Figure
1.
Geometry of both real and image moving point dislocations.
| Citation: | Zhihua Ouyang, Derek Elsworth, Jianlong Sheng. Theory of Volumetric Moving Dislocation in Poroelastic Halfspace and Characterization of Magma Intrusion Events. Journal of Earth Science, 2005, 16(1): 58-65.
|
The undrained change in pore fluid pressure that accompanies dike intrusion may be conveniently represented as a moving volumetric dislocation. The concept of a dilation center was developed to represent the field of undrained pressure change in a saturated linear elastic medium. Since instantaneous pore fluid pressures can be developed to a considerable distance from the dislocation, monitoring the rate of pressure generation and subsequent pressure dissipation in a fully coupled manner enables certain characteristics of the resulting dislocation to be defined. The principal focus of this study is the application of dislocation-based methods to analyze the behavior of the fluid pressure response induced by intrusive dislocations in a semi-infinite space, such as dike intrusion, hydraulic fracturing and piezometer insertion. Partially drained pore pressures result from the isothermal introduction of volumetric moving pencil-like dislocations described as analogs to moving point dislocation within a semi-infinite saturated elastic medium. To represent behavior within the halfspace, an image dislocation is positioned under the moving coordinate frame fixed to the front of the primary moving dislocation, to yield an approximate solution for pore pressure for constant fluid pressure conditions. Induced pore pressures are concisely described under a minimum set of dimensionless parameter groupings representing propagation velocity, and relative geometry. Charts defining induced pore fluid pressure at a static measuring point provide a meaningful tool for determining unknown parameters in data reduction. Two intrusive events at Krafla, Iceland are examined using the type curve matching techniques. Predicted parameters agree favorably with field data.
The undrained change in pore fluid pressure that accompanies dike intrusion may be conveniently represented as a moving volumetric dislocation. The concept of a dilation center (cleary, 1977) was developed to represent the field of undrained pressure change in a saturated linear elastic medium. Since instantaneous pore fluid pressures can be developed to a considerable distance from the dislocation, monitoring the rate of pressure generation and subsequent pressure dissipation in a fully coupled manner enables certain characteristics of the resulting dislocation to be defined. The principal focus of this study is the application of dislocation-based methods to analyze the steady and transient behaviors of fluid pressure response induced by intrusive dislocations in a semi-infinite space.
A volumetric dike injection within the upper crust is driven by buoyant ascent where density mismatch provides the motive force (lister, 1990; ryan, 1987), and motion may be controlled by rheological (rubin, 1993; Lister and Kerr, 1991) or brittle fracture (ingraffea, 1987) constraints. Regardless of the form of rheological (Parsons et al, 1992) or tectonic (Muller et al., 2001; Watanabe et al., 1999) constraints, the injection of magma results in a remanent volume change along the dike path, which in fluid saturated media may result in the generation of undrained pore fluid pressures (elsworth, 1992). These induced pore fluid pressures may be significant. In the case of two dike injection events at Krafla, Iceland, an overpressure generation of 70 m of head resulted, occurring over 4 km distant from the injection site. These significant effects have been implicated in the large-scale flank collapse of volcanoes (Elsworth and Day, 1999), where induced pore pressures may provide the necessary impetus for destabilization.
Rice and Cleary (1976) provided a fundamental solution for the disturbance of the pore pressure field due to localized inelastic deformation and within an infinite fluid-saturated porous elastic material. Localized inelastic deformation is represented by the dilation of an equiaxed plastic cube embedded within a saturated porous elastic medium, where compatibility is maintained between strains within the interior inelastic cube and the elastic exterior. The cube is shrunk to a point, while retaining a constant value of the dilation center strength. The pore fluid pressure that develops following the application of a point normal dislocation at the origin within an infinite medium is defined as
|
|
(1) |
with
|
|
(2) |
and
|
|
(3) |
where the pore pressure rise, Δp, above ambient pressure is spherically symmetrical within Cartesian (x, y, z) space at time t. K is the effective modulus of the interior region and the hydraulic diffusivity is c. The "undrained" Poisson 's ratio of the solid matrix is υu and ψ is the hydrostatic pore-pressure coefficient defined by Skempton (1954).
A consideration of the conservation of mass applied in both the interior plastic zone and the exterior elastic zone requires that the integrated volume change within the infinite medium be zero. Consequently, assuming that the edge length of the equiaxed cube is 2ε, then
|
|
(4) |
where the compressibility, cp, of the saturated medium is defined in terms of hydraulic diffusivity, c, intrinsic permeability, κ, fluid dynamic viscosity, μ, and porosity, ϕ, as cp=κ/(cϕμ). Since the volume change, Z, remains constant as ε approaches zero, equation (4) may be expressed as
|
|
(5) |
Substituting equation (1) into (5) and completing the integration yields
|
|
(6) |
Resubstituting equation (6) into equation (1) yields the pore pressure field resulting from the instantaneous insertion of volume, Z, into the porous medium at time t=0, and retaining the cavity inflated at this volume for t≥0. Replacing the dislocation potential, Z, with the more useful parameter of volume change, ΔV, the expression for the spherically symmetrical induced pore pressure field is represented as
|
|
(7) |
A volumetric dislocation, moving at constant velocity, U, in the direction of the negative x-axis, may be used to represent the insertion of a pencil-like form within a porous medium. The superposition of a colinear sequence of point volumetric dislocations may be assembled where a moving coordinate system is chosen to migrate with the advancing tip of the dislocation front. Considering the ith point dilation volume, ΔVi, generated in the specific time duration, Δτi, at time, τi, the incremental dilation volume, ΔVi, can be replaced by the product of the velocity of the moving dislocation, U, and cross-section area, a, as
|
|
(8) |
The location of the ith point dilation is determined as (U (t-τi), 0, 0) where it is assumed that n volumetric increments of the point dilation are generated both continuously and sequentially, and are inflated at times τi (i=1, 2, ……, n), respectively. As a result, the position of any dislocation volume increment, ΔVj, located relative to the primary dislocation volume increment at coordinates (x*, 0, 0) at current time t=τn would have been (x*-U (t-τj), 0, 0) at time t=τj. Substituting equation (8) into equation (7) yields pore pressure change due to volume change, ΔVi, at time τi as
|
|
(9) |
where
|
|
(10) |
and
|
|
(11) |
Since time, τ, varies from the time of initiation (τ=0) and the current time (τ=t), the total contribution to the pore pressure change at the monitoring point (x, y, z), resulting from volume changes of ΔVi, returns the integration
|
|
(12) |
where Δτi= (t-0)/n and
|
|
(13) |
and
|
|
(14) |
Noting RR2=x2+y2+z2 and substituting the dummy variable
|
|
(15) |
into equation (12) yields following some rearrangements
|
|
(16) |
representing the transient state. The integral must be evaluated numerically, except for the steady condition where t→ ∞ and the pressure distribution consequently reduces to
|
|
(17) |
Equations (16) and (17) may be rearranged, with respect to a minimum set of dimensionless parameters as
|
|
(18) |
and
|
|
(19) |
where the dimensionless parameters are defined as
|
|
(20) |
|
|
(21) |
|
|
(22) |
|
|
(23) |
|
|
(24) |
Physically, pD is dimensionless pore fluid pressure at the monitoring location; UD is dimensionless dislocation propagation velocity; tD is dimensionless time, and (xD, yD, zD) are dimensionless coordinates. The characteristic length chosen in these expressions is the minimum distance, l, measured from the measuring location to the path of the dislocation.
Both equations (18) and (19) may be appropriately transformed to represent the transient response at a static measuring location of (x, y, z) with reference to time, t*, defined as the time at which the path of the dislocation is the closest to the pore pressure measuring location. Then, the dimensionless coordinate, xD, is represented as
|
|
(25) |
where
|
|
(26) |
enabling dimensionless radius, RD, to be defined as
|
|
(27) |
where 
An approximate solution may be developed for a moving dislocation within a semi-infinite medium by applying an image dislocation to represent either the constant pressure, or zero flux boundary. The image dislocations may be viewed as fluid sources or sinks depending on the required boundary condition. A constant pressure boundary is applied through the use of an image dislocation opposing the real dislocation, and a zero flux condition is represented through the application of an equal dislocation. The path of the image dislocation remains symmetrical to the real dislocation when viewed relative to the surface. In the following, pore pressure response due to the real dislocation is denoted as (p-ps) R, and that due to the image is denoted as (p-ps) I. The solution for the linear system may be represented as
|
|
(28) |
As illustrated in Fig. 1, the real moving point dislocation is initiated at depth, H, below the surface. It propagates along the x-axis inclined θ degrees to the surface. Since the moving coordinate system is attached to the dislocation front, any point, R, on the inflated path of the real dislocation can be located as xR=U (t-τ), yR=0 and zR=0. Also, the coordinates of the reflected point "I" on the path of a moving point dislocation image may be determined as xI=u (t-τ)-2 (H-Uτsinθ) sinθ, yI=2 (H-Uτsinθ) cosθ and zI=0.
Considering induced pressures to be monitored at point, M, located at coordinates (x, y, z) in the moving coordinate system, the distances between points "R" and "M" and points "I" and "M" may be determined, respectively, as
|
|
(29) |
where RI2=RR2+q2, q2=4xsinθ (H-Utsinθ) +4 (H-Utsinθ)2-4ycosθ (H-Utsinθ) and b=4Usinθ (H-Utsinθ) +4xUsin2θ-4yUsinθcosθ-2xU.
The pressure response from the real point dislocation moving in an infinite space can be evaluated from equation (16) for transient state, or (17) for steady conditions, through the addition of appropriate subscripts. Similarly the image dislocation with negative volume change generates pore pressures at the monitoring point for a constant pressure boundary as
|
|
(30) |
The integral for the steady condition becomes
|
|
(31) |
All previously defined dimensionless parameters remain unchanged, as equations (20)-(24), excepting dimensionless distance from the monitoring point to any point on the path of the image
|
|
(32) |
|
|
(33) |
|
|
(34) |
|
|
(35) |
Rearranging equation (31), and substituting equations (20)-(24) and (32)-(35) enable the pore pressure behavior to be defined as
|
|
(36) |
Summing pore pressure responses from both image and real point moving dislocations, based on equation (28), yields
|
|
(37) |
for steady behavior where the definition of pRD is the same as pID except for the subscript. Pore pressures induced by the moving dislocation, and monitored remotely from the dislocation, may be used to define the form and rate of propagation.
A moving dislocation, such as magma intrusion, ascends at an inclination angle, θ≠0, and terminates on the surface at time t. It is clear that q=0 since H=Utsinθ, as shown in Fig. 1, and b becomes -2U (x-2hsinθ) in which h=xsinθ-ycosθ is the depth of a monitoring point below the free surface. The pore pressure response from equation (37) may be rearranged as
|
|
(38) |
where hD=h/l. The dimensionless pore pressure in equation (38) is a product of the dimensionless pore pressure from equation (19) and a coefficient, (1-e-2UDhDsinθ). Because UD is constant for a moving dislocation, the coefficient (1-e-2UDhDsinθ) in equation (38) does not influence the curvature of the pore pressure profile. Only equation (19) may be used to construct a series of type curves that represent the anticipated range of behavior. The matched results can be modified with the coefficient for the case of semi-infinite space. The minimum suite of dependent variables that control the pressure response areUD and tD-tD*, as illustrated in Fig. 2. In practice, the time, tp, at peak pressure is easily recorded. Dimensionless pore pressure from equations (19) and (38) may be appropriately transformed to represent the pore pressure response at a static measuring location of (x, y, z) with reference to time, tD-tDp. By definition, tD-tDp = (tD-tD*)-(tDp-tD*), where tDp-tD* can be solved from (∂pD/∂tD) =0. The dimensionless pressure in equation (19) is illustrated with reference to tD-tDp in Fig. 3.
The validity of the proposed moving dislocation models may be illustrated through direct comparison with pore fluid pressure responses recorded at remote monitoring points. Two intrusive events at Krafla, Iceland, with a constant pressure boundary, are used to demonstrate both the fidelity of the models, and their link to a real world situation.
Pore pressure data for two separate dike intrusion events at Krafla, Iceland are reported by Sigurdsson (1982), as illustrated in Figs. 4 and 5. The pressure pulses are recorded in a single open observation well, KG-5, completed to 1 300 m and cased to a depth of 600 m. The two magmatic events occurred on September 8, 1977 and July 10, 1978, and are identified by documented peak pressure increases (p-ps) in well KG-5 of 681 kPa and 358 kPa, respectively. The separation between the recording well and the surface phenomena associated with the two events was 4 300 m for the September event and about 9 300 m for the July event. Dimensions of the magma intrusion at the surface were 700 m long and about 70 cm width for the event of September 8, 1977, and several kilometers long and one meter width for the event of July 10, 1978. An eruption occurred in the September event (saemundsson, 1991). A block diagram aligned along the Krafla fissure swarm showing the magma intrusion system is illustrated in Fig. 6. Well KG-5 terminates in the upper single phase reservoir while the intrusive events, initiating from a magma chamber below 3 km depth (Ewart et al., 1990), also affects the lower two-phase reservoir. Pressure transients have not been detected in wells connected only with the lower zone.
The following field data were selected for the dislocation model based on field tests: transmissivity of 6.6×10-9 to 13.2×10-9 m3/(Pa·s) representing the "wellfield average" and "nearest well" analyses (Bodvarsson et al., 1984) and storativity of 3.7×10-11 m/Pa from the reservoir tests of Sigurdsson (1982). From these data the hydraulic diffusivity (c=transmissivity (κT/μ)/storativity (κT/(μc))) was calculated as 178 to 357 m2/s and the permeability, κ/μ, as 6.6×10-10 to 13.2×10-10 m2/(Pa·s) with T=10 m. The depth, h, of the monitoring point was documented as 900 m and a reasonable inclination angle for ascent path was prescribed as θ=80°.
Automatic curve matching using the least-square method was implemented using python language in this case study. Final matching is shown in Figs. 7 and 8 for the September event and Figs. 9 and 10 for the July event. For the September event, the approaching and departing pressure data match the curve of UD=10 on Fig. 7 and Fig. 8 respectively. A matching point was arbitrarily selected with the following values pD=0.606, p-ps=721.771 kPa, tD-tDp =0.435 and t-tp=11 880 s on the approaching part in Fig. 7. By definition, 
For the event of July 10, 1978, the approaching and departing pressure data match the curve of UD=4.01 on Fig. 9 and Fig. 10 respectively. Appropriate magnitudes from an arbitrarily selected match point on the approaching prepeak data are pD=0.206, p-ps=110.352 kPa, tD-tDp =0.89 and t-tp=72 000 s with UD=4.01 in Fig. 9. The factor (1-e-2UDhDsinθ) is evaluated as 0.48-0.60. The estimated cross-section area of the dislocation is 437-695 m2 for a dike a kilometer long and 1 m in width. The propagation velocity is evaluated as 0.18-0.26 m/s. Although independent estimates of the advance rate arenot available, the estimated velocity appears plausible compared to the results from the event of July 10, 1978. A minimum distance of l=7 589-10 748 m corresponds well with the reported separation between borehole KG-5 and the center of seismic activity at 9 300 m.
The theory of a moving dislocation has been developed to represent the process of intrusion into a saturated porous elastic solid of semi-infinite extent. This process is assumed to be displacement controlled and analogous to the insertion of a continuous intrusive dislocation at a constant rate. Attention is restricted to the analysis of the behavior of induced pore fluid pressure around a dislocation monitored at an observation point. Therefore, the proposed mod-els can be represented in the form of type curves for use in data reduction. Type curve matching may be used to decipher pressure response in characterizing the form of the moving dislocation.
The technique proposed here can more readily deal with complex problems including impermeable or constant pressure boundaries. The fidelity of the fit between the measured and the predicted responses is clearly encouraging. The two case studies show that a point dislocation model gives more satisfactorily predicted parameters.
| Bodvarsson, G. S., Benson, S. M., Sigurdsson, O., et al., 1984. The Krafla Geothermal Field, Iceland, 1, Analysis of Well Test Data. Water Resour. Res., 20(11): 1515-1530 doi: 10.1029/WR020i011p01515 |
| Brandsdottir, B., Einarsson, P., 1979. Seismic Activity Associated with the September 1977 Deflation of the Krafla Central Volcano in North-Eastern Iceland. J. Volcanol. and Geotherm. Res., 6: 197-212 doi: 10.1016/0377-0273(79)90001-5 |
| Cleary, M. P., 1977. Fundamental Solutions for a Fluid-Saturated Porous Solid. Int. J. Solids Struct., 13: 785-806 doi: 10.1016/0020-7683(77)90065-8 |
| Elsworth, D., 1992. Pore-Pressure Response due to Penetration through Layered Media. Int. Journal for Num. & Anal. Methods in Geomech., 16: 45-64 |
| Elsworth, D., Day, S. J., 1999. Flank Collapse Triggered by Intrusion: The Canarian and Cape Verde Archipelagoes. J. Volcanol. & Geotherm. Res., 95(1-4): 323-340 |
| Elsworth, D., Voight, B., 1995. Dike Intrusion as a Trigger for Large Earthquakes and the Failure of Volcano Flanks. J. Geophys. Res., 100(B4): 6005-6024 doi: 10.1029/94JB02884 |
| Ewart, J. A., Voight, B., Bjomsson, A., 1990. Dynamics of Krafla Caldera, North Iceland: 1975-1985. In: Ryan, M. P., ed., Magma Transport and Storage. John Wiley, New York. 225-276 |
| Ingraffea, A. R., 1987. Theory of Crack Initiation and Propagation in Rock. In: Atkinson, B. K., ed., Fracture Mechanics of Rock. Academic Press, San Diego. 71-110 |
| Lister, J. R., 1990. Buoyancy Driven Fluid Fracture: The Effects of Material Toughness and of Low Viscosity Precursors. J. Fluid Mechs., 210: 263-280 doi: 10.1017/S0022112090001288 |
| Lister, J. R., Kerr, R. C., 1991. Fluid-Mechanical Models of Crack Propagation and Their Application to Magma-Transport in Dykes. J. Geophys. Res., 96: 10049-10077 doi: 10.1029/91JB00600 |
| Muller, J. R., Ito, G., Martel, S. J., 2001. Effects of Volcano Loading on Dike Propagation in an Elastic Half-Space. J. Geophys. Res., 106(B6): 11101-11113 doi: 10.1029/2000JB900461 |
| Parsons, T., Sleep, N. H., Thompson, G. A., 1992. Host Rock Rheology Controls on the Emplacement of a Tabular Intrusion: Implications for Underplating of Extending Crust. Tectonics, 11: 1348-1356 doi: 10.1029/92TC01105 |
| Rice, J. R., Cleary, M. P., 1976. Some Basic Stress Diffusion Solutions for Fluid-Saturated Elastic Porous Media with Compressible Constituents. Rev. Geophys., 14(2): 227-241 doi: 10.1029/RG014i002p00227 |
| Rubin, A. M., 1993. Dikes vs. Diapirs in Viscoelastic Rock. Earth Planet. Sci. Lett., 119: 641-659 doi: 10.1016/0012-821X(93)90069-L |
| Ryan, M. P., 1987. Neutral Buoyancy and the Mechanical Evolution of Magmatic Systems. In: Mysen, B. O., ed., Magmatic Processes: Physiochemical Principles. Geochem. Soc. Spec. Publ., 1: 259-287 |
| Saemundsson, K., 1991. Jardfraedi Kroflukerfisins in Nattura Myvatns. Hid Islenska Natturufaedifelag, Reykjavik. 24-95 |
| Sigurdsson, O., 1982. Analysis of Pressure Pulses Resulting from Volcanic Activity in the Vicinity of a Well: [Dissertation]. University of Oklahoma, Noman, Oklahoma, USA. 75 |
| Skempton, A. W., 1954. The Pore-Pressure Coefficients A and B. Geotechnique, 4: 143-147 doi: 10.1680/geot.1954.4.4.143 |
| Watanabe, T., Koyaguchi, T., Seno, T., 1999. Tectonic Stress Controls on Ascent and Emplacement of Magmas. J. Volc. and Geophys. Res., 91: 65-78 |
Figures(10)
Copyright © 2013-2020 Journal of Earth Science 鄂ICP备15021562号-2
Tel: +86-27-67885075 Fax: +86-27-67885075 E-mail: xbb@cug.edu.cn
Address: Editorial Office of Journal, China University of Geosciences, Yujiashan, Wuhan, Hubei 430074, P. R. China
Supported by:
Beijing Renhe Information Technology Co. Ltd
E-mail:
info@rhhz.net
DownLoad:
DownLoad:
