** Numerical predictions on critical pressure drawdown and sand production for wells in weak formations**

**9.4 CASE 2: NUMERICAL PREDICTION OF CVPDD FOR WELL WITH CASING COMPLETION**

The procedure for predicting the CVPDD of a cased well is similar to that used for an open-hole completion, but with an additional step performed to simulate the perforation tunnel, as follows:

• Step 1: Establish the initial geostress and boundary conditions and apply gravity load;

make a 3D FEM calculation with porous flow coupled with deformation.

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• Step 2: Remove the borehole elements (simulating drilling) and apply the mud weight pressure on the borehole surface.

• Step 3: Remove the tunnel elements (simulating perforating) and apply the initial pore pressure and surface pressure/mud weight pressure on the perforation tunnel surface.

• Step 4: Reduce/replace the mud weight pressure with the BHP, which equals the reservoir pressure.

• Step 5: With reference to the safe pressure drawdown, further reduce the pressure on the borehole surface as well as the pore pressure at the boundary of the borehole surface.

• Step 6: Check the area for plastic deformation and the value of plastic strain, and make a recommendation for the CVPDD design.

An additional variable for the analysis of CVPDD with a casing completion is the number of shots per ft alongside the pressure drawdown value; the sanding potential depends on both the number of shots per ft and the pressure drawdown value. Casing failure is another factor to be considered in some cases as an unfavorable result of pressure drawdown; however, it is not considered here for brevity.

Generally, the perforation will create plastic deformation around the perforation tunnel. The total plastic strain is the sum of plastic strain caused by the perforation and the plastic strain caused by pressure drawdown. Sanding will have less influence on the wellbore stability for a cased well section than for an openhole well section. Thus, casing completion allows a larger value of plastic strain than that allowed for a well with openhole completion.

*9.4.1 Modeling casing*

Simplifications made in the modeling of casing is an important aspect in this calculation. The function of casing in completion is to enhance the stability of the wellbore. A concrete ring between the casing and formation seals the oil/gas within the formation. In the production process, both the concrete ring and the casing are nonpermeable and have no influence on the porous flow occurring within the formation. Therefore, to reduce the computational burden, the details of the casing and concrete ring can be neglected. However, their nonpermeable property and boundary effect is essential in the model.

The membrane elements used here are to simulate the casing function. Its stiffness is rather low, and the thickness is only 1 mm. Zero radial displacement constraints are applied to all nodes of the membrane elements. Low stiffness values will reduce the shear stress in the formation connected to the membrane. In this way, the major mechanical properties of the physical phenomena have been ensured with the least computational burden.

The model of the membrane is built to the model by using ‘offset mesh’ technique of Abaqus CAE. Key sentences in the data of the model are listed below. Only one line of ele-ment nodal information is provided in this list. Metric units have been adopted in the expres-sion of mechanical variables and parameters.

*Element, type=M3D8R

5585, 14, 270, 2032, 313, 8126, 8203, 8204, 8205

… …

*Membrane Section, elset=OffsetElements-1, material=LOWSTIFF, controls=EC-1 0.001,

*9.4.2 Case 2A: Casing with perforation of 8 shots per 0.3048 m*
*9.4.2.1 Description of the model: Case 2A*

In this example, a well section with a perforation density of 8 shots per 0.3048 m (per ft) is chosen as Case 2A for the CVPDD calculation.

A model of reservoir formation with the following values of geometric parameters is adopted.

The thickness of the model is set at 0.1524 m (0.5 ft), with four shots in the model (i.e., 8 shots per 0.3048 m (or say per ft)). The diameter of the model is 7 m, and the diam-eter of borehole is 8.5 in. (0.2159 m). The diamdiam-eter of the perforation tunnel is set at 1 in.

(0.0254 m).

The displacement constraints on all surfaces except the inner borehole surface and perfo-ration tunnel surfaces are derived from the numerical results of the global model at the field scale, as shown in Fig. 9.1.

On the inner surface of the borehole, the membrane layer is used to simulate the imper-meable property of the casing. Perforation tunnels punch through the imperimper-meable membrane.

A set of pressure drawdown values will be applied on the surface of the perforation tunnels. This sets the pore pressure boundary condition at surfaces of the perforation tunnels.

Gravity load and initial stress are applied to the entire model, and pressures are applied on the surface of perforation tunnels corresponding to various values of pressure drawdown.

Figure 9.16. Geometry of the model: Case 2A (8 shots per 0.3048 m).

Figure 9.17. Boundary conditions and loads of the model: Case 2A (8 shots/0.3048 m).

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*9.4.2.2 Numerical results of Case 2A*

The following cases of pore pressure boundaries have been simulated numerically:

• Initial geostress and pore pressure field.

• Stress and pore pressure field after the wellbore is drilled.

• Status as mud weight was replaced by BHP, which is 38 MPa.

• Status as pressure drawdown was set at 3447380 Pa (500 psi).

• Status as pressure drawdown was set at 4136856 Pa (600 psi).

Fig. 9.18 and Fig. 9.19 show the numerical results for the distribution of equivalent plastic strain around the perforation tunnel.

*9.4.3 Case 2B: Casing with perforation of 4 shots per 0.348 m (per ft)*
*9.4.3.1 Geometry of the model: Case 2B*

The values of the parameters of the submodel for Case 2B are almost the same as those used in Case 2A, except that the number of shots per 0.3048 m (per ft) of the perforation tunnel is 4 rather than 8 as previously used (Fig. 9.20).

Figure 9.18. Distribution of equivalent plastic strain around perforation tunnel with drawdown at 3447380 Pa (500 psi) for Case 2A.

Figure 9.19. Distribution of equivalent plastic strain around perforation tunnel with drawdown at 4136856 Pa (600 psi) for Case 2A.

Figure 9.20. Model of Case 2B: with 2 shots in 0.1524 m (1/2 ft) of thickness.

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*9.4.3.2 Numerical results of Case 2B*

The following cases of pore pressure boundaries have been simulated numerically:

• Initial geostress and pore pressure field.

• Stress and pore pressure field after wellbore is drilled.

• Status as mud weight was replaced by BHP, which is 38 MPa.

• Status as pressure drawdown was set at 3447380 Pa (500 psi).

• Status as pressure drawdown was set at 4136856 Pa (600 psi).

Fig. 9.21 through Fig. 9.23 show the numerical results for the distribution of equivalent plastic strain around the perforation tunnel for the situations with 0 Pa, 3447380 Pa (500 psi), and 4136856 Pa (600 psi) pressure drawdown, respectively.

Fig. 9.21 shows that the perforation process could result in equivalent plastic strain of as much as 0.392 around the perforation tunnel. However, it exists only around the tunnel and does not propagate far away. Conversely, the perforation process is a squeezing process, which should cause stiffening of the sand around the tunnel hole. Therefore, the influence of this plastic strain on the stability of the tunnel hole is negligible.

Fig. 9.22 shows that the maximum plastic strain after a 3447380 Pa (500 psi) pressure drawdown is 0.4075. However, the increment resulting from this pressure drawdown is only:

0.4075–0.392 = 0.0149.

The amount, 0.392, is caused by the perforation process.

Fig. 9.23 shows that the maximum plastic strain after 4136856 Pa (600 psi) pressure draw-down is 0.4128. However, the increment resulting from this pressure drawdraw-down is only:

0.4128–0.392 = 0.0202.

The 100 psi increase in pressure drawdown results in a plastic strain increase of 0.0053 which is the difference between 0.0202 and 0.0149.

Figure 9.21. Distribution of equivalent plastic strain around perforation tunnel with drawdown at 0 Pa for Case 2B.

Figure 9.22. Distribution of equivalent plastic strain around perforation tunnel with drawdown at 3447380 Pa (500 psi) for Case 2B.

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After considering all of the previously mentioned factors, the 4136856 Pa (600 psi) pressure drawdown is concluded to be safe for oil production without obvious sand production.

*9.4.4 Remarks*

With the fully coupled, poro-elastoplastic finite element method, 3D numerical analyses were performed to predict the CVPDD for wells with openhole completion and wells with casing completion. A submodeling technique was adopted to address the discrepancy between the scale of the oil field and that of the wellbore section. A global model at the field scale was used to establish the geostress field of the wells. The pressure drawdown and poro-elastoplastic behavior of the formation near the wellbore was calculated with the submodel, which has a diameter of 7 m.

Based on the results provided by the 3D finite-element calculation, the critical pressure drawdown before the onset of sanding was predicted, and a CVPDD of 2757904 Pa (400 psi) was suggested for the well with openhole completion and 4136856 Pa (600 psi) for well with casing completion. Distributions of the equivalent plastic strain, along with a plastic region where plastic strain occurs, were illustrated.

The visualization of the 3D numerical results illustrates the values of plastic strain and shows the size of the plastic region under the given pressure drawdown, which are useful in selecting the CVPDD. The results presented here indicate that the 3D FEM is a highly effi-cient tool for predicting CVPDD.

9.5 NUMERICAL PREDICTION OF SANDING PRODUCTION