Equivalence between 2D flow around impermeable barriers and 1D flow through low-permeable barriers

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Hydrodynamics in UHS

As mentioned before both aquifers and depleted gas reservoirs can be used to develop an UHS. However, the governing processes during the development period will be different for aquifers which are initially saturated only by water or brine and depleted gas reservoirs which can have a residual gas saturation.
In aquifers a gas bubble has to be created during the development period, hence, the aquifer water has to be displaced. Dependent on the external factors a continuous or step-wise expansion of the gas bubble is conceivable. The efficiency of the displacement between two almost immiscible fluids depends on several factors which are focused in section 2.1.1.
In depleted gas reservoirs some residual gas remained in the reservoir. If the influx of aquifer water into the reservoir was weak, the residual gas saturation could almost corresponds to the initial gas saturation and only the pressure was depleted. In this case the residual gas has to be displaced by hydrogen and the pressure has to be re-increased. Different schemes are suggested in the literature for the transformation of gas storages from one gas to another, which can be also applied for the development of an UHS [123]. A simple transformation could be performed by cyclic injection and production using the same wells. In other transformation schemes hydrogen is injected on one edge of the reservoir. Thereby, the residual gas is pushed to the other side of the reservoir or could be simultaneously produced on the opposite side. This displacement process between two completely miscible fluids is reviewed in section 2.1.2.
The operation of UHS will be done in a cyclic way with alternating periods of injection, withdrawal and idle. Depending on the energy production and demands the periods can be longer or shorter. At least a seasonal operation, where the storage is charged during the summer months and discharged during the winter months, is supposable. More frequent changes in the operation schedule are possible when the storage intends to balance electrical energy production. The storages have to provide high production rates, usually one or two orders of magnitude higher than during the depletion of a reservoir. Hence, the main driving force during the operation will be compression and expansion of the gas bubble. A certain amount of gas remains always in the reservoir as cushion gas. During this period mixing processes between different gases can still be important, e.g. when the residual gas was not completely displaced or when an alternative gas is used as cushion gas (e.g. N2 or CO2 as suggested in [123, 107, 97]). The displacement of gas by water from an aquifer potentially only plays a minor role as drive mechanism during withdrawal.

Gas-water flow

Gas and water are almost immiscible fluids which form a two-phase system within the porous rock. On pore scale the phases are separated by abrupt interfaces as illustrated in Fig. 2.1. The interaction between the phases is related to forces at the fluid-solid and fluid-fluid interfaces. For a gas-water system reservoir rocks are typically water-wet. This means that water tends more to adhere to the solid surfaces than gas [6]. As a consequence of the wettability water is rather present in smaller pores while the gas phase will inhibit larger pores or channels [6]. In addition to the adhesive forces, the behavior is influenced by the interfacial tension between gas and water [6]. The combination of these forces leads to a pressure difference across the gas-water interface which is referred to as capillary pressure [6]. For a gas-water system the capillary pressure results that the gas pressure is higher than the water pressure. The mentioned pore scale effects influence also the flow of the fluid phases. Not the entire pore space is available for each phase and consequently the flow of the phases interferes with each other [6]. To be able to flow each phase needs to have a continuous pathway through the porous medium [6].
On macroscopic scale the two-phase system can be averaged over an repre-sentative elementary volume (REV) (cf. Fig. 2.2). The pore space occupied by each phase divided through the total pore space is referred to as satu-ration [6]. The ability for each phase to flow through the porous medium is described by the concept of relative permeability which was derived from laboratory experiments [6]. The relative permeability is the ratio between the effective permeability for one fluid phase and the absolute permeability as a function of saturation [6]. As a consequence of the necessity for a con-tinuous flowpath for a phase, the relative permeability will become zero at some critical value for the saturation which is still larger than zero. This sat-uration is referred to as residual saturation [59]. The displacement processes when the wetting phase (water) displaces the non-wetting phase (gas) and vise versa have different characteristics [6]. During imbibition the wetting phase penetrates into the larger pores or channels while during drainage the non-wetting phase displaces the wetting phase in the smaller pores [59]. As a result the relative permeability curves as a function of saturation have dif-ferent shapes for the wetting and for the non-wetting phase. In both cases some amount of the displaced fluid phase will be left behind.
The displacement problem between two immiscible fluids on macroscopic scale was solved by Buckley and Leverett in a simplified form [19]. They derived an analytical solution for the saturation in a one-dimensional domain when the displacing fluid is injected from one side. The solution consists of a displacement front followed by a rarefaction wave (cf. Fig. 2.3). The value Figure 2.3: Saturation versus distance in two-phase displacement process: The example shows the displacement of oil by water what is qualitatively equal to the displacement of water by gas [31] of saturation behind the displacement front depends on the mobility ratio between the phases. The characteristics of this behavior were also confirmed by laboratory flooding experiments [19].

Mixing in gas-gas displacement

The mixing between gases with different composition plays a major role when a depleted gas reservoir is transformed to an UHS or when an al-ternative cushion gas is used. The principle difference compared to the mechanism of gas-water flow is the fact that there is no abrupt transition between the different gases at pore scale. In a one-dimensional displacement this leads on one hand to a complete or piston-like displacement where no initial gas is left behind the front [15]. On the other hand the miscibility leads to a smearing out of the front due to diffusive or dispersive flux at the front [123]. In Fig. 2.7 the behavior of a miscible displacement is shown for different Peclet numbers. A high Peclet number means that the diffusive flux has a low influence and the displacement front is piston-like. In con-trast, a small Peclet number means that the diffusive flux has an significant influence which results in a strong smearing out of the front. In a more-dimensional system the overall process is influenced by heterogeneities and anisotropies of the porous medium and mobility ratios, density differences, molecular diffusion and mechanical dispersion between the different gases [123].
Mobility differences in a gas-gas displacement arise mainly due to different dynamic viscosities. Hydrogen has a very low viscosity which results in a mobility ratio around 1.5 for the system H2-C H4 and 4 for the system H2-C O2. This could result again in an instable displacement when hydrogen is injected to displace another gas. However, this effect is much less than is the system of gas-water displacement because the miscibility leads to a high dispersion of the front which acts as stabilizing force [62].
Density differences are in the range of ∼80% for the system H2-C H4 and ∼95% for the system H2-C O2. The effect can have a negative influence when the injection aims to displace another gas but instead gravity override oc-curs. However, the effect can be also used to keep different gases segregated, e.g. when C O2 is used as cushion gas [123].
Ddiff 1⋅10−6m2/s. The effective molecular diffusion coefficient depends on poros-ity, saturation state and tortuosity of the porous medium and will be less. As molecular diffusion is proportional to the concentration gradient, it will be fast at the beginning but when the concentration gradients decrease its influence will also decrease. It is independent of advective/convective trans-port, thus, it could become the governing process during idle periods.

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Cell structure of archaea and bacteria

Microorganisms are simple constructed single cell organisms without cell nucleus [30]. The main types of microorganisms are archaea and bacteria which are similar in many aspects but have some differences. Subsequently the term microorganism is used whenever a distinction between archaea and bacteria is not necessary. The usual size of a microbial cell is 0.5 to 1 µm in diameter and 1 to 2 µm in length [84]. The cell contains a full organism with all functionalities like orientation, mobility, nutrient intake, digestion and reproduction [30]. In Fig. 2.8 the structure of a microbial cell is sketched. The cell is constructed out of a cyst which encloses the cytoplasm. The cyst consists of a cell wall and membranes. The cell wall is relevant for the shape and compression resistance of the cell [30]. The cytoplasm membrane is impermeable for most chemical components but let some components which are required for the bio-chemical processes pass. The cytoplasm encloses the chromosome, ribosomes and additional substances which are used for different bio-chemical processes [30]. The chromosome which is in contrast to eukarya not enclosed by a nucleus is referred to as nucleoid and contains the genetic information [30]. The difference between archaea and bacteria is the material out of which the cell walls and membranes consist. Some microorganisms have a flagellum on one side which makes a target-oriented movement possible [84]. Other cell appendixes, i.e. pili, can be used for attachment processes (cf. section 2.2.5) [47].

Cell duplication and metabolism

Microbial cells reproduce themselves by cell division as illustrated in Fig. 2.9 [30]. The process consists of several steps. At first the chromosome is duplicated [30]. Then, the cell is growing to the double size [30]. Finally, a ring-shaped contraction of the cell wall starts in the center and the cell is separated into two individual cells [30].
The process of cell duplication requires a series of chemical reactions within the microbial cell which is referred to as microbial metabolism [84]. The aim of metabolism is the increase of cell mass and finally the cell duplication. However, it has to be differentiated between growth metabolism and non-growth metabolism [84]. During non-growth metabolism the microbial cell does not increase its mass or duplicate but substrates and nutrients are still consumed for the maintenance of the cell structure.
For both types of metabolism the microorganism needs a source of energy and a source of carbon [84]. Under atmospheric conditions microorganisms tend to use a redox reaction (chemotrophic) by using oxygen or light (pho-totrophic) as the source of energy [84]. Under anaerobic conditions, e.g. in subsurface formations, they need to use a less energy efficient redox reaction. Different organic or inorganic substances can be used in this process as elec-tron donor [84]. In UHS it appears reasonable that the microorganisms use hydrogen as the electron donor. In the literature these types of metabolism are referred to as hydrogenotrophic. The most important types are sum-marized in section 2.2.4. The source of carbon can be also differentiated as organic source (heterotrophic) or as inorganic source (autotrophic) [84]. An inorganic source of carbon would be C O2.

Table of contents :

1 Introduction 
1.1 Motivation
1.2 Underground hydrogen storage
1.3 State of the art (applications)
1.4 Recent related research projects
1.4.1 H2STORE
1.4.2 HyUnder
1.4.3 Hychico
1.4.4 ANGUS+
1.4.5 Underground Sun Storage
1.4.6 HyINTEGER
1.5 Outline of the thesis
2 Literature review 
2.1 Hydrodynamics in UHS
2.1.1 Gas-water flow
2.1.2 Mixing in gas-gas displacement
2.2 Microbiology in UHS
2.2.1 Cell structure of archaea and bacteria
2.2.2 Cell duplication and metabolism
2.2.3 Microbial populations and growth
2.2.4 Relevant microbial species in UHS
2.2.5 Microbial transport and structures in porous media
2.3 Modeling of bio-reactive transport in porous media
2.3.1 Single-phase models for groundwater applications
2.3.2 Two-phase models for gas storage applications
2.3.3 Review of tools for (bio-)reactive transport modeling .
3 Analytical modeling of gravity-driven displacement 
3.1 Case study description
3.2 Balance equations for compositional two-phase flow
3.3 Reduction to a canonical model
3.4 Initial and boundary conditions
3.5 Hugoniot conditions, stability conditions and continuity of fractional flow
3.6 Two-component flow
3.6.1 Structure of the generalized fractional flow function .
3.6.2 Graphical construction of the solution
3.6.3 Solution before reaching the barrier
3.6.4 Solution after reaching the barrier
3.6.5 Characteristic points of the solution
3.6.6 Solution after reaching the second barrier
3.6.7 Gas rising velocity and growth velocity of gas accumulations
3.6.8 Comparison with immiscible two-phase flow
3.7 Three-component flow
3.7.1 Thermodynamics of three-component mixtures
3.7.2 Canonical model for three-component flow
3.7.3 Structure of the solution
3.7.4 Solution of the problem before reaching the barrier
3.7.5 Moment of reaching the barrier
3.7.6 Evolution of the reverse wave under the barrier
3.8 Comparison to two-dimensional problem
3.8.1 Formulation of 2D problem
3.8.2 Numerical implementation
3.8.3 Equivalence between 2D flow around impermeable barriers and 1D flow through low-permeable barriers
3.9 Summary and conclusions
4 Mathematical model for bio-reactive two-phase transport 
4.1 Physico-chemical processes
4.2 Bio-chemical processes
4.3 Review of models for microbial growth and decay
4.4 Coupling of processes
4.5 Transformation to dimensionless form
4.6 Parameters for microbial population dynamics
4.7 Summary and conclusions
5 Stability of the dynamic system 
5.1 Reduction to a system of two ordinary differential equations .
5.2 Linear stability analysis
5.3 Turing conditions
5.4 Numerical simulations
5.4.1 Limit cycle behavior
5.4.2 Turing instability
5.5 Summary and conclusions
6 Numerical modeling of storage cycles 
6.1 Numerical implementation
6.1.1 Spatial and temporal discretization
6.1.2 Iterative procedure
6.1.3 Adaptation of the governing equation system
6.1.4 Phase equilibrium and hydrodynamic parameters
6.1.5 Grid generation and rock parameters
6.1.6 Adjustments for heterogeneous corner-point grids
6.1.7 Storage initialization
6.1.8 Modeling of storage wells
6.1.8.1 Rate-controlled storage wells
6.1.8.2 Pressure-controlled storage wells
6.1.9 Operation schedule and time stepping
6.1.10 Material balance error
6.1.11 Post-processing and visualization
6.2 Case studies
6.2.1 Gas-water displacement in a 2D synthetic reservoir .
6.2.2 Storage scenario in a 2D synthetic reservoir
6.2.3 Storage scenario in a 3D realistic reservoir
6.3 Summary and conclusions
7 Conclusions 
Bibliography 

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