Determination of Hydrogen–Water Relative Permeability and Capillary Pressure in Sandstone: Application to Underground Hydrogen Injection in Sedimentary Formations

Abstract

To provide quantitative data for the development of underground hydrogen storage in porous sedimentary rocks, capillary pressures and relative permeabilities have been measured for the hydrogen–water system. The tests have been performed on a Triassic sandstone. Two potential underground hydrogen storage conditions (“shallower”: 55 bar, 20 $^{\circ }\hbox {C}$ and “deeper”: 100 bar, 45 $^{\circ }\hbox {C}$) have been investigated. Capillary pressure curves have been measured following a modified semi-dynamic technique. The data have been combined with mercury injection capillary pressure measurements to derive a model for capillary pressure valid over almost the entire water saturation range. Interfacial tensions and contact angles for the hydrogen–water system have been also derived. Relative permeability curves measured with the steady-state technique yield low values for minimum water saturations of $\sim$ 40%. When combined with the capillary pressure data, the relative permeability of hydrogen in sandstone can be evaluated for almost the total range of water saturation. Capillary numbers calculated for our relative permeability experiments indicate a capillary-limited flow regime for the hydrogen–water system. Despite the two differing sets of conditions investigated and this flow regime, the relative permeability curves stay very close from each other, an effect attributed to the almost constant viscosity of hydrogen under our pressure and temperature conditions. This is in contrast with other fluid pairs (e.g., $\hbox {CO}_{2}$–water system) where capillary numbers can strongly vary with pressure and temperature. Similarly, capillary pressure data vary little between the experimental conditions. The interpretation of the results would suggest that the relative permeability and capillary pressure results from this study are applicable to a wide range of pressure and temperature conditions.

Appendices

Appendix A: Protocol to Compute the Core Inlet Saturation from Core-Flooding Capillary Pressure Data

This protocol has been proposed by Ramakrishnan and Capiello (1991) for evaluating the non-wetting phase saturation at the core inlet with core-flooding capillary pressure measurements. As explained before, during capillary pressure measurement just gas is injected to the saturated core sample. Therefore, Darcy’s law (Eq. 2) at steady-state conditions will be:

$$\begin{aligned} -\frac{\mathrm{d}P_\mathrm{c} }{\mathrm{d}x}=\frac{\mu _\mathrm{g} }{Kk_\mathrm{r,g} \left( S \right) }\frac{Q_\mathrm{g} }{A} \end{aligned}$$

(A1)

where x is the position variable along the core, $\frac{\mathrm{d}P_\mathrm{c} }{\mathrm{d}x}$ the variation of the capillary pressure along the core, $k_\mathrm{r,g}$ the relative permeability of the gas phase which is a function of the water saturation, s, $\mu _\mathrm{g}$ the viscosity of the gas, $Q_\mathrm{g}$ the flow rate of the gas phase, K the absolute permeability, and A the cross-sectional area of the core. Since saturation is varying along the core, the capillary pressure and relative permeability to gas are changing as well. However, since the pressure drop through the core is very small compared to the fluid pressure, the volumetric flow rate and the viscosity can be assumed constant along the core. Therefore, integrating equation (A1) along the length of the core gives:

$$\begin{aligned} \mathrm{QL}=-\frac{\mathrm{AK}}{\mu _\mathrm{g} }\int \nolimits _{P_\mathrm{c,x=0} }^{P_\mathrm{c,x=L} } k_\mathrm{r,g} \left( {P_\mathrm{c,x} } \right) \mathrm{d}P_\mathrm{c,x} \end{aligned}$$

(A2)

where $P_\mathrm{c,x=0}$ corresponds to the capillary pressure measured during the measurements and $P_\mathrm{c,x=L}$ to the capillary pressure at the outlet of the core, considered as the entry capillary pressure of the rock. By noting $\Delta P_\mathrm{c,x} =P_\mathrm{c,x} -P_\mathrm{c,x=L}$ (the capillary pressure difference between one location along the core and the core outlet), Eq. (A2) becomes:

$$\begin{aligned} \mathrm{QL}=-\frac{\mathrm{AK}}{\mu _\mathrm{g} }\int \nolimits _{\Delta P_\mathrm{c,x=0} }^0 k_\mathrm{r,g} \left( {\Delta P_\mathrm{c,x} } \right) \mathrm{d}\Delta P_\mathrm{c,x} \end{aligned}$$

(A3)

Assuming a homogeneous relative permeability law along the core, differentiation of Eq. (A3) relatively to $\Delta P_\mathrm{c,x=0}$ gives:

$$\begin{aligned} \frac{\mathrm{d}Q}{\mathrm{d}\Delta P_\mathrm{c,x=0} }=\frac{\mathrm{AK}}{\mu _\mathrm{g} L}\cdot k_\mathrm{r,g} \left( {\Delta P_\mathrm{c,x=0} } \right) \end{aligned}$$

(A4)

This equation means that the relative permeability of gas can be obtained if the relationship between the flow (or injection) rate and the differential pressure measured during the capillary pressure experiments is known.

Fig. 11

Relationship between hydrogen volumetric rate, saturation and the differential pressure in experiments 1 and 2

Fig. 12

Relationship between gas volumetric rate and the differential pressure in experiments 1 and 2

In parallel, the average saturation in the core can be computed as:

$$\begin{aligned} {\bar{S}}_\mathrm{w} =\frac{1}{L}\int \nolimits _{x=0}^{x=L} S_\mathrm{w}\,\mathrm{d}x, \end{aligned}$$

(A5)

Combined with Eq. (A1), this leads to:

$$\begin{aligned} Q{\bar{S}}_\mathrm{w} =\frac{\mathrm{AK}}{L\mu _g }\int \nolimits _{\Delta P_\mathrm{c,x=0} }^0 k_\mathrm{r,g} \left( {\Delta P_\mathrm{c,x} } \right) S_\mathrm{w} \left( {\Delta P_\mathrm{c,x} } \right) \mathrm{d}\Delta P_\mathrm{c,x} \end{aligned}$$

(A6)

The differentiation of this equation relatively to $\Delta P_\mathrm{c,x=0}$ gives:

$$\begin{aligned} \frac{\mathrm{d}\left( {Q{\bar{S}}_\mathrm{w} } \right) }{\mathrm{d}\Delta P_\mathrm{c,x=0} }=\frac{\mathrm{AK}}{\mu _\mathrm{g} L}\cdot k_\mathrm{r,g} \left( {\Delta P_\mathrm{c,x=0} } \right) \cdot S_\mathrm{w} \left( {\Delta P_\mathrm{c,x=0} } \right) \end{aligned}$$

(A7)

Equations (A7) and (A4) can be changed into:

$$\begin{aligned} S_\mathrm{w} \left( {\Delta P_\mathrm{c,x=0} } \right) =\frac{\mu _\mathrm{g} L}{Akk_\mathrm{r,g} \left( {\Delta P_\mathrm{c,x=0} } \right) }\cdot \frac{\mathrm{d}\left( {Q{\bar{S}}_\mathrm{w} } \right) }{\mathrm{d}\Delta P_\mathrm{c,x=0} }\,\mathrm{with}\,\frac{\mathrm{AK}}{\mu _\mathrm{g} L}\cdot k_\mathrm{r,g} \left( {\Delta P_\mathrm{c,x=0} } \right) =\frac{\mathrm{d}Q}{\mathrm{d}\Delta P_\mathrm{c,x=0} }\nonumber \\ \end{aligned}$$

(A8)

In other words (with $v_\mathrm{inj}$ being the Darcy velocity, equal to Q / A):

$$\begin{aligned} S_\mathrm{w} \left( {\Delta P_\mathrm{c,x=0} } \right) =\frac{L}{{\Lambda }\left( {\Delta P_\mathrm{c,x=0} } \right) }\cdot \frac{\mathrm{d}\left( {v_\mathrm{inj} {\bar{S}}_\mathrm{w} } \right) }{\mathrm{d}\Delta P_\mathrm{c,x=0} } \end{aligned}$$

(A9)

where

$$\begin{aligned} {\Lambda }\left( {\Delta P_\mathrm{c,x=0} } \right) =\frac{K}{\mu _\mathrm{g} }\cdot k_\mathrm{r,g} \left( {\Delta P_\mathrm{c,x=0} } \right) =\frac{\mathrm{d}v_\mathrm{inj} }{\mathrm{d}\Delta P_\mathrm{c,x=0} }\cdot L \end{aligned}$$

(A10)

and $S_\mathrm{w} \left( {\Delta P_\mathrm{c,x=0} } \right)$ is the water saturation at the inlet. Therefore, with information on the injection rate, the differential pressure between the inlet and outlet of the core, and the average saturation in the core during the measurements, it is possible to retrieve the water saturation at the inlet.

However, in order to use Eq. (A9), precise knowledge of the relationship between Darcy velocity ($v_\mathrm{inj} )$, average saturation (${\bar{S}}_\mathrm{w} )$ and measured capillary pressure ($\Delta P_\mathrm{c,x=0} )$ is required. This relationship was obtained by fitting second-order polynomials of $v_\mathrm{inj} {\bar{S}}_\mathrm{w}$ as a function of $\Delta P_\mathrm{c,x=0}$ for experiments no1 and no 2, respectively (Table 2; Fig. 11). These polynomials were used to calculate the water saturation at the inlet for each experiment (Fig. 12).

Appendix B: Calculation of Relative Permeability of the Non-wetting Phase from Core-Flooding Capillary Pressure Data

Relative permeability data for the non-wetting phase were extended toward low water saturations by using the capillary pressure data obtained from the core-flooding measurement. This method was suggested by Pini and Benson (2013) based on the work of Ramakrishnan and Capiello (1991). The relative permeability of the non-wetting phase can be calculated from the relationship between the Darcy velocity ($v_\mathrm{inj} )$ and the capillary pressure ($\Delta P_\mathrm{c,x=0} )$. Starting with Eq. (A10), we have:

$$\begin{aligned} \frac{K}{\mu _\mathrm{g} }\cdot k_\mathrm{r,g} \left( {\Delta P_\mathrm{c,x=0} } \right) =\frac{\mathrm{d}v_\mathrm{inj} }{\mathrm{d}\Delta P_\mathrm{c,x=0} } \cdot L \end{aligned}$$

(B1)

which can be rewritten as:

$$\begin{aligned} k_\mathrm{r,g} \left( s \right) =\frac{\mathrm{d}v_\mathrm{inj} }{\mathrm{d}\Delta P_\mathrm{c} }\cdot L \cdot \frac{\mu _\mathrm{g} }{K} \end{aligned}$$

(B2)

Relative permeabilities were obtained by fitting second-order polynomials of $v_\mathrm{inj}$ as a function of $\Delta P_\mathrm{c}$ for experiments no 1 and no 2, respectively (Table 2). These polynomials were used to extend the relative permeability data for experiments no 3 and no 4 (Fig. 8). Relative permeabilities calculated using the polynomials for experiments no 1 and no 2 are given in Table 4.