Nuclear magnetic resonance (NMR) relaxometry measurements are commonly used to characterize the storage and transport properties of water-saturated rocks. Estimations of these properties are based on the direct link of the initial NMR signal amplitude to porosity (water content) and of the NMR relaxation time to pore size. Herein, pore shapes are usually assumed to be spherical or cylindrical. However, the NMR response at partial water saturation for natural sediments and rocks may differ strongly from the responses calculated for spherical or cylindrical pores, because these pore shapes do not account for water menisci remaining in the corners of desaturated angular pores. Therefore, we consider a bundle of pores with triangular cross sections. We introduce analytical solutions of the NMR equations at partial saturation of these pores, which account for water menisci of desaturated pores. After developing equations that describe the water distribution inside the pores, we calculate the NMR response at partial saturation for imbibition and drainage based on the deduced water distributions.

For this pore model, the NMR amplitudes and NMR relaxation times at partial water saturation strongly depend on pore shape, i.e., arising from the capillary pressure and pore shape-dependent water distribution in desaturated pores with triangular cross sections. Even so, the NMR relaxation time at full saturation only depends on the surface-to-volume ratio of the pore. Moreover, we show the qualitative agreement of the saturation-dependent relaxation-time distributions of our model with those observed for rocks and soils.

Understanding multi-phase flow processes in porous rocks and soils is vital for addressing a number of problems in geosciences such as oil and gas recovery or vadose zone processes, which influence groundwater recharge and evaporation. Effective permeability, which is defined as the permeability of a fluid in the presence of another fluid, is the decisive parameter for fluid transport, and depends on fluid saturation, wetting condition, and pore structure. In addition, saturation history influences the fluid content and the effective permeability (for a specific pressure), which are different for imbibition and drainage.

A method considered suitable for determining water content of rocks non-invasively is nuclear magnetic resonance (NMR), because the NMR initial signal amplitudes are directly proportional to the hydrogen content in the pore space, and the NMR relaxation times are linked to the size of the water-containing pores in the rock. In a two-phase system of water and air, only the water contributes to the NMR signal response. Therefore, NMR is widely used for estimating transport and storage properties of rocks and sediments (Kenyon, 1997; Seevers, 1966; Fleury et al., 2001; Arnold et al., 2006).

In recent years, several researchers have studied the relationship between NMR and multiphase flow behavior on the pore scale to better understand and infer the storage and transport properties of partially saturated rocks or sediments (e.g., Chen et al., 1994; Liaw et al., 1996; Ioannidis et al., 2006; Jia et al., 2007; Al-Mahrooqi et al., 2006; Costabel and Yaramanci, 2011, 2013; Talabi et al., 2009). As an extension of this research, we study the relationship between the water distribution inside the pores of a partially saturated rock and the system's NMR response by using bundles of pores with triangular cross sections. While Al-Mahrooqi et al. (2006) used a similar modeling approach to infer the wettability properties in oil–water systems, this study investigates the evolution of the NMR relaxation-time spectra during drainage and imbibition. For this purpose, we consider a capillary pore ensemble that is partially saturated with water and air. Traditionally, the pores within this ensemble are assumed to have a cylindrical geometry. Depending on pressure, cylindrical capillaries are either water- or air-filled, and thus they either contribute to an NMR response or not. Consequently, the NMR relaxation times of partially water-saturated capillary pore bundles always remain subsets of the fully saturated system's relaxation-time distribution; i.e., they are a function inside the envelope of the distribution curve at full saturation (see Fig. 1). However, in porous rocks, which are formed by the aggregation of grains, the pore geometry is usually more complex (Lenormand et al., 1983; Ransohoff and Radke, 1987; Dong and Chatzis, 1995) and may exhibit angular and slit-shaped pore cross sections rather than cylindrical capillaries or spheres (Fig. 2a). For example, in tight gas reservoir rocks, Desbois et al. (2011) found three types of pore shapes that are controlled by the organization of clay sheet aggregates: (i) elongated or slit-shaped, (ii) triangular, and (iii) multi-angular cross sections. The relaxation-time distribution functions derived from NMR measurements for such partially saturated rocks are frequently found to be shifted towards shorter relaxation times outside the original envelope observed for a fully saturated sample (Fig. 2b) (e.g., Applied Reservoir Technology Ltd., 1996; Bird et al., 2005; Jaeger et al., 2009; Stingaciu, 2010; Stingaciu et al., 2010; Costabel, 2011).

In angular pores, water will remain trapped inside the pore corners even if the gas entry pressure is exceeded. Standard NMR pore models that assume cylindrical or spherical pore ensembles (e.g., Kenyon, 1997), however, do not account for such residual water (Blunt et al., 2002; Tuller et al., 1999; Or and Tuller, 2000; Tuller and Or, 2001; Thern, 2014). To overcome this limitation, we adopt a NMR modeling approach initially proposed and discussed by Costabel (2011) and present numerical simulations and analytical solutions of the NMR equations for partially saturated pores with triangular cross sections to quantify NMR signal amplitudes and relaxation times. The NMR response of a triangular capillary during drainage and imbibition depends on the water distribution inside the capillary, which is subject to pore shape and capillary pressure. Thus, in the next chapter, we present the relationship between capillary pressure and water distribution inside cylindrical and triangular pore geometries during drainage and imbibition. For this purpose, the reduced similar geometry concept introduced by Mason and Marrow (1991) is used. Subsequently, based on the spatial water distribution, an analytical solution of the NMR diffusion equation (Torrey, 1956; Brownstein and Tarr, 1979) for partially saturated triangular capillaries is derived and tested by numerical simulations (Mohnke and Klitzsch, 2010). The derived equations are used to study the influence of pore size distribution and pore shape of triangular capillaries on the NMR response, in particular considering the effects of trapped water. Finally, an approach for simulating NMR signals during imbibition and drainage of triangular pore capillaries is introduced and demonstrated using synthetic pore size distributions.

In a partially saturated pore space, a curved liquid–vapor interface called
the arc meniscus (AM) arises due to the pore's capillary forces. In addition,
adsorptive forces between water and matrix lead to the formation of a thin
water film at the rock–air interface. Such water films with a thickness
typically below 20 nm (e.g., Toledo et al., 1990; Tokunaga and Wan, 1997)
exhibit very short NMR relaxation times. Although water films to some extent
may influence transport properties and water distribution of a partially
saturated porous system (Tuller and Or, 2001), the contribution of the film
volume to NMR amplitudes is very small with respect to the NMR signal
amplitudes arising from the water trapped in the menisci; i.e.,

Cross sections of a partially saturated triangular tube. The arc
meniscus of radius

In the following discussion, we consider a triangular capillary, initially
filled with a perfectly wetting liquid, i.e., contact angle

From an original triangle ABC, a new smaller triangle A

During imbibition of a (tri-)angular pore, the radius of curvature

The permeability of a porous system of such triangular capillaries is
strongly influenced by the shape factor

Combining Eqs. (1)–(3) with the concept of reduced similar geometry discussed
above, the degree of water saturation (

In the following section, analytical solutions for respective NMR responses that arise from partially saturated arbitrary triangular tubes are derived and matched against numerical simulations by means of the generalized differential NMR diffusion equations introduced by Brownstein and Tarr (1979).

The measured NMR relaxation signal

Following derivations of Brownstein and Tarr (1979), the inverse of the
longitudinal relaxation time

Longitudinal relaxation times

Upon consideration of a long (triangular) capillary, its surface-to-volume
ratio equals its perimeter-to-cross-section ratio, i.e.,

Saturated corner with active boundaries, i.e.,

Supposing the air–water interface to be a passive boundary with respect to
NMR surface relaxivity, i.e.,

Water (black) and air (white) distributions within a triangular
pore

In contrast, each water-filled corner of a partially saturated
non-equilateral triangle, i.e.,

Water (black and grays) and air (white) distributions within
a right-angled triangular pore (

To test the analytical (fast diffusion) models for partially saturated
triangular capillaries derived above, the calculated longitudinal NMR
relaxation times and amplitudes are compared to solutions obtained from 2-D
numerical simulations of the general NMR diffusion equation (Mohnke and
Klitzsch, 2010):

NMR response of an equilateral triangular capillary pore
model (with a side length of 1

As shown in Fig. 10, analytically (

Comparison of analytical and calculated NMR relaxometry data
originating from saturated pore corners (e.g., see Fig. 7) of varying
apertures (5

The model was also matched against numerical simulations for pores with
arbitrary angles. Figure 11 illustrates 2-D
finite-element simulations using saturated pore corners with angles

The goal of this section is to evaluate how pore shape affects the forward-modeled NMR response of a partially saturated system of pores (a pore size distribution). As discussed earlier, the NMR relaxation time of a single water-filled capillary pore is inversely proportional to its surface-to-volume ratio. Thus, at full water saturation, the relaxation-time distribution obtained from a multi-exponential NMR relaxation signal represents the pore size distribution of the rock. At partial water saturation it is often assumed that the NMR relaxation signal still represents the pore size distribution of the water-saturated pores (e.g., Stingaciu et al., 2010). We are going to demonstrate that this is valid for cylindrical but not for (tri-)angular pores.

In contrast to cylindrical pores, capillaries with (tri-)angular cross sections may be partially water-saturated during drainage or imbibition (cf. Figs. 8 and 9) because of the water remaining in the corners. Thus, they show a different water retention behavior, and the “desaturated” pores, i.e., their arc menisci, contribute to the NMR signal. Consequently, with increasing pressure (i.e., decreasing water saturation), the NMR relaxation behavior of the partially water-saturated triangular capillary pore bundle successively shifts to signal contributions with shorter relaxation times, exceeding the original distribution at full saturation. This shift reflects the fast relaxation of residual water trapped in the pore corners (Fig. 12). This behavior in angular pore geometries is demonstrated in Fig. 13. Here, the NMR relaxation components for a fully (blue line) and partially saturated (red and green) distribution of triangular capillaries are plotted. The green and red peaks show the signals of the residual water in the pore corners. As a consequence of the reduced geometry concept, the remaining water in the corners can be considered similar in size and shape due to the same NMR relaxation time, and thus only depends on pressure and not on pore size. Therefore, with decreasing saturation, i.e., increasing pressure, the NMR signal of the arc menisci increases and shifts towards smaller relaxation times. If the non-wetting phase (air) has entered all capillaries, only one single relaxation time remains for the pore bundle of equilateral triangles. For arbitrarily shaped triangular pores, three relaxation times would remain for the desaturated pore system. Hence, the concept of a relaxation-time distribution assumed in conventional NMR inversion and interpretation approaches would be no longer valid.

Relaxation components of fully (blue line) and partially desaturated triangular pore size distribution. At a specific saturation level, all pore corners with residual saturation exhibit the same NMR magnetization and relaxation behavior, thus superposing to a single fast relaxation component (e.g., red and green bars).

We applied the concept of fitting multi-exponential relaxation-time distributions to NMR transients calculated for pore bundles of circular and equilateral triangle cross sections in order to study how pore shape affects the typically shown relaxation-time distributions.

Water drainage and imbibition with water as the wetting and air as the non-wetting fluid were investigated by simulating water retention curves and corresponding NMR relaxation signals for a log-normal distributed pore size ensemble as shown in Fig. 14.

Pore size distribution model (log-normal distribution:

Herein, to clarify the subsequent discussion, we focused only on the
equilateral triangular capillary model. Other angular pore shapes (e.g.,
right angular triangles or squares) will exhibit a similar behavior.
Capillary pressure curves presented in Fig. 15a were calculated from
Eqs. (1), (5), and (6) for pore bundles with circular and equilateral
triangle cross sections. In contrast to water retention curves calculated for
the cylindrical capillary model, significant hysteresis between drainage and
imbibition can be observed for the triangular capillary model, i.e., in terms
of initial amplitudes (

The NMR

In contrast, inversion results for equilateral triangular capillary ensembles (Fig. 15f–h) – both for imbibition and drainage – show a similar shift to shorter relaxation times with decreasing saturation but also shift towards the outside the initial distribution at full saturation due to NMR signals originating from trapped water in the pore corners of the desaturated triangular capillaries. The effect of the pore corners on relaxation times at low saturations is also recognizable when comparing the (geometric) mean relaxation times, normalized on the values observed at full saturation (Fig. 15b): both, the drainage and the imbibition hysteresis branch of the triangular pore bundle, show smaller mean relaxation times than the cylindrical pore bundle.

In conclusion, the calculated inverse models for the triangular capillary bundle qualitatively agree with the behavior of the inverted NMR relaxation-time distributions at partial saturation that are frequently observed in experimental data, e.g., of the Rotliegend sandstone shown in Fig. 2.

Experimental NMR relaxometry data and corresponding relaxation-time distributions obtained at partial water/air saturation were explicated by a modification of conventional NMR pore models using triangular cross sections. The derived analytical solutions for calculating surface-dominated (fast diffusion) NMR relaxation signals in fully and partially saturated arbitrary angular capillaries were consistent with respective results obtained from numerical simulations of the general NMR diffusion equations.

Shape and size of triangular pores can strongly influence both NMR amplitudes and decay time distribution and the rock's flow properties, i.e., saturation and (relative) permeability. At full saturation the NMR relaxation time depends on the surface-to-volume ratio, which in turn depends on shape if considering angular pore capillaries. However, at partial saturation, the pore shape even more strongly influences the water distribution inside the pore system, and thus the NMR signal. In contrast to cylindrical capillaries, angular capillaries also contribute to the NMR signal even after desaturation of the pore due to the water remaining in the pore corners.

In this regard, non-equilateral triangular capillaries at partial saturation
exhibit a three-exponential relaxation behavior due to different
perimeter-to-surface (

Moreover, we studied the NMR response of a triangular pore bundle model by
jointly simulating the water retention curves for drainage and imbibition
and the corresponding NMR

Ongoing research will include further experimental validation and implementation of the introduced approach in an inverse modeling algorithm for NMR data obtained from partially saturated rocks to predict absolute and relative permeability on laboratory and borehole scales. Without considering angular pores, the NMR signal of trapped water cannot be explained; i.e., using the classical approach of circular capillaries, one cannot find a pore size distribution that explains the relaxation-time distributions at all saturations sufficiently (e.g., Mohnke, 2014). On the other hand, angular pore models can account for the trapped water and thus overcome the limitation of the classical approach. Moreover, following the approach of Mohnke (2014) but considering angular pores, we strive to estimate surface relaxivity, pore size distribution, and pore shape by jointly inverting NMR data at different saturations. Based on the obtained pore size distribution and triangle shape, we expect to improve the prediction of the absolute and relative permeabilities considerably.

The study was supported by the German Research Foundation (DFG) in the framework of the Transregional Collaborative Research Centre 32 (SFB TR 32) and Wintershall AG in the framework of the Wintershall Tight Gas Consortium at RWTH Aachen University. Edited by: M. Giudici