Numerical simulation of hydrodynamic and gas mixing processes in underground hydrogen storages

In aquifers a gas bubble is created during the development period; hence, the aquifer water is displaced. This displacement process could be inefficient because hydrogen has a low viscosity and very low density in contrast to water. The mobility ratio between hydrogen gas and water is estimated to be in the order of 2–5 which is unfavorable (Ho and Webb 2006). Consequences could be the occurrence of strongly pronounced viscous fingering and gravity override of the water phase, which was investigated in Tek (1989) and Paterson (1983). Paterson (1983) concluded from theoretical considerations that the viscous fingers could spread very far laterally below the cap rock and hydrogen could get lost beyond the spill point of the structure. In Tek (1989) the conditions for stable and instable displacement were derived mathematically. Both works conclude that the injection rate can be used as a control parameter for viscous fingering or gravity override. If the injection rate is low, the gravitational and capillary forces will be stronger than the viscous forces and hence, the displacement becomes more stable. In Hagemann et al. (2015) these findings were confirmed by numerical simulations. However, the simulation results show that the development period, using a low injection rate, could take several years. The required time depends on the aquifer geometry, the hydraulic properties of the reservoir rock and the hydrodynamic properties of the fluids. All these phenomena are well known from natural gas storages in aquifers, but with hydrogen they will be more prominent.

In depleted gas reservoirs, some residual gas remains in the reservoir. If the influx of aquifer water into the reservoir was weak, the residual gas saturation almost corresponds to the initial gas saturation and only the pressure was depleted. A problem could be the displacement of the residual gas by hydrogen. Different schemes are suggested in the literature for the transformation of gas storages from one gas to another, which can also be applied for the development of an UHS (Tek 1989). A simple transformation by cyclic injection and production using the same wells could result in a highly contaminated gas production, especially in the first years of operation. In other transformation schemes, hydrogen is injected on one edge of the reservoir. Therefore, the residual gas is pushed to the other side of the reservoir or could be simultaneously produced on the opposite side. This sweeping strategy potentially results in a more pure hydrogen production; however, a mixing between the different gases is inevitable.

In Tek (1989) and Pfeiffer and Bauer (2015) it was suggested to inject nitrogen during the development period of an aquifer storage site. Despite the lower investment costs for nitrogen as cushion gas, it has a higher viscosity and density than hydrogen and even than methane. Hence, the displacement of water is more efficient. The disadvantage is the intensive mixing of hydrogen and nitrogen when the cyclic operation is started. The simulation study in Pfeiffer and Bauer (2015) has shown that the hydrogen concentration in the produced gas stream is only around 50–80 % during the first years. In the same way as mentioned before, a different cushion gas can also be used in depleted gas reservoirs. It seems reasonable to use the already available natural gas as part of the cushion gas, while further required cushion gas will be delivered by the injected gas. The density difference could be utilized to separate the gases by injecting hydrogen into the top of the structure (Tek 1989). In Oldenburg (2003), carbon dioxide was suggested to be used as cushion gas. In this case, the density segregation would be relatively strong, because at typical reservoir conditions, carbon dioxide is very dense in comparison with hydrogen.

UHS will be operated 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. Similar to underground gas storages a seasonal operation, where the storage is charged during the summer months and discharged during the winter months, is conceivable. If the UHS is used to balance electrical energy, more frequent changes in the operation are possible dependent on the weather conditions. The storage process has to provide high production rates, usually one or two orders of magnitude higher than the depletion process 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 always remains in the reservoir as cushion gas. The displacement of gas by water from an aquifer potentially plays only a minor role as drive mechanism during production. It must be ensured that the produced gas stream fulfills the requirements of pureness and additionally a low water cut is favorable.

A completely different operation strategy, called the “selective technology” was suggested in Hagemann et al. (2015) and Panfilov (2010). The idea is to use an aquifer where horizontal impermeable or almost impermeable barriers exist. The storage is operated by two different well systems. The first system of wells is used to inject hydrogen into the bottom of the reservoir. The hydrogen starts to rise, because of buoyancy forces in the water phase. The rising gas is retarded at several barriers until it flows around or through these barriers. When the hydrogen reaches the cap rock of the reservoir, it is produced by the second system of wells, before it has the chance to spread laterally. The complexity of this operation strategy is the planning of storage time which corresponds to the characteristic time of hydrogen rising from the bottom to the top.

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–CH₄ and 4 for the system H₂–CO₂. This could result in an instable displacement when hydrogen is injected to displace another gas. However, this effect is much less than in a case of a gas–water displacement because the miscibility leads to a high dispersion of the front which acts as stabilizing force (Ho and Webb 2006).

Hydrogen has a very small molecular mass which results in large density differences compared to other gases. The effect can have negative influence when the injection aims to displace another gas but gravity override occurs. However, as mentioned before, the effect can be also used to keep different gases segregated, e.g., when CO₂ is used as cushion gas (Tek 1989).

Molecular diffusion is generally considered as a slow process when compared to advective/convective transport (Tek 1989). The molecular diffusion coefficient \(\tilde{D}_{\text {diff}}\) for hydrogen in the gaseous state is relatively high, in the order of \(1\times 10^{-6}\) m\(^{2}/\)s. The effective molecular diffusion coefficient depends on porosity, saturation state and tortuosity of the porous medium. 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 transport, thus, it could become the governing process during idle periods.

Mechanical dispersion, in contrast, is a mixing process which takes place due to the movement of fluids in the porous medium. It arises from variations in the velocity which can occur on different scales, ranging from microscopic to reservoir scale (Tek 1989). The mechanical dispersion coefficient \(\tilde{D}_{\text {disp}}\) depends on the velocity and direction of flow and can be formulated as follows (Scheidegger 1958):where a is the dispersivity in (m), \(\Vert v\Vert\) is the flow velocity in the principle direction, the subscript L refers the longitudinal direction and the subscript T refers the transverse direction. The dispersivity of the porous medium depends on its tortuosity and heterogeneity. However, the experimental measured values vary by several orders of magnitude. As the process is strongly depending on the considered scale, laboratory measurements cannot be directly transferred to reservoir scale (Tek 1989). Tracer tests, which have been performed on reservoir scale, have shown that the longitudinal dispersivity is between 1 and 100 m (Tek 1989; Carriere et al. 1985; Laille et al. 1986). The transverse dispersivity is usually one or two orders smaller. Assuming flow velocities of several meters per day, which are common in gas storages, the longitudinal mechanical dispersion coefficient will be around \(5\times 10^{-4}\) m\(^{2}/\)s. Hence, the mixing by mechanical dispersion is expected to be much more pronounced than only by molecular diffusion.

In a porous medium, it is possible to quantitatively estimate the efficiency of a displacement process by considering the microscopic and macroscopic displacement efficiency. The dimensionless measure E is therefore the volume fraction of the displacing fluid to the total pore volume.

The microscopic displacement efficiency \(E_{\text {D}}\) describes the displacement of the initial reservoir fluid at pore scale. According to Terry (2001), this factor is mainly influenced by the interfacial and surface tensions between the displacing and displaced fluid, the reservoir wettability as well as the shape of the relative permeabilities curves.

The simulation cases implemented in this paper are focusing on the macroscopic displacement efficiency, \(E_{\text {V}}\), which expresses the capability of the displacing fluid to mobilize the initial fluid in a volumetric sense. In this context, the allocation of the wells, the reservoir permeability distribution, gravity forces and the mobility of the initial and injected fluid are significant. The injection of a less viscous and dense fluid into a reservoir can provoke the appearance of at least two physical effects which interfere with the macroscopic displacement efficiency.

As a result of buoyancy, two immiscible fluids inside a closed system tend to dispose according to their density. In comparison with cavern storage operations, gravity segregation in a porous subsurface medium is normally attenuated. The sequence of geological deposition typically leads to a much higher horizontal than vertical permeability, which restricts fluid migration across the layering. However, the very small molecular mass of hydrogen causes a large density contrast of the occurring fluids so that an amplified hydrogen accumulation below the highest sealing layer has to be assumed. In order to obtain a qualitative impression of gravity override, two examples of hydrogen injection into a gas saturated (Fig. 1a) and water saturated (Fig. 1b) reservoir were simulated. Thereby, a two-dimensional cross section of the later introduced reservoir model was adapted by standardizing the grid dimensions and increasing the amount of grid cells in z-direction. The horizontal permeability is defined as 100 mD, while the vertical permeability is 10 times smaller. Hydrogen is constantly injected into the source placed at the left part of the reservoir, while the native fluid is produced from the sink placed at the right part of the structure. The simultaneous operation of an injection and a production well allows the pressure balancing. In the first case, the reservoir is initially saturated with 98 % gas, composed of approximately 80 mol% nitrogen and 20 mol% methane.

The injection of hydrogen leads to a relatively uniform displacement although the effect of gravity segregation is already indicated. In the second case, hydrogen is injected into a fully water saturated reservoir, which causes a very poor and unilateral displacement of the native reservoir fluid. The effect of gravity override is a function of the occurring density contrast. At the initial reservoir conditions of 170 bar and 125 \(^\circ\)C, hydrogen is roughly 75 kg/m\(^3\) less dense than methane and approximately 125 kg/m\(^3\) lighter than nitrogen. This relation becomes even more unfavorable in case hydrogen is injected into an aquifer, where a density difference between hydrogen and water of 937 kg/m\(^3\) has to be expected (NIST 2016). Besides the density contrast, gravity override is furthermore promoted by small injection rates and large vertical permeabilities (Terry 2001).

Besides gravity override, viscous fingering is another undesired physical phenomenon which is commonly linked to gas injection processes. The adverse relation between a displacing highly mobile fluid and a displaced sluggish native fluid promotes a unilateral displacement. The arising finger-shaped front can be provoked by several factors and can additionally occur on different stages. Besides viscous fingering at pore and volumetric scale, it is possible to further distinguish between viscous fingering in miscible and immiscible displacement processes.

The multiphysics simulator COMSOL was used to numerically simulate the displacement front in a hydrogen–gas (Fig. 2a) and a hydrogen–water (Fig. 2b) system. In both cases, hydrogen is constantly injected from the left side of the model, while the native fluid is produced from the opposite side. The distance between the injection and production side amounts 10 m, while the pressure difference results into 0.05 bar. The arising displacement process is disturbed by a small reservoir heterogeneity which can be obtained by the four small squares next to the injection boundary. This model heterogeneity is numerically required in order to initiate viscous fingering, but simultaneously it is small enough to not restrict the flow potential. The permeability disturbance initially interferes with the hydrogen–methane displacement, but the displacement front is then stabilized by diffusion and dispersion forces. Consequently Fig. 2a shows a stable displacement front between the two gases. In case of the hydrogen–water displacement (Fig. 2b), the permeability heterogeneity generates viscous fingers which dominate the subsequent displacement process. While the gas–gas displacement is characterized by a front velocity of 5 m/day, the hydrogen–water front migrates with 0.4 m/day through the model.

DuMu\(^{x}\) offers two different spatial discretization schemes. In the previous publications, the default box method was used which is appropriate for conforming grids. For the processing of geological more complex reservoir structures in corner-point format, the more conventional cell-centered finite volume method is recommended.

Since DuMu\(^{x}\) was initially developed to compute near surface ground water processes, the expected default permeability range does not cover the very small permeability values of deeper hydrocarbon reservoirs. In this case, an automatic request of a DuMu\(^{x}\) control function aborts the simulation since a physically incorrect result is misleadingly assumed. By adapting the equivalent permeability threshold value, larger value ranges can be computed.

During the adjustments from structured 2D grids to unstructured 3D reservoir applications, further numerical challenges occurred. Comprehensive geological reservoir models that are created in Petrel (Schlumberger 2016), typically consist of very irregular grid geometries and structures. Individual cells can for example deviate from the cuboid cell shape by instead assuming a rectangular prismatic shape which are also known as “degenerate cells”. These cells especially occur in those regions, where grid horizons are merging into each other. At geological faults, where an offset between the layers exits, it is additionally possible that the grid model becomes non-conforming. To meet the complexity of these irregular grid structures, some fundamental computation settings were modified. The flux between two neighboring cells depends on the difference in pressure potential and the transmissibility between the cells. The standard method in DuMu\(^{x}\) of calculating the transmissibility is accomplished by taking the face area, the harmonic mean of the two cell permeabilities and the distance between the cell centers. However, the deviating sizes and shapes of neighboring cells lead to a certain inaccuracy which is resulting in numerical instability. A better solution includes the initial calculation of the two transmissibilities from each cell center to their common face center, referred to as “half-block transmissibility” (cf. “Appendix 2”). Subsequently, the total transmissibility between the cells is calculated by taking half of the harmonic mean of these half-block transmissibilities. This option was implemented by the DuMu\(^{x}\) developers and is available since the DuMu\(^{x}\) release 2.8. Additionally, an adjustment for the cell and face center determination within the dune-corner point methods was required. While Petrel (Schlumberger 2016) exports the grid geometry by specifying the eight corner-point coordinates of a single cell, simulation programs initially determine the cell and face center points to ensure the mass balance calculation between neighboring cells. The standard setting to determine the cell and face centers is the calculation of its centroids. In contrast, our experience in DuMu\(^{x}\) indicates that the numerical accuracy and stability can be improved by determining the face center coordinates by calculating the arithmetic mean of its four corner points. The cell centers are subsequently calculated by taking the arithmetic mean of the center points of the upper and lower faces (cf. “Appendix 3”). This method is also suggested in Nilsen et al. (2012) and is believed to be common in commercial reservoir simulators. This adjustment especially helped to improve the mass transfer between degenerate cells and its neighboring cells.

The initialization of the reservoir was done in hydrostatic equilibrium where the gas–water contact (GWC) was defined at 3452 m with a pressure of 170 bar. The phase pressures and saturations were calculated by using the pressure gradients and capillary pressure relation. All performed simulations use the Neumann conditions to specify the spatial derivative of the partial differential flux equation at the boundary of the reservoir. Defining this value as zero, the boundary of the grid is set to a no-flow boundary.

Wells are represented by source or sink terms. From a mathematical point of view, they represent specific points or lines in the reservoir, where mass is either added or extracted from the closed system. Since the geographic well coordinates naturally deviate from the calculated cell center positions, it is necessary to find a reasonable approach to identify the cells which contain a well. We developed therefore a Matlab code, which automatically assigns the geographic well coordinate to the closest occurring cell center coordinate. By invoking these positions and defining a mathematical well model, the activity of wells can be numerically executed.

The injection and production of fluids is realized by implementing Peaceman’s well model. In contrast of using a constant injection or production rate, Peaceman’s well model automatically adjusts the amount of injected or extracted mass to the reservoir response. This can for example avoid an unrealistic high pressure in the near wellbore area and takes furthermore the mobility of the occurring fluids into account (Chen 2007). Certainly the size of an invoked grid cell significantly exceeds the real well diameter which sophisticates the well impact on the reservoir. Grid refinement around the well position allows a better representation of the reality but indeed requires considerable numerical expanse. In contrast, the implementation of Peaceman’s well model ensures a more simple comprise between representing the reality and defining a mathematical representation. By introducing an equivalent radius, Peaceman was able to find a connection between the occurring grid cell pressure and the bottom-hole flowing pressure. It is possible to approach the quantity of the equivalent radius, by either solving the analytical well flow model, numerically solving the pressure equation or directly calculating the yielding pressure between the well and its neighboring grid cells. The quantity of the equivalent radius amounts to approximately one-fifth of the average grid cell length. The simplest expression of Peaceman’s well model is listed below and is valid for a homogeneous reservoir and single-phase fluid flow:where Q is the injection or production rate in (kg/s), \(\rho\) is the fluid density in (kg/m\(^3\)), \(K_{xy}\) is the horizontal permeability of the grid cell containing the well in (m\(^2\)), \(h_{z}\) is the grid cell height in (m), \({\mu }\) is the fluid viscosity in (Pa s), \(r_{\text {e}}\) is the equivalent radius in (m), \(r_{\text {w}}\) is the geometrical well radius in (m) and s is the wellbore skin factor. Besides the physical behavior of the fluid density and viscosity, the difference of a defined bottom-hole flowing pressure \({P_{\text {wf}}}\) and the actual reservoir pressure P in the well grid cell adjusts the amount of the injected or extracted mass to the arising reservoir response. The multi-compositional two-phase flow formulation of our model for UHS requires the additional consideration of phase mobilities and concentrations of the components. Since our model is based on the balance of moles, a modification of the units is necessary. For injection, it is sufficient to consider the gas phase:where \(\hat{Q}\) is the injection or production rate in (mol/s) and \(c_{\text {g}}^{k,\text {inj}}\) is the composition of the injected gas. Molar density, dynamic viscosity and relative permeability are the actual values in the well grid cell. For production, both phases need to be considered:where \(c_{\text {g}}^{{k}}\) and \(c_{\text {w}}^{{k}}\) in this case are also the actual values in the grid cell containing the well. A scaling of the Dirac delta function is required to transform the point or line source to a volume source, where \(V_{\text {c}}\) is the volume of the well grid cell (cf. Eq. 18).

In order to numerically assess the technical feasibility of underground hydrogen storages, an appropriate reservoir grid was chosen whose properties resembles the typical reservoir shape of conventional gas storages. The used geological model is part of one of the largest on-shore gas fields in Europe. Due to the enormous dimension of the entire sandstone reservoir, a large amount of hydrogen is required to achieve a sufficient storage deliverability.

By cutting a much smaller prismatic fragment, whose cross sectional shape resembles a geological anticline structure, a more suitable grid model was created. With the help of Petrel (Schlumberger 2016), the initial slightly coarse grid structure was improved by reducing the grid volume by the factor 9. The arising fragment model has an approximate ground plan of 800 m times 1200 m and a gross reservoir thickness of 50 m which yields to an average grid cell size of 32 m in x, 31 m in y and 3.8 m in z-direction. Four independent, highly porous and permeable sandstone layers represent the potential storage horizons which are separated by tight clay layers. The sandstone layers are defined according to their original porosity (in average 13.08 %) and permeability (in average 22.40 mD) values. Above the gas–water contact at 3452 m, the reservoir is up to 90 % gas saturated. The major gas components are represented by roughly 80 mol% nitrogen and 20 mol% methane. A cross section through the reservoir is displayed in Fig. 3 where the figure is vertically stretched by the factor 7.

The implemented hydrogen underground storage scenario includes a total period of 10 years. During the development period, the reservoir is step-wise charged with hydrogen, while the subsequent 5 years simulate the annual cyclic operation period.

Using a depleted gas reservoir as storage structure, initially an appropriate storage charging period is necessary in order to obtain a sufficient storage deliverability. The deliverability is a function of the total gas inventory and is increasing with increasing reservoir pressure. In view of porous underground storage applications, Tek (1989) subdivides the total gas inventory into two types. First a substantial fraction of the gas inventory has to remain within the storage structure in order to guarantee a required minimum reservoir pressure. This so-called cushion gas is either intentionally not produced or is physically non-recoverable since it is trapped within the geological structure. The actual gas of interest is referred to as working gas. This gas can be withdrawn without damaging the long-term deliverability of the gas storage. After the establishment of a high and homogeneous hydrogen concentrated region in the well drainage area, the gas production during the operation period ensures the recovery of hydrogen.

Qualitative observation parameters for the evaluation of the hydrogen storage scenario are plotted in Fig. 4. The first 5 years schedule an alternating shape of 6 months of injection and 6 months of idle. As a result of the step-wise hydrogen injection, the initial reservoir pressure is increased from 170 to 370 bar. Thereby the injection rate is controlled by Peaceman’s well model (constant injection pressure value) which explains the curved reservoir pressure increase (Fig. 4a). Initially, a large amount of hydrogen is injected but since the well grid pressure progressively approaches the set injection pressure, the amount of mass injection declines (Fig. 4b). The average daily injection rate is targeted at roughly 250.000 Sm\(^{3}\)/day which corresponds to an cumulative annual hydrogen addition of roughly 47 million Sm\(^{3}\). After 5 years, the storage inventory consists of 194.5 million Sm\(^{3}\) of the native nitrogen–methane mixture and 236.6 million Sm\(^{3}\) of hydrogen.

The storage operation period starts again with an injection period in which further hydrogen is added into the storage structure. The average reservoir pressure is thereby increased to a final pressure of 400 bar, which should guarantee the storage deliverability. As this value is still 20 bar below the initial reservoir pressure, a safe storage operation is achievable. At this point of the maximum gas storage level, the reservoir contains more than 271 million Sm\(^{3}\) hydrogen. The cross section through the reservoir (Fig. 5a) shows a high and homogeneous hydrogen concentration in the vicinity of the injection well. This allows the completion of the storage charging period and the first commencement of a 4-month long production period. This extraction process is controlled by a production pressure of 300 bar which yields to a total cyclic average gas extraction of 107.7 million Sm\(^{3}\). As a result of the gas withdrawal, Fig. 5c shows the hydrogen concentration reduction within the formation. Accordingly, the nitrogen concentration within the top region of the structure increases (Fig. 5d). The hydrogen fraction of the extracted gas is plotted in Fig. 4d and shows a seasonal hydrogen fraction increase from an initial value of 82 mol% to a final value of 85.2 mol%. This indicates a closed reservoir system, where gas losses, due to migration or gas dissolution, are insignificant. Furthermore nitrogen and methane are seasonally extracted from the reservoir as impurities of the produced gas, while the injection gas purely consists of hydrogen. Consequently, the hydrogen concentration inside the storage formation is increasing. A common method to monitor gas losses in porous underground storage is given by plotting the reservoir pressure versus the total gas inventory (Fig. 4c). In a physical ideal and technical gas loss-free case, the resulting hysteresis slope of a closed reservoir will be identical during the gas injection and withdrawal. In contrast, gas losses can be identified by a scattered pressure response and a cyclic shifting of the hysteresis course. A parabolic shape of the injection and withdrawal line characterizes water driven reservoirs (Tek 1989). The resulting hysteresis of the last four operation cycles are plotted in Fig. 4c and indicate a gas loss-free operation. In this context, it is important to remind the mathematical no-flow boundaries of the reservoir system. The simulation focuses on the numerical investigation of gas mixing processes in underground hydrogen storages and does not contain scientific significance about possible hydrogen diffusion into sealing layers. Before implementing a numerical approach, the investigation of hydrogen diffusivity into caprocks initially requires essential experimental research. Furthermore Fig. 6a illustrates that the initial reservoir pressure is in hydrostatic equilibrium. The comprehensive hydrogen injection predominantly leads to the pressure increase within the storage horizons, while the pressure is slightly increasing in the interlayers (Fig. 6b). Since water is hardly compressible and the gas is injected into a closed system, the pressure duplication leads only to a slight shifting of the GWC in vicinity of the operation well (Fig. 7).