A case study of mud weight design with finite element method for subsalt wells
6.3 GLOBAL MODEL DESCRIPTION AND NUMERICAL RESULTS .1 Model description
Figure 6.1 shows a sectional view of the field to be investigated. The pay zone is at the salt base formation. The trajectory of the wellbore penetrates the salt body and enters the pay zone at a high angle.
Figure 6.2 shows the field scale model as a block with a height of 9000 m, width of 8000 m, and length of 8,000 m. The boundary conditions of the global model include the following:
all four lateral displacement constraints are applied, along with zero displacement constraints to its bottom. The top surface is free. The gravity load balanced with the initial geostress field is applied to every element of the model.
Figure 6.3 shows the salt body in red. The formation thickness below the salt to the base of the model ranges from 2200 to 2850 m. To simplify the example without losing accuracy, the
Figure 6.2. Global model at the field scale: geometry and boundary conditions.
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Figure 6.3. Relative position of salt body to the formations in the model.
Figure 6.4. Geometry of the salt body in the model.
Well trajectory North
West
Salt exit, Position of first submodel
Figure 6.5. Wellbore trajectory and the direction of salt’s central axis along with salt exit of the wellbore.
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formation details below the TVD interval of the reservoir are neglected. The salt geometry is a key factor that controls the stress patterns in its area. Figure 6.4 shows the salt body. Its thickness along wellbore trajectory is 5300 m. The width is 6 km, and its axis is in the direc-tion of N30°W, as shown in Figure 6.5 and Figure 6.1.
The salt geometry adopted in the model is only a part of the actual salt body, which is far larger than that of the model. The reason for selecting only a portion of the salt body is that the other portion does not influence the stress pattern within the formation investigated.
Consequently, it is necessary to have the current simplified model to reduce the computa-tional burden. The salt geometry of the model was also selected with reference to the geom-etry of the trajectory: the entire area penetrated by the trajectory has been included in the model. This is necessary and essential for calculating the submodeling; the submodels require the stress and displacement solutions from the global model as their boundary conditions.
The details of the trajectory geometry are omitted from the global model for simplification without accuracy loss of the field scale modeling.
It is important to include the geostructure characteristics, such as syncline or anticline, in the model; these geostructural traits determine the stress pattern at the salt base formation.
If all other conditions are fixed, the variation of the geostructure from syncline to anticline will cause obvious different stress pattern distributions at the salt base formation.
As shown in Figure 6.6, the model consists of four kinds of materials: top layer, surround-ing rock, salt, and base formation. The reservoir is a part of the base formation. Table 6.1 lists the values of the material parameters.
Pore pressure data was generated with the user subroutine. Figure 6.7 shows the variation of pore pressure with TVD.
Top layer
Surrounding rock
Salt
Base formation
Figure 6.6. Various materials used in the model.
Table 6.1. Values of material parameters.
Materials Density kg/m3
Young’s
modulus Pa Poisson’s ratio CS/Pa
Friction angle
Top layer 1,900 1 × 1010 0.3 elastic
Salt 2,250 1.3 × 1010 0.22 4 × 106 20°
Surrounding rock 2,350 Depth dependent Depth dependent 1 × 106 25° Base formation 2,350 Depth dependent Depth dependent 4 × 106 25°
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The code of the subroutines used in the calculation is listed below:
SUBROUTINE UVARM(UVAR,DIRECT,T,TIME,DTIME,CMNAME,ORNAME, 1 NUVARM,NOEL,NPT,LAYER,KSPT,KSTEP,KINC,NDI,NSHR,COORD, 2 JMAC,JMATYP,MATLAYO,LACCFLA)
INCLUDE ’ ABA_PARAM.INC’
C FUNCTION OF THIS SUBROUTINE IS TO CALCULATE THE TOTAL STRESS VALUE
CHARACTER*80 CMNAME,ORNAME CHARACTER*3 FLGRAY(15)
DIMENSION UVAR(NUVARM),DIRECT(3,3),T(3,3),TIME(2)
DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),COORD(*) C THE DIMENSIONS OF THE VARIABLES FLGRAY, ARRAY AND JARRAY C MUST BE SET EQUAL TO OR GREATER THAN 15.
C FUNCTION OF THIS SUBROUTINE IS TO INTRODUCE PRESSURE/DEPTH DEPENDENT PROPERTY OF E
15000 16000 17000 18000 19000 20000 21000
Pressure gradient /Pa/m
TVD depth /m
Figure 6.7. Distribution of pore pressure with TVD depth.
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DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),COORD(*)
C FUNCTION: TO ASSIGN INITAL PORE PRESSURE IN FORMATION.
C
* U(1) = 147.22E6+(1407-ZC)/(1407–889)*(158.5E6–147.22E6) IF(ZC.LE.889.0) U(1) = 158.5E6
RETURN END
The depth unit used in the user subroutines DISP and UPOREP is m. Its coordinate is the global Cartesian coordinate, which has the z-axis upward, rather than the downward coor-dinate usually used in the petroleum industry. This is used to match the convention used in Abaqus, which is a version of solid mechanics.
c
C FUNCTION OF THIS SUBROUTINE IS TO INTRODUCE BOUNDARY CONDITION OF PORE PRESSURE
* UW0 = 147.22E6+(1407-ZC)/(1407–889)*(158.5E6–147.22E6) IF(ZC.LE.889.0) UW0 = 158.5E6
RETURN END
The depth-dependent Young’s modulus and Poisson’s ratio are actually addressed in the model as mean-stress-dependent. The following data lines used in the model provide the model details. Accordingly, the following sentences must be used to introduce the user sub-routines into the calculation.
*Drucker Prager Creep, law=TIME 2.5e-22, 2.942, -0.2
*Drucker Prager Hardening, type=SHEAR 1.56e+06,0.
*Drucker Prager Creep, law=TIME 2.5e-22, 2.942, -0.2
*Drucker Prager Hardening, type=SHEAR 4e+06,0.
*Initial Conditions, type=Stress, GEOSTATIC Rock-2.top-layer,-1.18e7,8777.,-1.6396e7,8135.,0.8,0.8
****Initial Conditions, type=Pore pressure
*Initial Conditions, type=Pore pressure, user
**
_PickedSet19, GRAV, 9.8, 0., 0., -1.
** Name: TopPressure Type: Pressure
*Dsload
Surf-1, HP, 1.2e+07, 9777., 8557.
**
** OUTPUT REQUESTS
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**
*Step, name=Step-2, nlgeom=YES, amplitude=RAMP
*Soils, utol=2e+01
FV, MFR, SDV, STATUS, STATUSXFEM, UVARM
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
For the salt formation, this analysis uses the modified Drucker-Prager yielding criterion.
Cohesive strength and frictional angle of the Drucker-Prager model are provided by the fol-lowing values: d = 4 MPa, β = 30°.
The creep law, given in the following equation (Dassault Systems 2008), is adopted by:
εcr A
(( )
σσcrcr ntm (6.4) where εcr represents the equivalent creep strain rate; σcr represents the von Mises equivalent stress; t is total time variable; and A, n, and m are three model parameters, which are given the following values:A = 2.5 × 10−22, n = 2.942, m = −0.2
6.3.2 Numerical results of the global model
Figure 6.8 through Figure 6.10 show the numerical results obtained with the global model.
Figure 6.8 shows the distribution of stress σx. Figure 6.9 and Figure 6.10 provide the distribution of stress σy and σz. The plane in the sectional view is chosen at the place where
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wellbore trajectory is included. The stress shown here is the effective stress in which the amount of pore pressure is not included. The sign convention of solid mechanics is followed, which is positive for tensile stress and negative for compression.
6.4 SUBMODEL DESCRIPTION AND NUMERICAL RESULTS