Terzaghi Compaction
Introduction
Fluids that move through pore spaces in an aquifer or rervoir can shield the porous medium from stress becau they bear part of the load from, for instance, overlying rocks, diments, fluids, and buildings. Withdrawing fluids from the pore space
increas the stress the solids bear, sometimes to the degree that the rervoir
measurably compacts. The reduction in the pore space loops back and alters the fluid pressures. The feedback brings about more fluid movement, and the cycle continues.
Terzaghi Compaction describes a conventional flow model and us the results in postprocessing to calculate vertical compaction following Terzaghi theory (Ref. 2).
Model Definition
This example analyzes fluid and solid behavior within three dimentary layers
overlying impermeable bedrock in a basin. The bedrock is faulted, which creates a step near a mountain front. The diment stack totals 420 m at the centerline of the basin (x = 0 m) and thins to 120 m above the step (x > 4000 m). The top two layers are each
20 m thick.
Upper aquifer - constant head
Compressible confining unit
绝对禁忌Bedrock step
EXPLANATION
Boundary gment
identifier
Vertical exaggeration x 5
Figure 1: Model geometry showing boundary gments (from Leake and Hsieh, Ref. 1).
Pumping from the lower aquifer reduces hydraulic head down the centerline of the basin by 6 m per year. The head drop moves fluid away from the step. The middle layer is relatively impermeable. The pumping does not diminish the supply of fluids in the unpumped rervoir above it. The flow field is initially at steady state. The period of interest is 10 years.
This example ts up a traditional flow model and analyzes the vertical displacement during postprocessing. The flow field is fully described using the Darcy velocity in an equation of continuity
(1)
where S h is the storage coefficient (m −1), K equals hydraulic conductivity (m/s), and H reprents hydraulic head (m). In most conventional flow models, S h reprents small changes in fluid volume and pore space in that it combines terms describing the fluid’s compressibility, the solids’ compressibility, and the rervoir’s porosity. In the original rearch (Ref. 1) and in this model, S h is the specific storage of the solid skeleton, S sk .
Instead of solving Darcy’s law in the hydraulic head formulation, we solve Equation 1 in the pressure formulation
here, the storage coefficient S (1/Pa) is related to the fluid density, acceleration of gravity and the storage coefficient given in Equation 1 by the relation S = S h /ρg . Also, the hydraulic head is related to the fluid pressure and elevation H = p /ρg + D , and the hydraulic conductivity is related to the permeability and dynamic viscosity of the fluid K = κρg /μ.
Becau the aquifer is at equilibrium prior to pumping, you t up this model to predict the change in hydraulic head rather than the hydraulic head values themlves. The main advantage of this approach lies in establishing initial and boundary conditions. Here you specify that the hydraulic head along the centerline of the basin decreas linearly by 60 m over ten years, then simply state that the hydraulic head initially is zero and remains so where heads do not change in time.
The boundary and initial conditions are
S h H ∂t ∂-------∇+K ∇H –()⋅0=ρS p ∂t ∂-----∇+ρκμ
--p ∇ρg D ∇+()–⋅0=
where n is the normal to the boundary. The letters A through E , taken from Leake and Hsieh (Ref. 1), denote the boundary (e Figure 1).
Terzaghi theory us skeletal specific storage or aquifer compressibility to calculate the vertical compaction Δb (m) of the aquifer diments in a given reprentative volume as
where b is standard notation for the vertical thickness of aquifer diments (m).Model Data The following table gives the data for the Terzaghi compaction model:
Results and Discussion
Figure 2 shows a Year-10 snapshot from the COMSOL Multiphysics solution to the Terzaghi compaction example. The results describe conventional Darcy flow toward the centering of a basin, moving away from a bedrock step (x > 4000 m). The shading reprents the change in hydraulic head brought on by pumping at x = 0 m. The streamlines and arrows denote the direction and magnitude of the fluid velocity. The flow goes from vertical near the surface to horizontal at the outlet. Where the
TABLE 1: MODEL DATA VARIABLE DESCRIPTION VALUE
ρf
橙色玫瑰花语
Fluid density 1000 kg/m 3 S sk
Skeletal specific storage, aquifer layers 1·10-5 m -1Skeletal specific storage, confining layer 1·10-4 m -1 K s
Hydraulic conductivity, aquifer layers 25 m/d Hydraulic conductivity, confining layer 0.01 m/d H (0)
Initial hydraulic head 0 m H 0(t )Declining head boundary (6 m/year)·t n K H ∇⋅0
=Ωba ∂A n K H ∇⋅0
=Ωother ∂B H 0
=Ωupper edge ∂C H 0
=Ωsurface ∂D H H 0t ()
=Ωoutlet ∂E
H 0()0=ΩΔb S sk b H –()
=
霎时的反义词
diments thicken at the edge of the step, the hydraulic gradient and the fluid velocities change abruptly.
Figure 2: COMSOL Multiphysics solution to a Terzaghi flow problem. The figure shows
change in hydraulic head (surface plot) and fluid velocity (streamlines). References
mw是什么单位1. S.A. Leake and P.A. Hsieh, Simulation of Deformation of Sediments from Decline of Ground-Water Levels in an Aquifer Underlain by a Bedrock Step, U.S. Geological Survey Open File Report, 97-47, 1997.
2. K. Terzaghi, Theoretical Soil Mechanics, John Wiley & Sons, p. 510, 194
3.
Application Library path: Subsurface_Flow_Module/
Flow_and_Solid_Deformation/terzaghi_compaction
Modeling Instructions
From the File menu, choo New.
N E W
1In the New window, click Model Wizard.
M O D E L W I Z A R D
广东肠粉的正宗做法1In the Model Wizard window, click 2D.
2In the Select physics tree, lect Fluid Flow>Porous Media and Subsurface Flow>Darcy's Law (dl).
3Click Add.
4Click Study.
5In the Select study tree, lect Pret Studies>Time Dependent.
6Click Done.
G E O M E T R Y1
Rectangle 1 (r1)
1On the Geometry toolbar, click Primitives and choo Rectangle.
2In the Settings window for Rectangle, locate the Size and Shape ction.
3In the Width text field, type 5200.
联想超极本4In the Height text field, type 440.
5Locate the Position ction. In the y text field, type -440.
耳塞式耳机Rectangle 2 (r2)
陈欣和
1Right-click Rectangle 1 (r1) and choo Build Selected.
2On the Geometry toolbar, click Primitives and choo Rectangle.
3In the Settings window for Rectangle, locate the Size and Shape ction.
4In the Width text field, type 1200.
5In the Height text field, type 320.
6Locate the Position ction. In the x text field, type 4000.
7In the y text field, type -440.
8Right-click Rectangle 2 (r2) and choo Build Selected.
9Click the Zoom Extents button on the Graphics toolbar.
