Proceedings of the 23rd UK Conference of the Association for Computational Mechanics in Engineering 8 - 10 April 2015, Swansea University, Swansea
THE CHALLENGE OF COMPLEXITY IN SEDIMENTARY AND RESERVOIR SIMULATIONS Luca Formaggia1 , Anna Scotti1 and Alessio Fumagalli1 1
MOX Laboratory, Department of Mathematics, Politecnico di Milano, Pza Leonardo da Vinci 32, 20133 Milan, Italy
[email protected],
[email protected],
[email protected]
ABSTRACT In this work we will present part of the activity of the compgeo@mox group on the numerical modeling and simulation of flow in sedimentary basins and reservoirs. We will focus on a particular aspect, namely geochemical compaction processes and flow in fractured media. Key Words: porous media flow; fractured media; geochemical compaction processes
1. Introduction Exploitation of oil and water resources, as well as the safe storage of nuclear waste or the sequestration of CO2 calls for an accurate simulation of underground flows. The challenge is manifold. Properties of the underground rock and of its geometrical configuration are known with a very high level of uncertainty. Different coupled processes are involved: mono or multiphase-flow, chemical reactions, compaction phenomena, heat transfer. Material is usually heterogeneous with the presence of network of fractures and/or faults which typically act as preferential path for the flow (yet, sometimes, when filled with impervious material, they can act as flow barriers). In this presentation we will focus on two aspects, geochemical compaction phenomena and flow in fractured media. Compaction processes occur in rock layers during sedimentation, and their study is relevant to assess possible over-pressures (i.e. areas where the pore pressure exceeds the hydrostatic value) and the variation of porosity and permeability. Together with the mechanical processes, the fluid ↔ solid conversion, induced by the degradation of solid organic matter or dissolution/precipitation of minerals, affects flow through two basic mechanisms. On the one hand fluid production/consumption by chemical reactions acts as a source/sink term and may cause changes in the fluid pressure and thus in the effective stress experienced by the rock matrix. On the other hand, dissolution/precipitation mechanisms alter the solid matrix porosity, and consequently permeability, in the areas where the reactions occur. A numerical scheme for the solution of this complex coupled problem, outlined in Figure 1, has been developed [12], and it comprises chemical reactions, rock displacement and flow in the porous matrix, with permeability depending on porosity. It enables to treat, within the same framework, different geochemical processes such as dissolution/precipitation of minerals but also oil and gas generation from solid organic matter. We have used mixed finite elements for the Darcy equations and a mass preserving monotone scheme for the saturation equation. Figure 2 shows the result for the evolution of quartz mineral. The medium is subject to mechanical compaction and, at the same time, quartz may dissolve in water. The dissolution process produces extra porosity and therefore a stronger compaction in the corresponding portions of the domain. Another challenging application is the simulation of underground flow in the presence of fractures. Fractures (and faults) may affect greatly the effective permeability and storage capacity of the medium and, since their distribution may presents preferential directions, they also introduce anisotropies. They usually display significant variations in connectivity and size over the formation. Large and strongly connected fractures are typically located near bedding planes and fault zones, while small and disconnected
Initial Data C 0 φ−1 φ0 σ 0 Uw 0 p0w γ0
Quartz in Rock ∂C ∂t
= −VQrφ
C n+1
Quartz in Water ∂ ∂t (γφJ)
+ ∇ · γUw = ∇ · (DφF−T ∇γ) + rφJ
γ n+1
ρn+1 s Uw n+1 pn+1 w
φ
n+1
Bulk Pressure ∂s ∂ξ
= − [(1 − φ)ρs + φρf ] gJ Fixed Point Iterations
Pressure and Velocity ∂ ∇ · Uw = − ∂t (φJ) ˜
Uw = − JµK (∇pw − ρw FT g) w
φn+1
Effective Stress
σ n+1 = sn+1 − pn
sn+1
Figure 1: The coupled geochemical compaction problem in the case of mineral precipitation
Figure 2: Concentration of quartz in the rock at t = 0 My, t = 15 My, t = 30 My, t = 45 My, and t = 60 My during compaction.
fractures are usually located away from those regions. Traditional approaches based on heuristic modification of the permeability or semi-empirical transfer functions, see [18, 4, 3], are insufficient in the presence of a complex fracture distribution, making it impossible to compute transfer function analytically. Thus, numerical upscaling procedures based on realistic fracture characterization need to be applied to compute transfer functions accurately, for instance by solving local problems at grid blocks level[7, 14, 13, 15]. Due to their small aperture compared to the typical dimension of the computational domain it is unfeasible to use a grid capable of resolve fractures. Thus, fractures are modeled as a set of intersecting one co-dimensional manifolds immersed in the porous matrix, and reduced models are derived by projecting the differential equations on the tangent plane, while suitable conditions are imposed at the intersection among fractures[9]. In the presence of realistic fracture distribution the construction of a matching grid can be challenging. One possible strategy is to intersect a regular grid with the fracture network. This gives rise to a mesh of irregular polygons and finite volume or the mimetic finite difference (MFD) method can be used[5, 16, 1, 17, 8, 2]. An alternative, best suited for the study of single fractures or simple networks is to employ extended finite elements[6, 11]. In figure 3 we show an example of fractured media and the solutions obtained by the mimetic and the XFEM scheme described in [8], compared with a reference solution obtained by employing a finite volume scheme on an extremely refined mesh. In figure 4 we show the result on a complex network of 3D fractures. The computation is part of the nu-
Figure 3: An example of 2D fractured porous media (top-left). Pressure field for a reference solution on a grid with more than 106 elements (top-right). Solution obtained by mimetic finite differences (bottom-left) and by the XFEM method (bottom-right).The latter two computations were carried out on a grid of approximately 2000 elements.
merical upscaling procedure described in [10], where equivalent upscaled transmissibilities are computed numerically by solving high resolution local problems.
Figure 4: An example of simulation in a complex fracture network. Used within a numerical upscaling procedure to compute equivalent transmissibility.
Acknowledgements The support of Eni SpA is gratefully acknowledged, as well as that of the Italian MIUR and IndamGNCS. We also thank Paola Antonietti, Bianca Giovanardi, Nicola Verzotti, Stefano Zonca and all the other members of the compgeo@mox activity group. References [1] F.O. Alpak. A mimetic finite volume discretization method for reservoir simulation. SPE Journal, 15(2):436–453, 2010. [2] P.F. Antonietti, L. Formaggia, A. Scotti, M. Verani, and N. Verzotti. Mimetic finite difference approximation of flow in fractured porous media. (submitted), 2015. [3] T. Arbogast, J. Douglas, Jr., and U. Hornung. Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal., 21(4):823–836, 1990.
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