Erosion and Gas Content in Subsoil

Transcription

1 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 1 Erosion and Gas Content in Subsoil Curt Koenders Department of Mathematics; Kingston University; Penrhyn Road, Kingston on Thames, Surrey; KT1 EE; UK Abstract The mechanism that leads to fluidization in a partially saturated soil system at the bottom of a river or canal is explored. It is demonstrated that a simple analysis is capable of yielding the time it takes to first reach the fluidized state. This phenomenon takes place in a thin layer. The motion of this layer is further analysed, both ith and ithout an externally introduced shear field. It transpires that time constants that rule this regime are very small. It is shon that a shear field has the effect of negating the fluidixation. Assumptions that are made in this one-dimensional analysis are discussed. Introduction The presence of small air bubbles in a non-cohesive medium at the bottom of a ateray (river or canal) is an important issue hen large and fast external pressure variations may be expected. The factors that determine the onset of failure have been ascertained by theoretical means in the past by Roussell et al (). and experimental verification of the predictions have been shon to be quite accurate. The theory predicts quite simply that fluidization of an unprotected bed first takes place after a time t 1 reckoned from the moment hen the external pressure had started to fall at a rate of. This quantity is of eminent importance for the practising t is a key design parameter for unprotected beds. Basically it engineer. An estimate of 1 states: if one expects the external pressure fall at a rate to continue for less than a period of t 1 then the bed ill not fluidize. If it goes on for a period of time that is greater than t 1 then fluidization ill occur. The consequence of fluidization is that in this state the grains do not transmit forces, implying that an external fluid flo ill be the only driving force. So, in the fluidized state the granular bed is susceptible to erosion, much more so than in the state in hich enduring contacts persist. The estimate for the time t 1 is i t1 y c, (1) a here y is the pressure equivalent ater depth - that is the actual ater depth x plus the equivalent ater level associated ith the external pressure p : y p / x, i c the critical gradient (measured in engineering units as a fraction of the fluid specific eight ) and a a system parameter ith the dimension of the inverse of the square root of a velocity. The latter is an intriguing quantity as it is the only variable that contains information about the particle/fluid/gas mixture. Other than a numerical factor, the result could have been obtained by means of dimensional analysis, coupled ith some heuristic arguments. The presence of the gas bubbles is represented through the saturation measure, s i, stating the fraction of the fluid that is not in bubble form. Note that the bubbles tend to adhere to the grains; in fact it is very difficult to get rid of them and a typical figure for Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

2 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite the saturation in natural deposits at ater depths that are less than 1 m is some 95-99%. The source of high bubble gas deposits is most likely due to bacterial life. These thrive in beds ith organic matter. The latter ill reduce the permeability, making the time t 1 comparatively shorter, hich is associated ith a higher risk of fluidization. The saturation is embedded in the parameter a, as ill again be demonstrated belo. Previously (ibid) it as found that 4n1 s a i, () k here n is the porosity of the deposit and k its permeability. The saturation may be a very position-dependent parameter, especially depending on the depth. The post-fluidization development has so far received no attention. This aspect of the problem is addressed at the end of this paper. The fluidized state, coupled ith a shear stress exerted by a fluid flo parallel to the bed, is prone to shearing motion. This motion is ruled by viscous fluctuation theory, see McTigue and Jenkins (199). Once shearing motion of a layer at the top of the bed is established a particle pressure results and the response of the system to the appearance of this part of the stress is analysed. The salient outcome is that the fluidized state cannot persist unless the external ater level subsides at a sufficiently large rate. Broadly speaking this result states: the medium may fluidize, but motion due to the fluidization is only possible if the fluidization is sufficiently fast. Naturally, bed-loading may occur due to the usual factors, see for example Jenkins and Hanes (1998). The emphasis in this paper is on the fluidization due to unsaturated fluid. The paper has the folloing structure. First the physical theory that leads to the estimate of t 1 is reproduced, using the simplifying assumption that there is no depth variation of the fluid/gas mixture compressibility of the fluid. The theory employed is Biot's famous consolidation equation: Biot (1941). Previously, a more sophisticated analysis as explored in hich the compressibility depends on depth. The reason for the simplification is that a post-failure analysis ill be presented. The starting point of the post-failure analysis is obviously the pre-failure state and it is convenient to have this state in analytical form. No insight is lost in this simplification, though numerical factors may be slightly different in the more sophisticated exploration. The actual post-failure analysis assumes movement of the fluidized region due to a slo external horizontal flo. The granular temperature theory of McTigue and Jenkins (199) is used to estimate the stresses that are associated ith a shearing flo. The simplest version is employed to attain insight in first order effects. The stresses introduced by the horizontal flo perturb the original Biot problem. The perturbation is implemented and a solution is presented. Finally, concluding remarks are given, especially addressing future ork that may be desirable. Simple model of one dimensional consolidation The consolidation phenomena that take place during an externally imposed pressure drop are modelled mathematically. The constitutive behaviour of a soil/fluid/gas mixture has been studied in the past. Most effects that are relevant are discussed by Theunissen (198), draing on earlier ork by Barends (198). These authors have demonstrated that for bubbles beyond a certain diameter (roughly 1 m at atmospheric conditions), the compressibility ' of a gas/fluid mixture can be Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

3 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 3 approximated and expressed in the gas solubility, the initial saturation s i and the initial pressure pi 1 si 1 s p p 1 '. (3) p pi i i The parameter is alays small compared to unity, typically in the order of.. For small variations from the initial pressure the compressibility satisfies: ' 1 s i. (4) pi Thus an estimate of the initial pressure and the initial saturation are required to be able to model the compressibility of the fluid/gas mixture and hence of the consolidation effects that follo an increase or decrease in the external pressure. The latter is described by Biot s Equation (1941), relating the soil permeability k, the excess pore pressure p, the porosity n and the skeletal volume strain e. For a homogeneous soil in a fluid ith specific eight and for one-dimensional problems, Biot s Equation reads k p p e n '. (5) x t t Various corrections to this equation have been put forard. These are all discussed by Barends (198). One significant correction pertains to free moving gas bubbles; this correction appears not to be relevant, as gas bubbles are nearly alays observed to adhere to the soil particles. The compressibility depends on the ambient pressure p i, hich consists of the atmospheric pressure at the fluid surface p and a term that depends on the ater depth. The latter varies according to the ater depth according to x, here x is measured donards from the top of the ater level. An expression for e for densely packed material under compressive stress is obtained from a deformation modulus F, such that the excess skeletal stress ' Fe. Using furthermore Terzaghi s stress principle relating total stress excess, the excess poreater pressure p and the skeletal stress (using the convention that compressive stresses are negative) such that ' p, the rate of change of the volume strain is seen to be e 1 p. (6) t F t t The total stress satisfies stress equilibrium in one dimension and is thus independent of the spatial co-ordinate x, hile its time-dependence is prescribed by the external loading programme. The analysis here is intended for small variations in the external fluid pressure, due to changing ater levels. The relevant equation that requires solution is k p n 1 si 1 p 1. (7) x p x F t F t Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

4 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 4 The soil occupies a region x x, here x is the ater depth. F is assumed to be a constant in the densely packed soil. The saturation s i is also assumed to be a constant, independent of depth. The folloing simplifying assumptions are implemented. 1. The region in hich appreciable excess pore pressures develop is small in size compared to the ater depth, so that for the purposes of the analysis p x p x. In this ay the depth-sensitivity of the fluid/gas compressibility is not noticed.. Typically the value of F is much greater than p, thus implying that a reasonable 1 approximation is achieved if the terms proportional to F in Equation (7) are 1 neglected for a solution in a region of magnitude x n s i F x 1. All these assumptions are open to further investigation and refinement. The thrust of the analysis here is to be able to dispose of simple, leading order expressions for the excess fluid pressure and its gradient, so that basic effects can be roughly quantified. In hat follos extensive use is made of temporal Laplace transforms; transformed variables are denoted by a ^ and the Laplace frequency is called. The solution to Equation (7) is obtained immediately in Laplace transformed form 1 axx y pˆ( x, ) Aˆ( ) e, (8) here A ˆ( ) is a parameter that is independent of the position. The Laplace transform of the pressure gradient is easily obtained. The folloing equation therefore holds y pˆ( x, ) pˆ ( x, ). (9) a x As a result one may rite t y 1 p( x, ) p( x, t) d. (1) a ( t ) x The excess pore pressure at the boundary is For t t1 : p( x, t) t. (11) The gradient on the boundary follos from Equation (9), using the functional forms of the Laplace transforms (Abramoitz and Stegun (1965)) p( x x, t) a t y. (1) The gradient reaches the 'critical' value hen fluidization takes place. This occurs hen the vertical gradient exceeds the unit eight of the soil, Terzaghi (1943). The unit eight is called and the fluidization criterion for the critical gradient i c, directly expressed as a fraction of the unit eight of ater, reads 1 p ic 1 n x. (13) A practical value is of the order of magnitude of unity. Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

5 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 5 From equations (1) and (13) the time to reach the critical gradient is derived. This is the value reported in the Introduction, Equation (1). The result is no different than the one reached before, but useful Equations (9) and (1) have been modified slightly, compared to the theoretical outcome of the less-simplified theory (here the compressibility of the fluid/gas obeys a linear dependence as a function of depth). The fluidized medium Once enduring contacts are lost the equations for the granular assembly become entirely different. For fluidized material ithout shear the stress-strain behaviour outlined in the previous paragraphs is incorrect: the elastic moduli collapse and the only skeletal stress that can still be transmitted via the grains is associated ith a strain rate: ' f e t, here f is a viscous modulus. Laplace transforming gives the folloing Biot equation, valid for t t 1 k pˆ n 1 si ˆ pˆ pˆ p( t1). (14) x p x f The total stress continues to vary according to the falling external pressure: p ˆ ( x) ˆ. Thus near the top the pressure amplitude may alays be approximated in terms of the gradient amplitude Â, as ˆ ˆ p ˆ( x) A x x. (15) In a region here A exceeds the critical gradient fluidization takes place. This region extends from x x x1. (The boundary x 1 ill naturally be time-dependent). The equation that rules the fluidized region for times t t reads ˆ ˆ A x x A x x ˆ A( t ) x x ( t ) n 1 si p x f Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober. (16) The time span over hich the analysis is valid is sufficiently short to let f be constant and t) ( t ). The latter implies that ( t ) and therefore: ( 1 s ˆ i ˆ A ( ) n p x A A t The immediate result is A t) A( t ) exp p fn f x t 1 s ( i ˆ. (17). (18) The gradient thus reduces ith a time constant approximately equal to fn1 si p x. For reasonable estimates of f (order Pas) this time constant is very small (order s-ms) compared to reasonable engineering time scales (order 1-1 s). This is an indication of the time it takes to restore the gradient to the critical value. (The result justifies the assumption that the external load may be regarded as a constant for the period of time over hich the analysis is valid). What this analysis shos then is that hen fluidization takes place, driven by phenomena that typically take a time in the order of seconds, the viscous effects in the medium ensure that the gradient never exceeds the critical value. In physical terms this implies

6 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 6 that the soil does not boil, but that fluidized effects - due to the presence of gas bubbles in the soil - take place ithout any externally visible features. The soil, hoever, may be in a fluidized state for a substantial period of time precisely as has been observed in the experiments. For fluidized media that are simultaneously sheared the above analysis requires adaptation. In shearing a dense granular medium, particles are forced to deviate from their straight mean trajectories and thus acquire fluctuational motion. This part of the motion is characterised by a granular temperature T, hich is the quadratic mean fluctuation velocity of the assembly. Furthermore, regime specification is required, specifying ho the grains interact in the shearing motion. The regime considered here is the viscous regime (as opposed to the collisional regime), hich presumes a small Bagnold number, giving the ratio of collisional stress to viscous stress expressed in the solids mass density s, the particle diameter D, the linear concentration, the typical shear rate u x and the fluid viscosity sd u Bg. (19) x The order of magnitude for a layer ith a thickness of, say, 1 sand grains, the top ro 1 moving at a velocity of some 1ms is of the order of 1. This value is ell belo the critical value of some 5, so the slo, viscous limit is applicable. For viscous flos, hoever, the theory is little-knon. A modelling approach by McTigue and Jenkins (199) is closely folloed here, though other approaches have been put forard: Nott and Brady (1994). McTigue and Jenkins (199) regard the particles in the suspension as a dense gas, hich has mechanical and thermal properties, see Chapman and Coling (197). The energy fluctuations of the particles due to shearing are captured in a scalar measure called the granular temperature. Particle diffusion takes place hen a temperature gradient is present. The approach involves a balance equation for the fluctuational energy, equivalent to the heat equation for a gas. The flux vector of fluctuational energy is called Q, the average particle velocity vector is u, the stress tensor for the particulate phase is t and the dissipation rate (hich needs to be added separately as this is a multi phase fluid) is. In equilibrium the balance equation reads Q x k k t ik u x i k, () here Einstein s summation convention has been applied. Furthermore, introducing the particle pressure p, the fluid density f and the solids volume fraction, the other steady-state equilibrium balance las are the stress equilibrium and the equation of continuity p p tij s f g, (1) x x x i i j vi. () xi These equations are made applicable by introducing constitutive las much like those of a dense gas, that is an isotropic fluid constitutive la for the stress tensor, expressed Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

7 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 7 in the deviator velocity gradient d and a Fourier-type la relating the heat flux to the temperature gradient t d d, (3) ik kk ik ik T Qk. (4) xk McTigue and Jenkins (199) introduce very simple, reasonable and physically ellfounded expressions for the constitutive parameters, and as ell as an approximation for the dissipation rate and the particle pressure p. Their vie is that in the viscous limit forces are mediated through the fluid at short range and therefore a lubrication analysis holds. The interparticle force in the lubrication limit depends on the gap idth h beteen the particles and their relative velocity. An analysis of ho the form of the interparticle force is associated ith macroscopic constitutive properties then leads to the folloing proportionality relations, expressed in the particle diameter D and the inverse relative gap idth D h D 1, (5) h D, (6) h D 3 T D, (7) h D 1/ p 4 T, (8) D h The nondimensional coefficients depend on the local geometry (that is the particle shape) and are also eak functions of the solidosity. 4 is furthermore proportional to the parameter 1 e, here e is a measure for the energy loss during an encounter beteen to grains. When the grain surfaces are very smooth e 1, but this is not relevant to sand grains here a distinct angularity is present. This theory then is highly appropriate to sheared fluidized media. While it still contains a number of coefficients as ell as the underlying assumptions of dense gas theory, the key physical mechanisms that take place in flo are represented. The theory orks out quite simple in that the main motion of the particles is in the direction of the surface of the bed. For thin layers, typically ith a thickness of fe particles, an averaging approach may be taken; the temperature gradient is neglected and the mean dissipation rate in the layer follos from the mean shear strain rate and the mean shear stress, as follos t 1. (9) The shear stress is imposed externally on the bed and is related to the mean flo profile of the ater in the channel. Assuming a Karman-Prandtl approach (see Landau and Lifschitz (1981)) the magnitude of this quantity is 1 f v * t, (3) here * v is a velocity that is characteristic for the fluid flo and such that the mean fluid velocity above the bed behaves as Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

8 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 8 v * f x x v* v.4 ln (31) b The shear stress t 1 is thus a given quantity that follos from a fit of the fluid velocity profile. The important ne element in this analysis is the particle pressure. Combining formulae (7), (8) and (9) it is seen that p L ; 4 D L t1. (3) h 3 This is again a mean, layer-thickness averaged value. The shear rate is assumed to be time-dependent. The motion takes time to establish itself. Ho exactly that takes place is not knon. The extra skeletal stress is inserted into Biot's equation to give for the fluidized medium k ˆ pˆ L ^ pˆ n 1 s i pˆ p( t1). (33) x p x f f The interest focusses here on effects that occur immediately after the motion has started. It as seen in the previous section that the time constants are so small that the ongoing decrease in the external ater level is not noticed. Belo it is shon that the effects that follo from the extra skeletal stress also become manifest in a very short time. Hoever, it is not clear hat exactly the time development of this stress is. In hat follos it ill be assumed that this stress comes into being at a much faster rate than the external ater level drop rate. This point ill be returned to in the discussion. The extra excess pore pressures are called q; they are averaged over the layer; the averaging is denoted by <>. This averaging is performed over the layer thickness ; the layer is very thin in terms of numbers of partaking particles, justifying the approximation k qˆ 1 L 1 qˆ ^. (34) x f f The solution for the extra pressure up to second order in x x is sought. This is possible by using equation (34) and to boundary conditions. The first is that the extra excess pore pressure vanishes at the top of the layer. The second follos from the boundary condition for a small extra stress in the packed layer; this pressure satisfies Equation (9) y qˆ(, ) qˆ (, ). (35) a x Both pressure and gradient are continuous. The solution for qˆ is qˆ Ax ˆ 1 Bˆ x, (36) ith Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

9 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 9 Aˆ a and Bˆ a 6L ^ a 4 y 1 1 fk 8 y 1 1L ^ a y 1 1 fk 8 y 1 ; (37) 3 fk. (38) 3 fk The form of these expressions suggests to time scales. The first is, hich is a very short time scale. The second is a time a / y, hich is also very short. For time scales longer than these milliseconds, hich are of interest for the practical 1 engineers, take,, to give ^ L Aˆ Bˆ 3. (39) 3 fk Thus, the extra pressure takes the form ^ 3L x qˆ. (4) x 1 3 fk Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

10 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite q/q (x-x )/ Figure 1: Pore pressure as a function of position before (solid line) and after (dashed Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

11 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 11 line) shear is applied to the fluidized bed. The axes are scaled to achieve nondimensonality. q is a reference pressure. Clearly both extra excess pore pressure and its gradient are proportional to the particle pressure; this a time dependent, but positive variable. The sign of the various contributions is important. In Figure 1 the before and after situations are depicted. The solid line represents the pressure profile before the particles start to move due to the shear. The gradient in the fluidized layer is equal to the critical gradient. When the particles start to move (strictly speaking, very shortly after this time point) the pore pressure takes the value plotted by the dashed line. It is seen that the gradient in the fluidized layer suddenly becomes much smaller. As a result the fluidization ill halt. But the gradient is position dependent. The pressure gradient remains unaltered at the fluidized/non-fluidized boundary. If any motion takes place it ill be in a very thin shear layer near this boundary. It is concluded that here gas in the fluid predicts fluidization, subsequent motion due to shear predicts the impediment of fluidization. Conclusions/discussion The conclusions that may be dran from the above analysis are summarised. First of all it transpires that the prediction of the time to onset of fluidization in an unsaturated system is relatively insensitive to the details of ho the fluid/gas compressibility varies as a function of the depth. An average value may therefore be supplied to obtain a reasonable first-order estimate for this important parameter. Secondly, a medium fluidized by the mechanism reported in this paper shos fe outard signs of the state it is in. Internal inspection of the medium by means of endoscopy shos the grains to be moving relative to one another as the gas bubbles inflate. This proves that the analysis predicts the correct event, but also that the fluidized material remains densely packed. Finally, it has been demonstrated that any shearing motion is tharted by the emergence of a particle pressure. The onset time constant of this quantity is unknon, but it has been assumed to be much faster than the pressure drop associated ith the general ater level fall. This really requires experimental verification, hich is left here as a suggestion for further research. It must be recognised that systems like these - properly controlled - are not easily reproduced in the laboratory. References Abramoitz, M., Stegun, I.A. (1965) "Handbook of mathematical functions", Dover, Ne York. Barends, F.B.J. (198) "Nonlinearity in groundater flo", Thesis, University of Technology Delft. Biot, M.A. (1941) "General theory of three dimensional consolidation", J. Appl. Physics, 1, Chapman, S. and Coling T.G. (197) "The mathematical theory of non-uniform gases" Cambridge, Cambridge University Press. Jenkins, J.T. and Hanes D. (1998) Collisional sheet flos of sediment driven by a turbulent fluid. J. Fluid Mechanics 37, 9-5. Landau, L.D. and Lifschitz, E.M. (1981) "Hydrodynamik". Berlin, Akademie-Verlag. Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober

12 M. A. Koenders: Erosion and Gas Contant in Subsoil Seite 1 McTigue, D.F. and Jenkins, J.T., (199) Channel flo of a concentrated suspension, in: Advances in Micromechanics of Granular Materials, Shen, H.H. et al, eds (Elsevier), pp Roussell, N., Köhler, H.J. and Koenders, M.A. () Analysis of erosion protection measures in partially saturated subsoils. Proc. Geofilters. Warsa, Poland Terzaghi, K. (1943) "Theoretical soil mechanics" Wiley, Ne York. Theunissen, J.A.M. (198) "Mechanics of a fluid-gas mixture in a porous medium" Mechanics of Materials 1, Druckdynamik in Erdbauerken, BAW Karlsruhe, 3. Oktober