CO 2 - Water System

The second example refers to a CO2leakage through a network of fractures. The example is again a fictitious test scenario. However, the soil parameters given in Table 5.5 are “more realistic” in the sense that the entry pressures in the matrix and fracture is slightly larger. For this reason, in comparison with the previous example,

Figure 5.7: Bristol problem: Statistical distribution of the matrix block following a log-normal distribution (from Geiger et al. [2011]).

Figure 5.8: Bristol problem: Pressure distribution after 1e5 seconds obtained from the intra-coarse block transmissibility determination.

the CO2is not able to enter the matrix blocks, as it can be observed from Figure 5.12.

The propagation of the CO2in gaseous phase through the interconnected fracture system is correctly approximated by the first fracture continuum. The non-wetting phase reaches the right boundary after approximately 740 seconds (Figure 5.12 a-d). After that the saturation continues to increase in the fracture without entering the blocks (Figure 5.12 e-h). For a more realistic soil parameters, meaning higher difference in the entry pressures of the fracture and matrix, the non-wetting fluid phase can not enter the rock matrix anymore and will propagate through the fracture network system. Several boundry and initial conditions, fluid systems and soil

(a) t=2541.7 sec (b) t=9117.8 sec (c) t=2541.7 sec

(d) t=9.84e+05 sec (e) t=1.641+06 sec (f) t=7.521+06 sec

(g) t=2.0167+07 sec (h) t=2.0167+07 sec (i) steady state

Figure 5.9: Bristol air-water problem: Non-wetting phase saturation profiles plotted over the diagonal for Bristol Channel fractured reservoir analogue

Table 5.4: Simulation times and grid specifications

Experiment Model Vertex No Element No Sim. Time [sec]

Ex.1: Permeameter 2pDFM 37269 73847 15517.6

2pMINC 21×21 20×20 421.3

Ex.2: Fivespot 2pDFM 37269 73847 122483.6

2pMINC 21×21 20×20 2046

Ex.3: Bristol 2pDFM 42490 84348 331364

2pMINC 19×9 18×8 180

(a) t=7000 sec

(b) t=5.90e+05 sec

(a) steady state

Figure 5.10: Bristol air-water problem: Non-wetting phase saturation spatial distri-bution with DFM and MINC model

Figure 5.11: Bristol air-water problem: Mass fluxes of wetting and non-wetting phase plotted over time at line (x = 1.8m) obtained with2pDFMand 2pMINCsimulators

Table 5.5: Domain and fluid properties for CO2-Water system problem Domain Properties

matrix fracture

Permeability,K [m²] 1.0e−15 1.0e−11

Eff. porosity,φ [-] 0.25 0.90

Entry pressure,pd [Pa] 2000 1000

Pore size dist. idx.λ [-] 2.0 2.0

Residual saturationSwr [-] 0.0 0.0

Residual saturationSnr [-] 0.0 0.0

Fracture apertureb [m] - 1.0e−3

TemperatureT [^◦K] 293.15 293.15

Fluid Properties Viscosity Water,µw [kg·m⁻¹·s⁻¹] 1.0e−3

Density Water,ρw [kg·m⁻³] 1000

Viscosity CO2,µn [kg·m⁻¹·s⁻¹] f(p,T)≈1.6e−5 Density CO2,ρn [kg·m⁻³] ^p_RT^nM≈1.9

Boundary Conditions Left[Dirichlet]:

pw,Sn [Pa], [-] pw=2.0e+07 Sn=0.90

Right[Dirichlet]:

pw,Sn [Pa], [-] pw=1.0e+07 Sn=0.10

ELSE [No Flow]

Initial Conditions [Pa], [-] pw=1.0e+07 Sn=0.10

parameters have been modeled in the process of2pMINCsimulator validation. Some of the setups, which were sometimes ”realistic problems” , were harder to be solved and lead to convergence problems in the DFM and rarely in the extended MINC model. However, the extended MINC method represents a trustworthy option as it is able to approximate the fluxes and the flow variables in each region of the domain and at each time.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 5.12: Bristol CO2 - Water system problem: Non-wetting phase saturation profiles plotted over the diagonal for Bristol Channel fractured reservoir analogue

Summary

As more and more engineering applications require the correct simulation of flow and transport processes in porous media, and while many of these media present a certain degree of fracturing, this work deals with the development of numerical models that can simulate two-phase flow in large-scale fractured reservoirs. Among the applications which these models are addressing to, there are the estimation of contaminant spreading and removal, the reservoir exploitation, or more recently the CO2sequestration, the geothermal reservoir exploitation, and the nuclear waste repositories.

Fractured systems can occur on a variety of lengths and scales which makes difficult the development of a general model that can handle easily all of them. First, the fundamental definitions and concepts of flow and transport in porous media are in-troduced. They refer to the properties of fluid (e.g. density, viscosity) and matrix (e.g.

permeability, porosity) and the ones resulting from their interaction (e.g. saturation, residual saturation, interfacial tension, capillary pressure and relative permeabil-ity). After that, the focus is set to give a general overview of fractured systems by reviewing the relatively large available literature. Several types of fracture system characterizations are presented. Generally, they are categorized according to the scale of concern, the way of formation, or to their geometrical features. Following that, the main fluid flow fracture models are described in detail, i.e. discrete (DFM) and continuum (CFM). The DFMs are more suitable for the near field scale for obtaining the most accurate results, while the CFMs are best used for far and very far field scale problems. The mathematical models are created by extending the general equations of multi-phase flow in porous media to fractured porous media. The method used for solving the two-phase flow equations in fracture-matrix systems is based on a phase pressure saturation formulation and is fully coupled.

The strongly coupled non-linear system of partial differential equations is discretized spatially with the BOX-method and temporally with the fully implicit Euler scheme.

In the DFM model the fractures are represented either with equi-dimensional entities or with lower-dimensional ones in a conforming finite element setting. With an interface condition the continuity of the fluxes and of the pressure is preserved while it allows the physically discontinuous saturation at nodes shared by fracture and

matrix. A special attention is given to the interface conditions that represent the saturation discontinuities in lower-dimensional elements which require to express the primary variables of the matrix domain in the shared nodes with the ones of the fracture.

For modeling large-scale fractured reservoirs it is important to come with efficient practical strategies that can reduce as much as possible the computation time while keeping the desired accuracy, therefore CFMs have proved to be the right approach for this cases. The CFMs use representative elementary volumes larger than the fracture length scale with averaged properties that can handle easily the matrix-fracture interactions through a source or sink term. For the purpose of this research the double-continuum models (DCMs) are chosen among the CFMs and are pre-sented in detail. It is shown how the limitations of the standard DCMs have lead to the development of the multiple interacting continua (MINC) method which is, in fact, a generalization of the double-porosity single-permeability model. The MINC method is able to take into account the hydrodynamic answer of the rock matrix by discretizing the flow domain inside each coarse MINC block into several nested volume elements associated with an average thermodynamic state. The standard MINC method is extended with an upscaling technique that allows the determina-tion of the effective parameters of the nested volume elements from discrete fracture representations. This procedure is performed once at the pre-processing level in two steps: determination of the inner-block transmissibilities and volumetric fractions and the determination of the inter-block transmissibilities. Each step requires solving with the DFM single-phase flow problems at the local scale which is a much simpler task.

The numerical models are implemented into computer code under the general frame program DuMu^x. Modeling two-phase flow with the lower-dimensional DFM is implemented into the numerical simulator2pDFM, and with the MINC method is implemented into the numerical simulator2pMINC.

After that, the workflow for efficient simulation of the two-phase flow in large-scale fractured porous media is described. It starts with the geometrical modeling of the fracture network system which is/can be created with a geostatistical fracture generator, then the generation of the optimal fine- and coarse-mesh, and ends with the numerical simulator and the post-processing.

Each of the numerical models is tested and verified. The DFM model is verified with laboratory experiments and other simulators. One important issue was to show that simulation results of lower-dimensional DFM are in good agreement with the trusted equi-dimensional DFM. Then it is used as a reference solution for the verification of the MINC approximate model. Another investigated issue is the influence of the grid refinement on the solution.

For the MINC model six verification examples are designed and prove that the

model can be trustworthy. Following that, the extended MINC model is applied to several field scale problems. The first test is an idealized periodic fracture system.

The effective parameters of the extended MINC method , as well as the reference solution are computed with the DFM. The tests investigate the influence of the flow direction with respect to fracture orientation on the saturation distributions and front propagation speed. In the first test, “permeameter type experiment”, the flow is parallel to the direction of the horizontal fractures, whereas in the second experiment, the ”quarter five-spot problem”, it is diagonal to the fracture orientation.

Both examples show a reasonably good match for the saturation distributions and the front propagation speed. The MINC method allows to predict the saturation inside the matrix block if the entry pressure is overcome. The next example is a naturally fractured reservoir from Bristol with a highly connected fracture system. The tests showed that the2pMINCsimulator can represent the fluxes and saturation profiles obtained with the discrete2pDFMsimulator, giving very good speedup factors. In the same time, both the matrix-fracture transfer and entry pressure effects are correctly estimated and are based on a physical approach.

The new and innovative points of this research are:

• the DFM-L accounts correctly for the storage term in the fractures which was not specifically taken into account in many of the the previous models (e.g.

Reichenberger et al. [2006], Kim and Deo [2000]) together with the extended capillary pressure-saturation interface condition.

• the classic DP and MINC models were used especially in oil reservoir engi-neering for estimating the recovery rates and were not specifically looking at the fluids distributions in the reservoir. The current study extends the view by looking at the spatial distribution of saturations, at front propagation speed and at the recovery rates/fluxes.

• in the oil reservoir engineering the main process is the water imbibition and therefore the entry pressure effects do not play a major role. This research extends the applicability range of the upscaling procedure investigated by Karimi-Fard et al. [2006] and Gong et al. [2008] to processes of non-wetting fluid phase infiltration in an initially wetting phase saturated or partially saturated domain.

• a key point of the work dealt with constructing a new set of comprehensive model tests for the DFM-L and extended MINC method which are able to test, verify and validate them. In the literature there is no clear description of techniques to validate the MINC method therefore this research can be also used as a guide.

• the constructed workflow creates a flexible frame for modeling two-phase flow in fractures of arbitrary sizes, orientation and shapes. Another novelty aspect

of this work is that it tries to incorporate all stages involved in the modeling process: geometry creation, handling, meshing, flow simulation and result interpretation.

Concluding Remarks

• The DFM-L can accurately reproduce the reference solutions of DFM-E in terms of saturation distribution, pressures, fluxes, front propagation speed and entry pressure effects.

• The DFM-L yields systems which are much easier to solve than those of equi-dimensional DFM. The DFM-L does not require gridding inside the fracture and, therefore, it avoids small elements inside the fractures.

• The DFM-L can model flow in fractures of arbitrary sizes, orientations (e.g.

perpendicular or parralel to flow direction) and shapes or in fracture network systems.

• The BOX method is locally mass conservative and can easily be applied to unstructured grids.

• The MINC method can take into account the matrix influence which plays a great role in the large-scale multi-phase flow problems. This is important for the evaluation of the storage capacity and the behavior of a reservoir.

• The extended MINC method is flexible because it benefits from both the geo-metrical complexity of the DFM and the computational simplicity of the CFM, which is a compromise between accuracy and computation speed. This is left at the will of the modeler as the fractures can be mapped on computationally efficient grids. The ability to combine the advantages of the continuum and discrete fracture model concepts allows to appropriately transfer the flow char-acteristics of individual fractures to effective parameters assigned to coarse blocks while preserving the flow and transport features. The flexibility also refers to the ability of choosing the accuracy of the solution, the computation speed, the subgridding method (e.g. constant volume fractions, equi-distant volume fractions, etc.), the ability of working with spatial information about the fracture system of various complexity and detail.

• The extended MINC simulation method is general because it allows handling of densely fractured systems with irregular geometries as well as of the regions where fractures have lower connectivity.

• The extended MINC method preserves the total flow characteristics and their spatial and temporal distribution.

• The extended MINC method can accurately reproduce the reference solutions of DFM-E and DFM-L in terms of saturation distribution, pressures, fluxes, front propagation speed and entry pressure effects.

• Even though the most accurate solutions are obtained with a DFM approach, which can represent the best choice for the future, considering the increase in computational power, an accurate, practical tool with high flexibility still remains a good alternative.

Questions and Future Work

There are a number of improvements and open questions that arise and should be answered in the future.

• For being able to run a fully realistic simulation of a storage reservoir the current 2pMINCand2pDFMsimulators should be extended to 3D.

• When dealing with more complex heterogeneities at the large-scale the MINC model should also be extended to unstructured grids.

• Many processes in reservoir engineering (e.g. CO2sequestration) are more complex and can not be fully described only with the two-phase flow equa-tions. Therefore, the2pDFMflow simulator should be extended to include non-isothermal, multi-component transport processes.

• Some fractured reservoirs present regions that have vugs and fractures ex-tending on multiple scales. For these cases the DPSP might no more be valid and the MINC method has to be extended to account for the multi-porosity multi-permeability systems. As shown the extended MINC method could be applied successfully to these systems however further testing is required.

• The assumption that the relative permeability functions can be transferred at large-scale MINC model from the fine-scale might require further investigation.

• In the shown examples the gravity has been neglected inside the coarse blocks.

A further extension should consider these effects.

• The DFM is working under the assumptions that the fractures have permeabil-ities which are order of magnitude higher than those of the matrix and that the fracture capillary pressure curves are always below the ones of the matrix.

However, in real applications negative or blocking fractures can occur. The interface condition has to be then reevaluated.

• Fractures can also exhibit regions where the Darcy law is not valid. For these cases better descriptions might be obtained with Forchheimer’s law (Frih et al.

[2008]) .

• Even though in principle the extended MINC model developed in this research can be applied for solving any kind of multi-phase flow problem in fractured porous media, it is not intended to be a universal approach. As in every other approach, it has advantages and disadvantages and there is room for improvement.

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