Reservoir Geomechanics with ABAQUS

Transcription

1 Reservoir Geomechanics with ABAQUS B. Bostrøm and E. Skomedal Statoil ASA, Norway Abstract: The coupled hydro-mechanical behavior of two North Sea high-pressure/hightemperature gas-condensate fields during depletion is studied using the soils consolidation procedure available in ABAQUS. Time dependent reservoir pressure fields obtained from threedimensional black oil reservoir models were transferred into finite element reservoir geomechanical models in ABAQUS as three-phase flow in deforming reservoirs cannot be performed within ABAQUS. Keywords: Petroleum Engineering, Reservoir, Coupled Analysis, Geomechanics, Failure. 1. Introduction With reservoir pressures of about 8 9 MPa and temperatures of degrees C the two gas condensate fields Kristin and Kvitebjørn are characterized as high-pressure/high-temperature fields (HP/HT). The fields are located offshore mid Norway. Production start is planned in The Kristin field comprises of two main reservoirs, Garn and Ile, separated by intra reservoir shale. Kvitebjørn on the other hand is highly faulted. The possible compartmentalization of the reservoir associated with the faults is a major uncertainty in developing such a field. Production from the reservoirs will be obtained by huge depletion as no gas or water injection is planned. Tremendous effective stress changes are therefore expected, which have an influence on the various aspects of field development and planning. This necessitates a reservoir geomechanical study. The focus of the models presented for the two HP/HT gas condensate fields are different. A fullfield model is presented for the Kristin field, while the results of a detailed analysis of an internal fault is given for the Kvitebjørn field. The full-field or global model addresses several issues important for production, like reservoir compaction, subsidence, well casing integrity and infill drilling options. This case also focuses on the coupling between the standard reservoir simulator ECLIPSE TM (trademark of Geoquest Schlumberger) and ABAQUS that is necessary due to the above-mentioned shortcoming in the coupled pore fluid flow and stress analysis procedure in ABAQUS. 24 ABAQUS Users Conference 117

2 The local analysis on the other hand is carried out to investigate possible fracturing (shear or tensile) of the formation through and around faults, crushing, reactivation of faults and changes in the sealing capacity of the interbedded shale layers during depletion, possible top seal leakage through faults and fractures. Finally this paper addresses the necessary enhancements of ABAQUS, so that it will be of even greater value for the reservoir modelers. 2. Full-field model of the Kristin field 2.1 Geometry and model A 2D plane strain model is sufficient due to the large extension of the reservoir normal to the plane of interest. The width of the reservoir at studied section is 4166 m, while the reservoir height is typically 16 m adding the thickness of the Garn and Ile reservoirs. Our geomechanical model includes in addition the over-, side- and under-burden. The total width of the model is 28 5 m and the total height is 1 m. The geometry of the reservoir layers is taken from a vertical cross section of the 3D ECLIPSE model. ABAQUS/CAE allow you to import parts from CAD system using industry-standard formats. Unfortunately, ECLIPSE does not support these formats, implying that ECLIPSE geometry printed in ASCII format must be transferred into a CAD format using a translator. Several minor faults have been mapped in the two main gas-condensate reservoirs. A limited number of drained triaxial tests on faulted reservoir material indicate that the strength and stiffness of these faults are higher than the surrounding material. Internal faults (barriers) are therefore modeled by a rough, small sliding contact formulation. In this way there are no relative motion between the surfaces, while pressure may still be discontinuous across the fault. Faults outside the reservoir are not included in the present model for the same reasons. The entire finite element mesh is shown in Figure 1.a. The bottom boundary, the left and right vertical boundaries are fixed. Displacement nodes at the top (seabed) boundary are free. The model consists of a total of nodes and 2639 elements. Reservoir mesh details are shown in Figure 1.b and c. Different analysis procedure and matrix strength criterion are used for the different parts of the model. The reservoir layers use the pore fluid diffusion/stress analysis procedure, while undrained (no pore fluid flow) behavior is assumed for the surrounding shale. The 6-node triangular displacement and pore pressure element CPE6MP is used for the reservoir layers, while the 6-node triangular displacement elements CPE6M is used for the surrounding shale. 2.2 Material The model is divided into 8 material layers: The hydrocarbon bearing sandstone formations Garn and Ile, the underlying water bearing sandstone Tofte, the intra reservoir shales Not and Ror and the shales in the overburden, sideburden and the underburden ABAQUS Users Conference

3 Reservoir sandstone stiffness and strength deduced from standard triaxial laboratory testing are made porosity dependent by a field dependent parameter, where the initial field parameter distribution is taken equal to the distribution of the initial porosity in the reservoir. Geomechanical input data for the shale material are taken from sonic logs and existing laboratory tests on shale samples from the exploration wells at the field. Further recall that a displacement finite element formulation (element CPE6M) is chosen for the shale material. This implies that undrained stiffness must be calculated. It may be shown that the undrained Poisson s ratio, ν u, and the undrained Young s modulus, E u, are related to the drained modulus through the following expressions ν u Eu E ( 1 2ν )( 1+ ν ) = 2 1 ( 2ν )( 1+ ν ) 3 1 = 2 1 ( 2ν ) + ( 2ν )( 1+ ν ) Eν + K a E + K E K a + a E K a Where of E is the drained Young s modulus and ν is the drained Poisson s ratio. The apparent compressibility of the pore water, K a, is given as K a= K n w where K w is the bulk modulus of the pore water and n is the porosity. The spatial varying stiffness is controlled by an artificial temperature distribution, i.e. temperature dependent stiffness. 2.3 Prescribed conditions The initial effective stresses are generated in the ABAQUS subroutine, SIGINI. The initial geostatic stress field must be in equilibrium with the applied loads and boundary conditions. Ideally, the loads and initial stresses should exactly equilibrate and produce zero deformations. This state is obtained performing an initial ABAQUS analysis fixing all displacements degree of freedoms. Calculated reaction forces written to the ABAQUS output file are then used to create nodal point forces, which are applied in the first step of the actual ABAQUS analysis. 24 ABAQUS Users Conference 119

4 The pore pressure depletion history within the reservoir is transferred from the ECLIPSE reservoir model to ABAQUS utilizing the user subroutine DISP. A file containing the pore pressure in each ECLIPSE block is read for each time step analyzed by ABAQUS. The initial porosity distribution is transferred from the ECLIPSE reservoir model to ABAQUS utilizing the user subroutine VOIDRI for reading initial porosity in ABAQUS. The initial void ratio e, defined as the ratio between the pore volume and the solid volume, is related to the porosity n through: e =n/(1-n). 2.4 Results and discussion Contour plots of applied pore pressure reduction, displacements, stress path parameter and xyplots of deformation along well paths are presented Pore pressure Contour plot of the applied pore pressure reduction for year 225 is shown in Figure 2. The maximum reduction is slightly more than 8 MPa in the western part of Garn at the end of production. The minimum reduction at the end of production is about 3 MPa in the eastern part of Ile. The maximum differential pressure across barriers is slightly smaller than 2 MPa. The maximum differential pressure occurs in Ile between year 29 and 214. The maximum differential pressure in Garn is about 16 MPa Displacement Horizontal and vertical displacement profiles at years 27, 29 and 225 are plotted in Figure 3. Three profiles are chosen and the placement of these is indicated in Figure 1.b. The predicted seabed subsidence is in general very modest. It is equal to.7 cm above the central part of the reservoir in the year 225 (Figure 3.b), and it increases slightly towards the reservoir flanks. Reservoir compaction on the other hand is largest in the western parts and decreases somewhat towards east. It is equal to 45 cm in the west, 38 cm in the central parts and 27 cm in east in the year 225. The overburden expansion is a Poisson s ratio effect, which is amplified due to the undrained behavior of the shale (ν u is larger than.47 in the overburden). The horizontal contraction of the reservoirs may be deduced from Figure 3.a and c. The resulting horizontal contraction is equal to 17 cm in the year Stress path Stress changes during depletion is a key issue for well planning after start up of production as the decline of the fracture pressure due to reservoir depletion is deduced form this. A common parameter to monitor the stress path during production is the stress path coefficient, defined as the ratio of the change in horizontal stress σ h to the change in pore pressure p, h γ = σ p ABAQUS Users Conference

5 The calculated spatial variation of the stress path coefficient in the year 225 is relative homogeneous. It is less or equal.65 in the two main reservoirs as shown in Figure 4. Similar numbers found using an established analytical model is.82 (uniaxial strain model), i.e. less favorable with regards to infill drilling. The advanced numerical model includes the effect of bounding shale (arching) and rock nonlinearities Well casing damage Planned well paths are projected into the studied cross section. Displacements in the horizontal and vertical direction along this well path are shown in Figure 5.a. The axial strain resulting from these displacements are shown in Figure 5.b. It is seen that the maximum axial strain along this path is about 4 mstrain in the year 225. This strain is larger than the strain (of about 2 mstrain) that normally causes yielding in the steel. The maximum strain takes place in the lower part of the Ile formation with high porosity. It is however possible that the full-field model is too coarse to capture localized deformations within the Ile formation. This means that the axial strain may locally be even slightly larger. It is also seen that the strains in the well outside the reservoir layers are small. This means that the uncertainties in the calculated strains of significance first of all are affected by uncertainties in the drained bulk modulus of the sandstone. Finally, the change in curvature is too small to affect the deformations of the well. 3. Detailed model of the Kvitebjørn field 3.1 Geometry and model A detailed 2D plane strain ABAQUS model of two segments with 4 m relative throw and 4 degrees fault dip is shown in Figure 6. The model consists of the 23 layers defined in the fullfield-reservoir-model established in ECLIPSE. The thickness of the reservoir layers is 175 m. The model includes a 25 m thick impermeable shale layer above and below the reservoir. Roller boundaries are assumed along the vertical boundaries, the bottom boundary is fixed vertically and horizontally and the top boundary is loaded with a constant vertical stress equal to the in-situ vertical stress of the model. The in-situ stress field is assumed constant throughout the model. The effective overburden stress is equal to σ v is equal to 11.5 MPa, while the effective horizontal stresses σ h = σ H are equal to 7.5 MPa. The loads and initial stresses equilibrate exactly and produce zero deformations as expected. Again the 6-node triangular displacement and pore pressure element respresents the reservoir sandstone in addition to the fault zone. A linear elastic ideal plastic Mohr Coulomb model gives strength. The shale material in burdens is represented by 6-node triangular displacement elements. A linear elastic ideal plastic Mises model gives the strength. The initial void ratio is an important parameter as the stiffness and strength of the reservoir sandstone depends on this value. 24 ABAQUS Users Conference 121

6 The fault is represented by continuum elements. Pressure within an internal fault varies linearly between the pore fluid pressures on each side of the fault. The top and bottom of the fault zone are assumed to have the strength of the shale, while the central part where the two reservoir segments are in contact is here assumed to have identical stiffness and strength as the reservoir sandstone layer with the lowest porosity. 3.2 Results and discussion The ultimate pressure reduction for a reservoir segment is approximately 6 MPa, starting up at 77.5 MPa and ending up with 18 MPa. Differential pressure across the fault is studied in Case A and B. The left segment is depleted in Case A, while unchanged pore fluid pressure is assumed for the right segment. Opposite assumptions for the two reservoir segments are assumed in Case B. Both the left and the right segment are depleted in Case C Case A depletion of left segment Evolution of tensile failure and compaction failure in the form of stress paths for selected points within the reservoir sandstone layers is shown in Figure 7. The placement of the three points chosen are indicated by the letters A, B and C in Figure 6. Note that the final pressure drawdown is equal to 1 MPa in this plot. Possible compaction failure is predicted for a typical stress path for the depleted left reservoir segment (curve A)). The stress ratio K defined as σ 3 / σ 1 is equal to.49 and the stress path coefficient γ defined as σ 3 / p is equal to.58 for this point. A limited increase in the degree of shear mobilization is obtained for the fault zone (curve B), while tensile failure is experienced for the right reservoir segment with unchanged pressure (curve C). The degree of shear mobilization, defined as the ratio of the mobilized friction tanρ and the friction at failure, tanφ, is plotted in Figure 8 for the left reservoir segment. All values are less than.6 after 6 MPa depletion. This implies that the state of stress at any point in the depleted segment is situated far below the Mohr Coulomb failure envelope. Focus in the rest of chapter will therefore be on the behavior of the reservoir surrounding shale and the fault zone. Figure 9 gives the evolution of plastic yielding expressed by the effective plastic strain in the burdens and fault. The effective plastic strain is given by p ε = 2 ε 3 pl ij ε pl ij Reservoir segments are colored grey in this plot, i.e. they are not displayed. Yielding is initiated in the shale below the acute corner of the depleted segment at 26 MPa depletion. A continuous failure zone between the left at right reservoir segment is established after 28 MPa depletion, see Figure 9.a. Shale failure above the top corner of the depleted reservoir is initiated at 54 MPa depletion. Failure in the intermediate fault zone is not initiated. An effective stress increase in the fault zone is actually specified due to the assumption of linear pressure variation through this zone ABAQUS Users Conference

7 3.2.2 Case B - depletion of right segment Contour plots of the effective plastic strain in burdens and fault zone are shown in Figure 1.a and b. Yielding is initiated above the acute corner of the depleted reservoir segment at 28 MPa depletion according to Figure 1.a Case C - depletion of both segments Effective plastic strain contours at 6 MPa depletion are shown in Figure 11. Plastic flow is concentrated to the shale part of the fault zone. This is believed to be a consequence of the free boundary at the top of the model. Uniaxial vertical strain is observed for large parts of the model and this is reflected in the limited plastification. A fixed boundary at the top will result in high shear stresses in the shale above and below the depleted reservoir segments Concluding remarks for internal faults The local finite element analyses of internal faults with ABAQUS predict Most likely increased communication between two reservoir segments isolated by an internal fault due to large pressure difference across the fault - no tensile failure or shear failure in the fault zone - however, there may be communication through the shale at the top and bottom corner of the depleted segment Top-seal leakage through faults and fractures is not expected A starting point for incorporating these results in ECLIPSE is to search for internal faults with differential pressure above 28 MPa, and introduce communication between these at the acute angle of the segment that is most depleted. This search may either be done manually or by a script. However increased communication between two segments are not a function of the differential pressure alone. The absolute pressure certainly also play a role as shown by Case C. It is also believed that the results are sensitive to fault geometry expressed by the dip. The ABAQUS results so far (Case A and B) indicate that the differential pressure necessary to lead to increased communication through the bounding shale decreases with decreasing dip, as plastic yielding is initiated in the bounding shale close to the acute angle of the depleted segment. 4. Necessary enhancement of ABAQUS Undrained (no pore fluid flow) total stress analyses are carried out for the shale material. Strength is given by the Mises criterion. However, undrained effective stress analysis is the preferred alternative for undrained rock behavior. Zero permeability pore pressure elements may be used for this purpose. This will be quite cumbersome as only continuous pore pressure elements are available in ABAQUS. The mesh must be opened up at the interface between the reservoir sandstone and the surrounding shale as pressure degree of freedoms should not be common at this interface. 24 ABAQUS Users Conference 123

8 Model generation will be simpler if discontinuous pore pressure elements were available in ABAQUS as these could be used to model undrained shale behavior. Zienkiewicz et al (1983) introduced the nine-noded displacement and linear pressure element for 2D application. High accuracy was obtained for incompressible fluid mechanics problems with this element type. In the compressible case pore pressure degrees of freedom may be eliminated at element level, i.e. resulting in a displacement element. Naylor (1974) exploded this formulation, showing how a standard displacement based finite element program can be modified so that stresses are separated into pore fluid at rock matrix components. Reduced integration was advocated in the nearly incompressible case. The computational costs related to this formulation are considerably less than the discontinuous pressure elements. The K w -formulation by Naylor (1974) is widely used in specialist geotechnical finite element programs. The implication of tensile or shear failure may be studied in more detail introducing interface elements (discrete crack analysis) at critical places. However, all interface contact elements was removed in ABAQUS version 6.1. Instead one should use the *CONTACT PAIR option with SMALL SLIDING parameter to model infinitesimal sliding and pore fluid interaction between deformable bodies. Unfortunately this new option do not account for tangential gap flow, which is essential for examining possible hydraulic communication between two isolated reservoir segments. From our standpoint it seems that reintroducing the interface elements again in ABAQUS is a good idea. The user subroutine option in ABAQUS makes the program to a very versatile tool. It is even possible to define your own finite element in ABAQUS. However developing and maintaining such computer code is not a part of our strategy. So for these topics we hope that the ABAQUS organization see them as useful suggestions. Material routines on the other hand may be something it is easier to get accept for internally to spend time on developing. 5. Conclusions Results from the studies of two North Sea HP/HT gas-condensate fields have been presented here. The full-field simulations of the Kristin field focus on reservoir compaction, subsidence, and possible well casing damage and stress changes during depletion. The evaluation of the well casing integrity is based on axial strain along the well path. Stress changes during depletion is a key issue for well planning after start up of production as the decline of the fracture pressure due to reservoir depletion is deduced form this. The local Kvitebjørn model on the other hand focuses on the mechanical behavior of faults and barriers during depletion. A realistic geometry, with interbedded shale layers and inclined fault zone was established. Evolution of shear failure as a consequence of increasing differential pressure across the fault zone was analyzed and the limiting value for start of cross between two separated reservoir segments was found. This contributes to better insight of the flow field during the production history. The full-field analysis of Kristin utilizes an external coupling between the reservoir simulator ECLIPSE and ABAQUS that has been established. Initial pressure, initial porosity variation and simulated pressure history are transferred from ECLIPSE to ABAQUS utilizing the user ABAQUS Users Conference

9 subroutine facilities in ABAQUS. ECLIPSE reservoir geometry is imported into ABAQUS/CAE reading AutoCAD files. Later experience indicates that a better approach is to incorporate geometry information directly in the Python script file that generates your ABAQUS model. Both commercial and research simulators that take the coupled nature of three-phase-flow and deformation into account exists today. It is then natural to ask if this will make the partly coupled approach described here superfluous? Our experience is however that a partly coupled approach between a conventional reservoir simulator and a stress simulator is the best approach for the near future when advanced geomechanical issues must be taken into account. The partly coupled approach benefits from the latest developments in physics and numerical techniques of both simulators. Large-scale 3D simulations, which a natural next step for us is another issue that favors the partly coupled approach, as it is believed that computational speed will be crucial. Summarizing our experience so far with ABAQUS for reservoir geomechanical simulations we may state that ABAQUS is very well suited for reservoir geomechanical simulations. However with some minor enhancements as pointed out previously ABAQUS will be of even greater value for the reservoir modelers. 6. Acknowledgements The management of Statoil s Kristin and Kvitebjørn projects are thanked for permission to publish this paper. Hans Petter Jostad at Norwegian Geotechnical Institute is acknowledged for valuable support on the Kristin analysis. 7. References 1. Naylor, D.J., Stresses in Nearly Incompressible Materials by Finite Elements With Application to the Calculation of Excess Pore Pressure, I.J.N.M.E., Vol. 8, pp , Zienkiewicz, O.C, Taylor, R.L. and Baynham, J.A.W., Mixed and Irreducible Formulations in Finite Element Analysis, Chapter 21 of Hybrid and Mixed Finite Element Methods (eds S.N. Atluri, R.H. Gallager, and O.C. Zienkiewicz), pp , Wiley, ABAQUS Users Conference 125

10 West Middle East 4166 m Garn Ile Figure 1. FE Mesh. (a) Entire model. (b) Close up of the two main gas-condensate reservoirs and (c) Reservoir details in the west. Figure 2. Pore pressure depletion in year 225, i.e. after 2 years of production ABAQUS Users Conference

11 Horizontal displacement, m Vertical displacement, m Vertical depth, m Horizontal displacement, m Vertical displacement, m Vertical depth, m Horizontal displacement, m Vertical displacement, m Vertical depth, m Figure 3. Horizontal and vertical displacement profiles at years 27, 29 and 225. (a) Western part, (b) Middle part and (c) Eastern part (see Figure 1.b for placement). 24 ABAQUS Users Conference 127

12 Figure 4. Stress path coefficient, defined as the ratio of the change in horizontal stress to the change in pore pressure, has an average value of.5-.6 during depletion. Displacements (m) Axial strain (mstrain) TVD m seabed TVD m seabed u_x u_y 48 Figure 5. Deformations along a given well path in year 225. (a) Horizontal and vertical displacements and (b) Axial strain ABAQUS Users Conference

13 overburden A Left segment ψ =4 B underburden underburden C 4m Right segment Figure 6. Geometry and boundary conditions. Deviatoric stress, MPa B A C 4 2 C B Minimum effective principal stress, MPa A Figure 7. Stress paths for points A, B and C in Figure 6, in σ 3 -q axes (Case A). Corresponding Mohr Coulomb failure envelopes are marked in the same manner. Figure 8. Degree of shear mobilization f=tanρ/tanφ for left reservoir segment after 6 MPa depletion, Case A. f is less than ABAQUS Users Conference 129

14 Figure 9. Evolution of effective plastic strain in reservoir surrounding shale for Case A. (a) 28 MPa depletion, (b) 6 MPa depletion and (c) 1 MPa depletion ABAQUS Users Conference

15 Figure 1. Evolution of effective plastic strain in reservoir surrounding shale for Case B. (a) 28 MPa depletion and (b) 6 MPa depletion. Figure 11. Contours of effective plastic strain after 6 MPa depletion for Case C. 24 ABAQUS Users Conference 131