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Summary The permeability field in a reservoir simulation greatly influences the resulting flow field and therefore a thorough knowledge of it is crucial. However, the permeability field is usually associated with a high degree of uncertainty since only few measurements of reservoir properties are available. Fractures can form highly conductive shortcuts through the matrix domain. Therefore, it is important to estimate fracture parameters such as location, orientation and size as precisely as possible. Ensemble Kalman filters (EnKF) are widely used for history matching (or data assimilation) in the context of sub-surface flows in order to estimate parameters, reduce uncertainty and improve simulation results. This work studies the evolution of a reservoir as it might occur e.g. during reservoir stimulation of a geothermal system. During the first stage, large isolated fractures with a preferred orientation arise one after the other. During the second stage, these fractures get connected by others, which have a different preferred orientation. We assume that location, orientation and length of all fractures are known a priori. The only uncertainty therefore lies in the hydraulic aperture of each fracture segment. Further we assume that prior probabilistic knowledge of the hydraulic aperture is available, e.g. from seismic measurements. We upscale the fractures and simulate the flow in the reservoir with a single-continuum model. We reduce the uncertainty of the hydraulic apertures with an iterative EnKF using empirical measurement data; here from a reference simulation. During the formation of the fractures, we use pressure and flow at in- and outlet boundaries as measurements. Once the whole reservoir is developed, a tracer is injected at the inlet and its concentration at the outlet boundary is used as measurement. In this context also the effect of different fracture-matrix permeability ratios is studied.

Summary Embedded Discrete Fracture Models (EDFMs) for fractured porous media are preferable over Discrete Fracture Models if complex fracture geometries are to be fully resolved and the fractures and matrix discretizations are conformal. Lagrangian particle-tracking schemes offer convenient means for solute transport modeling because in EDFM frameworks, an orthogonal grid can be used irrespective of the fracture geometries. However, the absence of resolved fracture-matrix interfaces and different dimensionalities of the matrix and fracture continua motivate the use of a stochastic framework for particle-tracking. In this work, we developed a stochastic, time-adaptive particle-tracking scheme for EDFM models of fractured media with a permeable matrix. We formulated the probabilities of inter-continuum particle transfer, which have dependency on the particle travel time through the matrix/fracture control volumes. We showcase the conservative nature of the proposed particle-tracking scheme and additionally, illustrate the estimation of averaged solute concentration field. Such an illustration hints at the potential extensions of the tracking scheme, e.g., modeling of solute transport with kinetic reactions, and its incorporation into random walk models for dispersion in fractured media.

A new approach for uncertainty assessment in porous media is devised. The goal is to study tracer or phase transport while assuming that the multi-point velocity statistics is known. The method depends on solving a transport equation for the joint probability density function (PDF) of velocity and concentration (or phase saturation). Similar PDF methods have been developed for turbulent flow simulations and proved extremely successful in providing joint statistics of species concentration and velocity required for turbulent combustion modeling. As in the case of turbulent flows, the joint PDF equation for uncertainty assessment of transport in porous media is defined over a high dimensional space (physical, velocity, concentration or phase, and time). This results in severe computational limitations, if a finite-volume, finite-difference or finite-element method is employed, which is the motivation for the use of a particle method. The crucial advantage of the PDF modeling approach is its flexibility and high level of closure. For example, no modeling is required for macro-dispersion (ensemble mean of the product of velocity-concentration fluctuations). Moreover, no assumptions are made regarding correlation length scales and unlike perturbation-based Statistical Moment Equation (SME) methods, the PDF method is not restricted to small input variance. Here, the PDF methodology is presented for incompressible single-phase tracer transport in heterogeneous porous media, but more general applications involving multi-phase transport are discussed. A detailed description of the PDF method is given and its accuracy is demonstrated by comparison with published Monte Carlo results.

A simulator with hexahedral multiblock grids has been developed to simulate flow in reservoirs with geometrically complex features (i.e., faults and wells). A reservoir model can always be gridded with multiblock grids with matching grid lines between blocks. However, this matching requirement may introduce many small cells that have little impact on the simulation results as well as unnecessarily increase the number of blocks. We thus relax the matching requirement by allowing for nonmatching grid lines at block boundaries. This generalization simplifies the construction of multiblock grids greatly as each block can now be gridded independently and patched together afterwards. For instance, a new grid around a well can be inserted into an existing stratigraphic grid. The result is much simpler grids with fewer blocks and/or cells as compared to original matched multiblock grids. The complexity shifts from the grid construction to the calculation of transmissibilities. To this end, we need an accurate, consistent discretization scheme for nonmatching grids. We first derive a flux continuous discretization scheme for convex polyhedra grids. For general geometries a prismatic interface grid is constructed that establishes cell connectivities across the unmatched interface. Applying a piecewise constant approximation, we compute transmissibilities for this interface grid and map them back to the original hexahedral cells. We analyze the accuracy of this method via comparisons to matched fine grids. The effectiveness and applicability of this new method is demonstrated with reservoir models with wells and faults.

A black oil reservoir simulator was developed for hexahedral multi-block grids. An object-oriented approach for the design and implementation of the simulator was adopted. We discuss the general features and object-oriented (O/O) design of the simulator.