6.68. Consiering first the incremental pore pressure, the tial loaing on the seafloor imposes the conition: ( i t ( pˆ = pt exp ω z = (6.69 The bottom bounary (z = - m is taken to be impermeable, which implies that: pˆ z z= = (6.7 It is expecte that the one imensional solution of Section will apply on the sie bounaries (x = ±5 m. If the - skinepth an - loaing efficiency for col water are labelle c an γ c respectively, then equation (6.7 implies the bounary conitions: ( i z pˆ ( ± 5, z, t = γ p ( i t ( p i t c T exp ω γ c T exp ω (6.7 c Following Van er Kamp & Gale (983 it is assume that the bounary conition for the vertical stress is the same on all four bounaries: σ = p T exp( iωt ( Finally, in orer to ensure that e ˆ =, the following conition must be met on all four bounaries: ( ν p ν α σ = σ 33 (6.73 ν ν Since the tial loaing is assume to occur at a single angular frequency ω, it follows that the incremental pressure an stress can be written in the form:

32 Chapter 6: Poroelasticity an tial loaing 94 ˆ σ ˆ σ pˆ 33 ( x, z, t = P( x, z exp( iωt ( x, z, t = Σ( x, z exp( iωt ( x, z, t = Σ (, exp( 33 x z iωt (6.74 The governing equations (6.63, (6.64, (6.68 an the bounary conitions (6.69 (6.73 can then be solve using stanar finite ifference techniques (Press et al., 986 to fin the complex amplitues P(x,z, Σ (x,z an Σ 33 (x,z. Unfortunately, only a very limite amount of computer time was available to perform calculations of this nature, an the parameters use (K m =.5 GPa, K g = 5 GPa, ν =., φ =., k = - m are not consistent with those assume elsewhere in this issertation (Chapter 5. Computer facilities were not available to rerun the simulations with consistent parameter values. The results (Figures shoul therefore be viewe as illustrative of a general principle rather than efinitive. Nonetheless, the simulations cover a sufficiently wie range of parameter values to illustrate the ifferent incremental pore pressure regimes which apply accoring to the size of the col skinepth c. The subscript c is use to enote the values of the loaing efficiency (γ c an skinepth ( c obtaine for col water from equations (6.6 an (6.8. Similarly, the subscript h is use to enote the values (γ h an h obtaine for hot (~4 C, gas-like water. In these numerical examples, γ c.93 an γ h.4. The seafloor parameter known to the least precision is the permeability (k, which can vary over several orers of magnitue (Table 6.. Equation (6.8 shows that col skinepth c epens on the ratio (k / ω, an so the effect of varying k is simulate numerically by varying ω. Figure 6.4 shows the calculate incremental pore pressure for tial loaing with k = - m, ω =. s -. In this case the col skinepth ( c =.6 m is much smaller than the imensions of the cell, an so the iffusive part of the incremental pore pressure is restricte to a region of lengthscale c = 5. m aroun the seafloor an the bounary between the liqui-like an gas-like regions. At istances greater than c from these bounaries in elastic properties, the solution takes a form equivalent to the instantaneous pressure signal of equation (6.. In other wors the solution is approximately pˆ γ p exp( iωt in the liqui-like region of the cell an p γ p exp( iωt h T c T ˆ in the gas-like region of the cell. Therefore the incremental pore pressure is in phase with the tial loaing over most of the convection cell.

33 Chapter 6: Poroelasticity an tial loaing 95 (a (b (c ( Figure 6.4: The complex incremental pore pressure amplitue P(x,z with p T = kpa, c =.6 m, γ c =.94 an γ h =.4. (a The real part of the solution, representing the incremental pore pressure at high tie. (b The imaginary part of the solution, representing the incremental pore pressure at rising half tie. (c The magnitue of the incremental pore pressure. ( The phase of the incremental pore pressure expresse as a phase lag behin the ocean tie in egrees. In all figures, the isotherms at 3 C, 4 C an 5 C are rawn in white to illustrate the structure of the unerlying convection cell (Figure 5.4. Figure 6.5 shows the incremental pore pressure results with k = - m, ω =. s -. In this case, the larger value of the col skinepth ( c = 6 m explains why the incremental pore pressure response of Figure 6.5 oes not follow the bounaries in flui properties (Figure 6.3 as closely as in Figure 6.4. The larger skinepth has le to a smearing of the response across the bounary between the liqui-like an gas-like regions.

34 Chapter 6: Poroelasticity an tial loaing 96 (a (b (c ( Figure 6.5: The complex incremental pore pressure amplitue P(x,z with p T = kpa, c = 6 m, γ c =.94 an γ h =.4. (a The real part of the solution, representing the incremental pore pressure at high tie. (b The imaginary part of the solution, representing the incremental pore pressure at falling half tie. (c The magnitue of the incremental pore pressure. ( The phase of the incremental pore pressure expresse as a phase lag behin the ocean tie in egrees. In all figures, the isotherms at 3 C, 4 C an 5 C are rawn in white to illustrate the structure of the unerlying convection cell (Figure 5.4. Finally, Figure 6.6 shows the incremental pore pressure results with k = - m an ω =. s -. In this case, the col skinepth ( c = 86 m is sufficiently large that incremental pore pressure response is ominate by the iffusive component over the entire convection cell.