14CSER009 Dr.ir. M. Gerritsma voorstel.pdf

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Shell-NWO/FOM

Computational sciences for energy research

Call 2014

PhD project proposal

Project number: 14CSER009

Title: Affordable high resolution reservoir modeling using a multiscale approach

Main applicant: Dr.ir. M. Gerritsma

Affiliation: TUD


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Affordable high resolution reservoir modeling using a multiscale approach

Dr.ir. M. Gerritsma, Dr. S. Hulshoff and Dr.eng. A. Palha

April 22, 2014

1 Title of the project

Affordable high resolution reservoir modeling using a multiscale approach

2 Applicant(s)

Naam, voorletter en titel: : Dr.ir. M.I. GerritsmaOrganisatie : Technische Universiteit Delft / Delft University of TechnologyTelefoonnummer : +31 15 2785903

E-mail : [email protected]/female : MaleAanstellingspercentage : 100%Vast dienstverband : ja

2a Co-applicants

Naam, voorletter en titel: : Dr. S.J. Hulshoff Organisatie : Technische Universiteit Delft / Delft University of TechnologyTelefoonnummer : +31 15 2781538E-mail : [email protected] : 100%Vast dienstverband : ja

Naam, voorletter en titel: : Dr.eng. A. PalhaOrganisatie : Technische Universiteit Eindhoven / Eindhoven University of TecnhologyTelefoonnummer : +31 6 39125130E-mail : [email protected] : 100%Vast dienstverband : Postdoctoral research assistant

Promotor Prof Fulvio Scarano, Head of the AWEP Department and Section Head of Aerodynamics of the Faculty of Aerospace Engineering at TU Delft will act as promotor for this project.

3 Abstract

In this proposal the mixed multi-scale finite element (MMSFE), [1, 4, 5, 10], and multiscale finite volume (MFVM), [15],methods are combined in an arbitrary order mimetic least-squares discretization, [6, 8, 17, 18, 19] to solve flow problemsin reservoir engineering. Mimetic methods satisfy conservation laws at the cochain level, without any reference to basisfunctions. The anisotropic, heterogeneous, material-dependent constitutive laws are satisfied in a point-wise sense. Inthis respect, mimetic methods resemble finite volume methods for the topological relations and finite element methods forthe metric-dependent relations. The topological relations are additive, so properties at the fine scale can be immediatelytransferred to the large scale by summation. The large scale heterogeneous properties are directly obtained from thesmall scale dynamics in the mimetic spectral least-squares formulation, yielding a systematic procedure for coarsening

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(upscaling) and refinement. The ultimate goal of this study is to produce an effective coarse grid model for the flow inporous media on rough grids.

4 Keywords

least-squares methods, mimetic methods, multiscale modeling, computational geoscience

5 Key publications1. Akkerman, I., Bazilevs, Y., Calo, V.M., Hughes, T.J.R., Hulshoff, S., The role of continuity in residual-based

variational multiscale modeling of turbulence , 41(3), pp 371-378, 2008.

2. Kreeft J. and Gerritsma M, Mixed mimetic spectral element method for Stokes flow: A pointwise divergence-free solution , J. Comput. Phys., 240, pp. 284-309, 2013.

3. Palha, A., Rebelo, P.P., Hiemstra, R., Kreeft, J. & Gerritsma, M., Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms , J. Comput. Phys., 257, pp.1394–1422, 2014.

6 Relevance, background, and state of the art

6a Relevant innovation contract

This proposal contributes to the top sector ‘Energy -Gas’. More specifically to the development and implementation of innovative exploration and production technologies to support natural gas production in the Netherlands.

6b Background and state of the art

The need to understand subsurface flow processes is large. The utilization of hydrocarbon reservoirs, of ground waterresources and of geological CO2 storage, to name a few, all make extensive use of this knowledge.

The accurate representation of the geological features, e.g. faults, fractures, erosion and horizontal wells, requiresgeometrically flexible grids. For example, the industry standard of stratigraphic grids (corner-point, PEBI, etc.) and themore recently employed unstructured grids have these properties but contain irregular cells with complex connectivities.The conventional two-point flux approximation methods are known to be only consistent and convergent on K -orthogonal

grids, [3]. Multi-point flux approximation (MPFA) methods, [2, 20], and mixed finite elements, [9], are being activelyresearched for the development of consistent and convergent methods on complex and rough grids, [14].

Additionally, highly resolved geological models, due to computational restrictions, cannot be used directly for flowsimulations. Nevertheless, small scale effects such as compressibility, gravity and capillary flow, faults, fractures, multi-phase flows and compositional displacements need to be taken into account for a proper representation of the large scaledynamics. Coarse-grid models based on non-trivial criteria for upscaling, [12], are generally used. Recently, multiscalemethods, [11, 13], have received much attention for their intrinsic definition of upscaling. In multiscale methods particularcare is needed, since, when a local conservation property is not satisfied, the fractional flow curve deviates significantlyfrom that obtained from well-resolved simulations. To address these issues, recent work has been done in the developmentof mixed multiscale finite element (MMSFE) approaches, [1, 4, 5, 10], and multiscale finite volume (MFVM) approaches,[15].

7 Project description

7a Objectives

The focus of this project is on the development of a novel discretization technique for the solution of the flow in complex,highly heterogenous and fractured porous media, for use in reservoir engineering. Therefore, we intend to construct anumerical method that is geometrically flexible, presents minimal grid-orientation effects, and constitutes an accurate androbust alternative for conventional upscaling techniques, [12]. This will allow a reduction in the number of degrees of freedom used. In order to achieve this, mixed multiscale finite element (MMSFE), [1, 4, 5, 10], and multiscale finite volume(MFVM), [15], methods will be combined with arbitrary order mimetic least-squares discretizations, [6, 8, 17, 18, 19].

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The objective is to take the geometric flexibility of the mimetic approach plus the robustness of least-squares methods,[7], and extend it to efficiently capture the coarse scale on the primal grid behaviour of the solution and the small scalefeatures on the dual grid, in order to bridge the gap between the fine scale details of the geological models and the globalflow simulation.

7b Scientific challenges

Accurate numerical simulations in reservoir engineering require excessively fine grids which are either adapted to thegeometrical features of the reservoir or enforce K -orthogonality, relaxing on geometric accuracy. K -orthogonality generally

leads to grids with high aspect ratios and imposes strict constraints on the mesh geometry, leading to considerablegeometrical errors in the representation of faults, for example. Traditional upscaling methods generally work well whenthe length scales are well separated, but when the length scales are not so well separated it becomes much more difficult,see [12], and a numerical solution tends to become unreliable. The scientific challenge in this project is to develop anumerical discretization which is reliable on rough grids and systematically incorporates the fine-scale information intothe coarse-scale equations such that it is consistent with the properties of the differential operators, reducing computationtimes. In this project we intend to address this challenge by developing a discretization based on a multiscale mimeticspectral least-squares formulation.

7c Innovative character and progress beyond the state of the art

The innovative character of this project lies in adapting the mimetic method to grids with industry-standard complexity andapplying it as a consistent and accurate building block for constructing a mimetic least-squares multiscale method. Due to

their intrinsic geometric flexibility, mimetic methods have been shown to eliminate grid-orientation effects, [19], allowingaccurate modelling of fault geometry without introducing discretization errors. Moreover, they combine the flexibilityof the finite element method and the conservation properties of the finite volume method. The intrinsic separation of topological relations (conservation laws) from constitutive relations in mimetic least-squares methods allows the naturalconstruction of distinct approximation scales for the different physical quantities. For example, in the solution of Darcyflow equations, see Figure 1, the pressure and velocity fluxes can be solved with independent resolutions (both h and p)in a mimetic least-squares setting, while retaining optimal convergence rates at the coarse and fine scales independently.

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Figure 1: Mimetic spectral element solution taken from [19] applied to the test case described in [20]. Left: a color plotshowing the pressure field with on top the velocity vectors. Right: h-convergence plots of the pressure and flux displayingoptimal convergence rates. [Courtesy of Jasper Kreeft, TU Delft]

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7d Scientific relevance

A proper coupling between small scale dynamics and large scale dynamics is not restricted to reservoir engineering butpertains to all physical models with a spectrum of length and time scales. The techniques developed in this project willbe useful in other branches of science.

7e Methods and techniques

Mimetic methods intrinsically separate topological relations (metric free) from constitutive relations (metric dependent),

see also [16]. The topological relations can be exactly represented at the cochain level, resembling the finite volumeformulation. The exact character of the topological relations (conservation laws) at the cochain level is independent of thegeometry of the grid. This allows highly curved and rough grids to be used, without loss of convergence properties. Onthe other hand, the constitutive relations can be represented as a projection between functional spaces associated withtwo distinct grids, a dual and a primal one. Typically this projection is restricted to spaces of equal dimension. In amimetic least-squares formulation (MS-LS), [6], it is possible to circumvent this restriction see Corollary 1 in [6]. In thisway, different approximations, in both h and p, can be used for the quantities that are related by a constitutive relation.For reservoir engineering this implies that one can use a coarse grid for pressure and a fine grid for velocity. The highlyheterogeneous material properties enter the constitutive laws through the permeability, mobility or transmissibility multi-pliers which are modelled at the small scales. In this way the constitutive equations systematically establish the relationbetween the fine- and coarse-grid scales. Upscaling of small scale quantities to large scales then becomes unambiguousthrough the summation of the small scale cochains.

7f Computational and data storage resources and requirements

7g Work plan for the requested PhD

Apart from the scientific work to be performed in this project, the Ph.D. candidate is expected to actively collaboratewith experts in the field. In particular we expect collaboration with

• Pavel Bochev, Sandia National Lab, NM, USA on least-squares formulations.

• Konstantin Lipnikov, Los Alamos National Labs, NM, USA, mimetic methods and flow through porous media.

• Prof Tom Hughes, University of Austin, TX, USA, multiscale modeling.

• Dr Jasper Kreeft, Shell Global Solutions, Amsterdam, Spectral and High Order Mimetic Methods.

Year Period Ph.D. candidate

1 i Literature research reservoir engineeringii Literature research multiscale modelingiii Literature research mimetic methodsiv 2D Modelling Darcy law using mimetic spectral least-squares method with Pavel Bochev

2 i MS-LS modelling of Vertical Fault test case 1, [16]ii Comparison with MATLAB Reservoir Simulation toolbox from SINTEF

MS-LS modeling of Varying multiplier test case 2, [16]iii, iv Multiscale modeling of incompressible Euler

3 i,i i Multiscale modeling flow p orous media with Konstatntin Lipnikoviii,iv Extension of MS-LS to 3D

4 i Test case 3 Faulted mo del with lenses of low-p ermeability, [16]ii Test case 4 Real field model, [16]iii Evaluation of test cases and feedbackiv Preparation of dissertation

8 Budget

The “standard budget” of 256 ke is requested

Personnel (cost for one Ph.D. candidate) 216 keRunning budget (bench fee) 20 keTravel cost (bench fee) 20 keAdditional budget 0 ke

Total 256 ke

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9 References

[1] J.E. Aarnes, On the use of a mixed multiscale finite element method for greater flexibilty nd increased speed of improved accuracy in reservoir simulations , Multiscale Model. Simul., 2, pp. 421-439, 2004.

[2] Aavatsmark, I., An introduction to multipoint flux approximations for quadrilateral grids , Comput. Geosci., 6, pp.405–432, 2002.

[3] Aavatsmark, I., Interpretation of a two-point flux stencil for skew parallelogram grids , Comput. Geosci., 11, pp.199–206, 2007.

[4] T. Arbogast and S.L. Bryant, A two-sale Numerical subgrid technique for waterflood simulations , SPE J. 7, pp.446-457, 2002.

[5] T. Arbogast, Implementation of a loaclly conservative numerical subgrid upscaling scheme for two-phase Darcy flow ,Comput. Geosci., 6, pp. 453-481, 2002.

[6] P. Bochev and M. Gerritsma, A spectral mimetic least-squares method , pp. 1-31, submitted to CAMWA, 2014.

[7] P. Bochev and M. Gunzburger. Least-squares finite element methods. Springer, 2009.

[8] Bouman, M.P., Palha, A., Kreeft, J.J. & Gerritsma, M.I., A conservative spectral element method for arbitrarydomains. In Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in ComputationalPhysics, 76, pp. 111-120, 2011.

[9] Brezzi, F., Fortin, M., Mixed and hybrid finite element methods , Springer series in computational sciences, vol. 15Springer, New York, 1991.

[10] Z. Chen and T. Hou, A mixed finite element method for elliptic problems with rapidly oscillating coefficients , Math.Comput. 72, pp. 541-576, 2003.

[11] Efendiev, Y. and Hou, T., Multiscale finite element methods: theory and applications , Vol. 4. Springer, 2009.

[12] Farmer, C.L., Upscaling: a review , Int. J. Numer. Meth. Fluids, 40, pp. 63–78, 2002.

[13] H. Hajibeygi and H.A. Tchelepi, Compositional Multiscale Finite-Volume Formulation , Society of Petroleum Engi-neers. doi:10.2118/163664-PA, 2014.

[14] J. M. Hyman and S. Steinberg, The convergence of mimetic discretization for rough grids , Computers & Mathematicswith Applications, vol. 47, no. 10-11, pp. 1565–1610, May 2004

[15] P. Jenny, S.H. Lee and H.A. Tchelepi, Multi-scale finite volume method for elliptic problems in subsurface flow simulation , J. Comput. Phys., 187, pp. 47-67, 2003.

[16] H.M. Nilsen, K.-A. Lie and J.R. Natvig, Accurate Modeling of Faults by Multipoint, Mimetic and Mixed Methods ,SPE J., 17, pp. 568-579, 2012.

[17] Kreeft, J.J., Palha, A. and Gerritsma, M.: Mimetic framework on curvilinear quadrilaterals of arbitrary order.http://arxiv.org/abs/1111.4304, 2011.

[18] Kreeft, J.J. and Gerritsma, M.I., Mixed mimetic spectral element method for Stokes flow: a point wise divergence-free solution , J. Comput. Phys., 240, pp. 284-309, 2014.

[19] Palha, A., Rebelo, P.P., Hiemstra, R., Kreeft, J. and Gerritsma, M., Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms , J. Comput. Phys., 257, pp. 1394–1422,2014.

[20] M. Wheeler, G. Xue, and I. Yotov, A multiscale mortar multipoint flux mixed finite element method , ESIAM: M2AN,46:759–796, 2012.

10 Appendices

The paper [6] will be added since it is not yet publically available.

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14CSER009 Dr.ir. M. Gerritsma voorstel.pdf

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