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ECMOR XV - 15th European Conference on the Mathematics of Oil Recovery

29 August – 1 September 2016, Amsterdam, Netherlands

Mo Gra 02Dynamic Unstructured Mesh Adaptivity forImproved Simulation of Bear Wellbore Flow inReservoir Scale ModelsP.S. Salinas* (Imperial College London), D. Pavlidis (Imperial CollegeLondon), A. Adam (Imperial College London), Z. Xie (Imperial CollegeLondon), C.C. Pain (Imperial College London) & M.D. Jackson (ImperialCollege London)

SUMMARYIt is well known that the pressure gradient into a production well increases with decreasing distanceto the well and may cause downwards coning of the gaswater interface, or upwards coning ofwateroil interface, into oil production wells; it can also cause downwards coning of the water table,or upwards coning of a saline interface, into water abstraction wells. To properly capture the localpressure drawdown into the well, and its effect on coning, requires high grid or mesh resolution innumerical models; moreover, the location of the well must be captured accurately. In conventionalsimulation models, the user must interact with the model to modify grid resolution around wells ofinterest, and the well location is approximated on a grid defined early in the modelling process.We report a new approach for improved simulation of nearwellbore flow in reservoirscale modelsthrough the use of dynamic unstructured adaptive meshing. The method is novel for two reasons.First, a fully unstructured tetrahedral mesh is used to discretize space, and the spatial location of thewell is specified via a line vector. Mesh nodes are placed along the line vector, so the geometry ofthe mesh conforms to the well trajectory. The well location is therefore accurately captured, and theapproach allows complex well trajectories and wells with many laterals to be modelled. Second,the mesh automatically adapts during a simulation to key solution fields of interest such as pressureand/or saturation, placing higher resolution where required to reduce an error metric based on theHessian of the field. This allows the local pressure drawdown and associated coning to be capturedwithout userdriven modification of the mesh. We demonstrate that the method has wideapplication in reservoirscale models of oil and gas fields, and regional models of groundwaterresources.



Introduction It is well known that the pressure gradient into a production well increases with decreasing distance to the well and may cause downwards coning of the gas-water interface, or upwards coning of water-oil interface, into oil production wells; it can also cause downwards coning of the water table, or upwards coning of a saline interface, into water abstraction wells. To properly capture the local pressure drawdown into the well, and its effect on coning, requires high grid or mesh resolution in numerical models; moreover, the location of the well must be captured accurately. In conventional simulation models, the user must interact with the model to modify grid resolution around wells of interest, and the well location is approximated on a grid defined early in the modelling process. We report a new approach for improved simulation of near-wellbore flow in reservoir-scale models through the use of dynamic unstructured adaptive meshing. The method is novel for two reasons. First, a fully unstructured tetrahedral mesh is used to discretize space, and the spatial location of the well is specified via a line vector. Mesh nodes are placed along the line vector, so the geometry of the mesh conforms to the well trajectory. The well location is therefore accurately captured, and the approach has potential for complex well trajectories to be modelled. Second, the mesh automatically adapts during a simulation to key solution fields of interest such as pressure and/or saturation, placing higher resolution where required to reduce an error metric based on the Hessian of the field. This allows the local pressure drawdown and associated coning to be captured without user-driven modification of the mesh. Moreover, the method automatically detects if a well stops producing/injecting or begins to function, adjusting the mesh resolution by either increasing it or reducing it respectively. We demonstrate that the method has wide application in reservoir-scale models of oil and gas fields, and regional models of groundwater resources. Methodology The numerical formulation presented builds on that reported by Jackson et al. (2015), Salinas et al. (2015) and Gomes et al. (under revision) so only a short overview is provided. For simplicity, the rock and fluids are considered incompressible so the mass balance equation can be written as

pppp qSt

S

v.

(1) in which ϕ represents the porosity, p is the phase, S is the saturation, v the saturation-weighted Darcy velocity and q a source term. Darcy's law for phase p can be written as:

gvσvK ppppp

rp

pp P

kS

(2)

where krp is the relative permeability for phase p, K is the permeability tensor, P is the pressure, g is the gravity vector and ρ is the density. By analogy with the Navier-Stokes equation, σ is considered as an absorption term, with coupled FE and CV representation of the nonlinear convolution of saturation, relative permeability and permeability tensor fields which is piecewise constant within each FE and CV. To close the system, the sum of the saturations has to be unity:

1 pS (3)



The method developed is based on the family of Control Volume Finite Element Methods (CVFEM). Many variants of this approach have been used to model multiphase flow (e.g. Huyakorn and Pinder, 1978; Dalen, 1979; Forsyth, 1991; Fung et al., 1992; Durlofsky, 1993, 1994; Helmig and Huber, 1998; Bastian and Helmig, 1999; Edwards, 2002; Geiger et al., 2004; Hoteit and Firoozabadi, 2006; Wheeler and Yotov, 2006; Matthai et al., 2007; Matringe et al., 2007; Schmid et al., 2013; Jackson et al., 2015; Mostaghimi et al., 2015). In this method, pressure and velocity are discretized using finite elements and properties that needs to be conserved, such as saturation, are discretized using control volumes. The element pairs used here are PM(DG)-PN and PN+1(DG)-PN (DG). The element pair PM(DG)-PN is a classical element pair. Velocity is discontinuous and has a discretization order M that has to be smaller or equal to the discretization order of the pressure N. Pressure is continuous across the elements. Saturation has the same discretization order as the pressure. Figure 1 shows an example of a P1(DG)-P2 element pair, in which velocity has a linear variation across elements and is discontinuous between elements, while the pressure is quadratic and continuous between elements. The PN+1(DG)-PN (DG) formulation has discontinuous velocity and discontinuous pressure representation between elements. This is the key difference compared with previous CVFE methods reported previously and enables the use of CVs that do not span to various elements. Instead, all CVs are inside a single element. Figure 2 shows a P2(DG)-P1(DG) element pair in which the velocity is quadratic and discontinuous between elements and the pressure is linear and discontinuous between elements. It can be seen how three CVs are defined per element. These CVs are equivalent to the ‘sectors’ of Afkan et al. (2011). Like in a classical CVFE formulation, viscosity, relative permeability and saturation are represented CV-wise. Permeability and porosity are represented element wise. Thus, each CV is associated with a unique permeability and porosity, in contrast to classical CVFE methods in which a single CV has different values of permeability and porosity. All the numerical methods presented here, including dynamic mesh optimization, are implemented within a single, integrated open-source code, available under the terms of the Lesser General Purpose License (LGPL): the Imperial College Finite Element Reservoir SimulaTor (IC-FERST).

Figure 1 2D representation of a P1(DG)-P2 element pair. On the left we show the FEs and on the right the CVs. The mesh optimization methodology used here is the same as that described by in Jackson et al. (2015) and Salinas et al. (2015) (and see also Adam et al., ECMOR XV), employing the surfaces defined in the geologic model and the hr-adaptive, unstructured mesh adaptivity library of Pain et al. (2001). This algorithm (see Pain et al., 2001 for a full description) projects the variables defined in



CV space into the FE space as necessary and parallelizes efficiently (see, for example, Piggott et al., 2008). Here, the pressure and velocity projection uses a conservative Galerkin interpolation technique (see Farrell and Maddison, 2011), while the saturation projection uses a new interpolation method presented recently in Adam et al. (2016). This interpolation technique is fully conservative and bounded, and has second order remapping. The mesh is adapted to the saturation and pressure field to ensure that the saturation and pressure are correctly represented. The resolution of the mesh can therefore automatically increase/decrease around a well depending on the local flow field. If, for example, if the well is shut, a low resolution around the well is sufficient to correctly represent the pressure field. However, if the well is flowing, a higher resolution may be required in order to accurately preserve the pressure and saturation fields.

Figure 2 2D representation of a P2(DG)-P1(DG) element pair. On the left we show the FEs and on the right the CVs. Results Production from an unconfined aquifer In this numerical experiment, our formulation is compared against the Thiem analytical solution for unconfined aquifers. The Thiem model applies to an infinite, homogenous aquifer with production of only water from the well. Figure 3 shows the numerical domain; the aquifer is modelled as a circular disk of radius 200m and thickness 20m. A vertical well is located at the centre of the domain. The permeability is 9.87 10-14 m2 and porosity is 0.2. The relative permeability is modelled using the Brooks-Corey model (Brooks and Corey, 1964) with end-point relative permeabilities of 0.3 for water and 0.8 for air; immobile and residual saturations are 0.2 for the water and air phases respectively and the saturation exponent is 2 for both phases. Fluid viscosity and density are 0.001 Pa s and 1000 kg/m3 respectively. Note that the Thiem solution ignores gravity except in maintain a sharp interface between water and air. The upper and lower boundaries are sealed; the sides have a constant pressure of 1 MPa. The well extracts water at a constant rate of 0.25 m3/day. Figure 4 (A) shows the initial configuration of the numerical experiment and the location of the well. A layer of air 5 metres thick overlies a water layer occupying the remaining 15 metres. Since the Thiem solution requires the well to only extract water, only the lower 5 metres of the well are open to flow. Figure 4 (B) and (C) show the water phase volume fraction and the corresponding fixed and adaptive meshes after 2240 days of extraction. The mesh is adapted to the phase volume fraction and pressure. The case using dynamic mesh optimization is able to model the flow behaviour more accurately and the pressure drawdown around the well is captured better. Figure 5 (A) and (B) shows



the pressure field at the same time. In the case using mesh adaptivity, the pressure drawdown around the well is steeper, and more closely matches the analytical solution (Figure 6). Figure 3 Numerical domain, the diameter is 200 m, the thickness is 20 m and the well is situated and the centre of the domain. Figure 4 2D slices of the domain showing the wetting (water) phase saturation at (A) initial conditions, and after 2240 days of production using (B) a fixed mesh and (C) a dynamically adaptive mesh.

(A)

(B)

(C)



Figure 5 2D slices of the domain showing the pressure field after 2240 days using (A) a fixed mesh (A) and (B) dynamic mesh optimization.

Figure 6 Drawdown profiles for the unconfined aquifer model using an adaptive mesh method and a fixed mesh method with a similar number of elements (70000) compared against the analytical solution. Production from a reservoir containing complex intersecting faults In this test case a more realistic heterogeneous domain is considered (Figure 7). The domain contains several layers of contrasting permeabilities (10-12 and 10-15 m2) that have been offset by two different generations of faults, which have different orientations and dips. The faults intersect because of their differing orientations, and because of their differing dips. Discretizing such a model using pillar grids is extremely challenging, but it is straightforward using unstructured meshes. The model was produced using a surface-based modelling algorithm that creates geologic models represented using Non-Uniform Rational B-Splines (NURBS) surfaces (see Jacquemyn et al, ECMOR XV) and the

(A)

(B)

Pressure



initial geometry-adaptive mesh used to discretize the surfaces and associated volumes was created using an automated meshing workflow with the NURBS surfaces as input (see Melnikova et al, ECMOR XV). The model boundaries have no flow conditions except for the side furthest away from the well that has a constant pressure of 1 MPa and injects water into the domain. The well extracts with a constant pressure of 0 Pa. The domain size is 80 (height) x 100 (width) x 200 (length) metres. The relative permeabilities for each phase are the same as used in the previous example, and the displacing water and displaced oil have the same viscosity and density. Again, the dynamic mesh is adapted to the saturation and pressure fields.

Figure 7 Numerical domain with the well position being the black line. The blue regions have a permeability of 10-12 and the red ones of 10-15 m2. Figure 8 shows the saturation profiles and the corresponding mesh at two different times using a fixed mesh and unstructured dynamic mesh with a similar amount of elements (24000). The case using dynamic mesh optimization properly focuses the injected water through the high permeable layers. Dynamic mesh optimization improves the quality of the solutions by focusing the resolution where it is required and reducing it elsewhere. In the vicinity of the well, the resolution is always high because there is a large pressure gradient that requires high mesh resolution to be properly captured.



Figure 8 Saturation profile and corresponding mesh at two different times using (left) a fixed mesh and (right) dynamic mesh optimization. Figure 9 shows a zoom of the area around the well for both numerical experiments. The well is in both cases correctly preserved. In the case using dynamic mesh optimization, it can be seen that the resolution around the well is significantly increased compared with the case using a fixed mesh. The flow is also much better captured.

Figure 9 Area around the well (dashed black line). Saturation profile using (left) a fixed mesh and (right) dynamic mesh optimization. Conclusions High resolution is often desirable to capture near wellbore flow. However, with fixed meshes this requires increased resolution during the whole simulation. Yet in many cases new wells are drilled and existing wells are closed. Maintaining a fixed and high mesh resolution is computationally inefficient. Unstructured dynamic mesh optimization gives the ability to change the mesh to fit the



flow behaviour in space and time to provide high resolution where and when required. This means that the mesh is no longer decided a priori. For near wellbore flow, it has been shown that the flow behaviour can be depending on the mesh resolution. Increasing the resolution around the well helps to accurately represent the flow behaviour. Acknowledgements Funding for Salinas from ExxonMobil, for Pavlidis from the EPSRC (‘Computational Modelling for Advanced Nuclear Power Plants’) and EU Horizon 2020 IVMR, for Adam and Melnikova from TOTAL, and for Xie from the EPSRC (‘Multi-Scale Exploration of Multiphase Physics in Flows’ – MEMPHIS) is gratefully acknowledged. TOTAL is also thanked for part funding Jackson under the TOTAL Chairs programme at Imperial College. Y. Melnikova and C. Jacquemyn are also thanked for providing the mesh and domain the second numerical experiment. References Adam, A., Percival, J.R., Pavlidis, D., Salinas, P., Xie, Fang, F., Pain, C.C., Muggeridge A.H. and Jackson, M.D. [2016] Higher-Order Conservative Interpolation Between Control-Volume Meshes: Application to Advection and Multiphase Flow Problems with Dynamic Mesh Adaptivity. Journal of Computational Physics, 321, 512-531. Doi: 10.1016/j.jcp.2016.05.058

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