Terzaghi Compaction
Introduction
Fluids that move through pore spaces in an aquifer or reservoir can shield the porous medium from stress because they bear part of the load from, for instance, overlying rocks, sediments, fluids, and buildings. Withdrawing fluids from the pore space
increases the stress the solids bear, sometimes to the degree that the reservoir
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 uses the results in postprocessing to calculate vertical compaction following Terzaghi theory (Ref. 2).
Model Definition
This example analyzes fluid and solid behavior within three sedimentary layers
overlying impermeable bedrock in a basin. The bedrock is faulted, which creates a step near a mountain front. The sediment 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
Bedrock step
EXPLANATION
Boundary segment
identifier
Vertical exaggeration x 5
Figure 1: Model geometry showing boundary segments (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 reservoir above it. The flow field is initially at steady state. The period of interest is 10 years.
This example sets 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  represents hydraulic head (m). In most conventional flow models, S h  represents small changes in fluid volume and pore space in that it combines terms describing the fluid’s compressibility, the solids’ compressibility, and the reservoir’s porosity. In the original research (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.
Because the aquifer is at equilibrium prior to pumping, you set up this model to predict the change in hydraulic head rather than the hydraulic head values themselves. 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 decreases 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 (see Figure 1).
Terzaghi theory uses skeletal specific storage or aquifer compressibility to calculate the vertical compaction Δb  (m) of the aquifer sediments in a given representative volume as
where  b  is standard notation for the vertical thickness of aquifer sediments (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 represents 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
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
=Ωbase ∂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 –()
=
sediments 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
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, choose New.
N E W
1In the New window, click Model Wizard.
M O D E L W I Z A R D
1In the Model Wizard window, click 2D.
2In the Select physics tree, select Fluid Flow>Porous Media and Subsurface Flow>Darcy's Law (dl).
3Click Add.
4Click Study.
5In the Select study tree, select Preset Studies>Time Dependent.
6Click Done.
G E O M E T R Y1
Rectangle 1 (r1)
1On the Geometry toolbar, click Primitives and choose Rectangle.
2In the Settings window for Rectangle, locate the Size and Shape section.
3In the Width text field, type 5200.
4In the Height text field, type 440.
5Locate the Position section. In the y text field, type -440.
Rectangle 2 (r2)
1Right-click Rectangle 1 (r1) and choose Build Selected.
2On the Geometry toolbar, click Primitives and choose Rectangle.
centering
3In the Settings window for Rectangle, locate the Size and Shape section.
4In the Width text field, type 1200.
5In the Height text field, type 320.
6Locate the Position section. In the x text field, type 4000.
7In the y text field, type -440.
8Right-click Rectangle 2 (r2) and choose Build Selected.
9Click the Zoom Extents button on the Graphics toolbar.

版权声明：本站内容均来自互联网，仅供演示用，请勿用于商业和其他非法用途。如果侵犯了您的权益请与我们联系QQ:729038198，我们将在24小时内删除。