Mandefro Belayneh

Finite Element - Node-Centered Finite-Volume Two-Phase-Flow Experiments With Fractured Rock Represented by Unstructured Hybrid-Element Meshes

Fractured-reservoir relative permeability, water breakthrough, and recovery cannot be extrapolated from core samples, but computer simulations allow their quantification through the use of discrete fracture models at an intermediate scale. For this purpose, we represent intersecting naturally and stochastically generated fractures in massive or layered porous rock with an unstructured hybrid finite-element (FE) grid. We compute two-phase flow with an implicit FE/finite volume (FV) method (FE/FVM) to identify the emergent properties of this complex system. The results offer many important insights: Flow velocity varies by three to seven orders of magnitude and velocity spectra are multimodal, with significant overlaps between fractureand matrix-flow domains. Residual saturations greatly exceed those that were initially assigned to the rock matrix. Total mobility is low over a wide saturation range and is very sensitive to small saturation changes. When fractures dominate the flow, but fracture porosity is low (10 to 1%), gridblock average relative permeabilities, kr,avg, cross over during saturation changes of less than 1%. Such upscaled kr,avg yield a convex, highly dispersive fractionalflow function without a shock. Its shape cannot be matched with any conventional model, and a new formalism based on the fracture/matrix flux ratio is proposed. Spontaneous imbibition during waterflooding occurs only over a small fraction of the total fracture/matrix-interface area because water imbibes only a limited number of fractures. Yet in some of these, flow will be sufficiently fast for this process to enhance recovery significantly. We also observe that a rate dependence of recovery and water breakthrough occurs earlier in transient-state flow than in steady-state flow.

Numerical simulation of multi-phase fluid flow in structurally complex reservoirs

Abstract Realistic simulation of structurally complex reservoirs (SCR) is challenging in at least three ways: (1) geological structures must be represented and discretized accurately on vastly different length scales; (2) extreme ranges and discontinuous variations of material properties have to be associated with the discretized structures and accounted for in the computations; and (3) episodic, highly transient and often localized events such as well shut-in have to be resolved adequately within the overall production history, necessitating a highly adaptive resolution of time. To facilitate numerical experiments that elucidate the emergent properties, typical states and state transitions of SCRs, an application programmer interface (API) called complex systems modelling platform (CSMP++) has been engineered in ANSI/ISO C++. It implements a geometry and process-based SCR decomposition in space and time, and uses an algebraic multigrid solver (SAMG) for the spatio-temporal integration of the governing partial differential equations. This paper describes a new SCR simulation workflow including a two-phase fluid flow model that is compared with ECLIPSE in a single-fracture flow simulation. Geologically realistic application examples are presented for incompressible 2-phase flow, compressible 3-phase flow, and pressure-diffusion in a sector-scale model of a structurally complex reservoir.

Numerical simulation of water injection into layered fractured carbonate reservoir analogs

Water flooding of fractured reservoirs is risky because water breakthrough can occur early, leading to a prohibitively high water cut. In mixed or oil-wet carbonates, capillary drive is negligible or absent. For this scenario, we investigate fluid-pressure-driven displacement of oil by water in two-phase flow numerical models based on naturally fractured limestone beds mapped along the British Channel coast. These reservoir analogs are represented by unstructured finite-element grids with discrete representations of intersecting fractures. We solve the governing equations for slightly compressible two-phase flow with our original control-volume finite-element method. This permits the direct examination of displacement patterns in fractures and rock matrix.We find that the irreducible saturation in the fractured carbonate is much higher than the value prescribed to the rock matrix. The shape of water invasion fronts is highly sensitive to the viscosity ratio of oil and water. When the Brooks-Corey relative permeability model is applied to the rock matrix at a viscosity ratio of 1, the total mobility, t, is low at intermediate saturations. This stabilizes displacement fronts where a girdle of reduced t develops, but this effect disappears as the viscosity ratio increases.For an idealized model with a water-wet matrix, we have also evaluated the effect of countercurrent capillary-pressure–driven flow across fracture-matrix interfaces. The rate of this countercurrent imbibition scales with the specific fracture surface area and decays exponentially as intermediate saturation zones develop adjacent to the fractures. The resulting reduced t feeds back into the fluid-pressure-driven displacement process.

Comparison of deterministic with stochastic fracture models in water-flooding numerical simulations

Determination of multiphase flow properties considering the variation of fracture patterns (i.e., number of fracture sets, their orientation, length distribution, spacing, and in-situ aperture) remains a key challenge in reservoirs. In reservoir engineering, one way is by studying outcrop analogs with comparable petrophysical properties and a similar geological history, and incorporating these data into model building, discretization, and numerical simulation. The limitation of directly incorporating attributes measured on outcrops is that this method is error prone because of postburial processes. Mineralized fracture (vein) attributes are good candidates to use as analogs for open fractures formed under in-situ conditions, to establish the relationship between fracture length and aperture and help to reveal the conditions at the time of their formation, and to quantify fracture-induced porosity in rock masses. Vein attributes determined from scan lines and window samples were combined to condition the stochastic generation of fractures using the discrete fracture network code FracMan. Comparison of water breakthrough time and oil saturation at breakthrough was then determined by applying a constant pressure gradient for each realization to simulate water-flooding numerical simulation using the combined finite element–finite volume method. The different stochastic realizations were compared with discrete fracture and matrix models, and we show how the uncertainty in these fracture attributes affects multiphase flow behavior in naturally fractured rocks. Uncertainty in quantifying these attributes has a profound impact for predicting the oil recovery and water breakthrough time based on limited information from boreholes.

Prediction of vein connectivity using the percolation approach: model test with field data

Evaluating the uncertainty in fracture connectivity and its effect on the flow behaviour of natural fracture networks formed under in situ conditions is an extremely difficult task. One widely used probabilistic approach is to use percolation theory, which is well adapted to estimate the connectivity and conductivity of geometrical objects near the percolation threshold. In this paper, we apply scaling laws from percolation theory to predict the connectivity of vein sets exposed on the southern margin of the Bristol Channel Basin. Two vein sets in a limestone bed interbedded with shales on the limb of a rollover fold were analysed for length, spacing and aperture distributions. Eight scan lines, low-level aerial photographs and mosaics of photographs taken with a tripod were used. The analysed veins formed contemporaneously with the rollover fold during basin subsidence on the hanging wall of a listric normal fault. The first vein set, V1, is fold axis-parallel (i.e. striking ∼100 ◦ )a nd normal to bedding. The second vein set, V2, strikes 140 ◦ and crosscuts V1. We find a close agreement in connectivity between our predictions using the percolation approach and the field data. The implication is that reasonable predictions of vein connectivity can be made from sparse data obtained from boreholes or (limited) sporadic outcrop.

Palaeostress orientation inferred from surface morphology of joints on the southern margin of the Bristol Channel Basin, UK

Abstract A field study of the orientation of surface features on joint planes has been carried out on the north Somerset coast along the southern margin of the Bristol Channel Basin (BCB), with the aim of inferring the palaeostress orientations. Surface features are very delicate structures on joint surfaces and when preserved give important clues regarding the direction of propagation of a joint and the interaction between the remote and local stresses. In this chapter surface features (i.e. plumose structures and twist-hackle fringes) observed on joints in Liassic limestone beds interbedded with shales from the BCB are described and their implications regarding the evolution of the stress field operating during their formation considered. The results are then compared with the generally accepted model for the region of an anticlockwise rotation of the stress field. The incompatibility between N-S-directed remote compression and continually changing local stress caused by the local structures (folds and faults), inhomogeneities and the anisotropy created by rapidly alternating limestone and shale beds resulted in the master joints breaking down into en echelon cracks (twist-hackle fringes). Joints in thicker limestone beds show well-developed plumose structures and twist-hackle fringes, whereas in thinner beds the fringe zones extend from the top to the bottom of the bed. It is found that regardless of the orientation of the parent joint in which the twist-hackle fringes are contained, they generally strike E-W and show both right- and left-stepping arrangements.

Fracture-pattern variations around a major fold and their implications regarding fracture prediction using limited data: an example from the Bristol Channel Basin

Abstract This chapter focuses on the evolution of fractures during the inversion of the Bristol Channel Basin, and examines the lateral and vertical consistency of the resulting fracture network within the alternating Liassic limestones and shales. The study has two principal aims. The first is to determine the reliability of fracture systems deduced using more limited data from less well-exposed regions or unexposed regions sampled only by drilling, and the second is to assess whether the fractures are linked to a regional stress field or are the result of a local stress field controlled by the geometry and mechanism of formation of a fold. The joint patterns were studied using a combination of scanline and window sampling, and the results indicate that there are considerable variations in the fracture systems between adjacent limestone beds and also lateral variation within the same bed. Although there is little doubt that the independent development of fracture patterns in adjacent limestone beds is facilitated by the intervening shale horizons, which allow them to become mechanically decoupled, the reasons for these variations are still unclear.

The implications of fracture swarms in the Chalk of SE England on the tectonic history of the basin and their impact on fluid flow in high-porosity, low-permeability rocks

Abstract The Upper Cretaceous (Senonian) Chalk in Kent, SE England, is considered with the aim of establishing the tectonic history of the basin in which it was deposited, based on the chronology of fractures and an understanding of the role of these fractures in controlling fluid movement in high-porosity-low-permeability sediments. The earliest brittle structures in the study area are NE-SW-striking, flint-filled shear fractures, with dips of c. 60°, which were formed when the maximum compression (σ1) was vertical and were utilized as channels for fluid movement during flint filling. Flint also occurs along bedding planes, suggesting a diagenetic source. This phase was followed by the development of NW-SE-striking fracture swarms containing fractures ranging between vertical joints and steeply dipping hybrid fractures with acute dihedral angles of c. 40°. The absence of flint along these fractures indicates that they formed after diagenesis of the Chalk. NW-SE-striking, subvertical, regularly spaced, through-going joints then formed as a result of a NW-SE regional compression linked to the Alpine collision. The final stage in the basin history relates to the formation of bed-parallel and vertical (i.e. bed-normal), bed-restricted, systematic and unsystematic fractures associated with uplift and unloading. To model fluid flow through the fracture network present in the Chalk, a finite-element-finite-volume modelling was carried out. The fracture geometries mapped in the field were discretized using unstructured hybrid element meshes with discrete fracture representations. The permeability of fractures was calculated from the cubic law and the petrophysical properties of the rock matrix were taken from Chalk reservoirs in the North Sea. In the models, a constant pressure was applied at the top of the oil-saturated, fractured Chalk while water was injected at the base. In spite of greater density, the water preferentially displaced the oil from the fractures and migrated faster through the fracture swarms and joints than through bed-restricted fractures and the rock matrix. Almost 830f the total flow within the model occurred through the fractures. The results of the field study, combined with those of the numerical modelling, suggest that fracture swarms have a strong impact on the movement of fluids in fractured and faulted reservoirs.

Massively Parallel Sector Scale Discrete Fracture and Matrix Simulations

We have been able to solve a reservoir simulation problem which was previously thought of as intractable: We simulated multiphase displacement, including viscous, capillary, and gravitational forces, for highly resolved and geologically realistic models of naturally fractured reservoirs (NFR) at the sector, i.e. kilometre, scale with very reasonable runtime. This has been possible because we used massive parallelisation and hierarchical solvers in conjunction with a new discrete fracture and matrix modelling (DFM) technique that is based on mixed-dimensional unstructured hybrid-element discretisations. High-resolution DFM simulations are important to resolve the non-linear coupling of small scale capillary - viscous and large scale gravitational - viscous processes adequately for sector scale NFR. Cross-scale process coupling in NFR controls oil recovery and NFR often exhibit power-law fracture length distributions, i.e. they do not possess an REV, and highly permeable fractures can extend over the full hydrocarbon column height. As a consequence, emergent displacement patterns have been observed which are difficult to quantify using traditional means of upscaling. However, such patterns could now be used as benchmarks to reach a better consensus on the correctness of promising new upscaling techniques. The parallel DFM technologies presented here allow us to to obtain these results much more efficiently and hence explore the parameter space in greater detail. We observed a linear scaling behaviour for up to 64 processes and a significant decrease in runtime when applying our parallel DFM approach to three highly refined NFR simulations. These contain thousands of fractures, up to 5 million elements, and have local grid-refinements below 1 m for model dimensions between I and 10 kilometres. We achieved this excellent speedup because we reduced inter-processor communication by minimising the overlap between individual domains and decreased idle time of individual processors by distributing the number of unknowns equally among the processors.

Comparison Between Natural and DFN Models in Water-flooding Numerical Simulations

this method is error prone due to post-burial processes. Mineralised fractures (vein) attributes are good candidates as analogues for open fracture formed under in-situ conditions and to establish the relationship between length versus aperture and help to reveal the conditions at the time of their formation. Vein attributes were used to condition stochastic generation of fractures using the Discrete Fracture Network (DFN) code, FracMan. Comparison of water breakthrough time and oil saturation at breakthrough were then determined by applying constant pressure gradient for each realisation to simulate water-flooding numerical simulation using the combined finite element – finite volume method (FEM – FVM). Stochastic realisations were compared with the actual system. The results indicate that, depending on the variability of the above attributes, fluid flow in fractured rocks vary from pervasive where the fractures play subordinate role to a highly localised flow where most of the flow occurs through single or connected network.