2.1 Radiotracer solution

The active component of the radiotracer used in this study is the radionuclide $^{11}$ C with a half-life $t_{1/2}=20.33$  min. The isotope was produced in the chemical form of [11C]CO₂ using a CTI-Siemens RDS111-11 MeV cyclotron located at the Biomedical Isotope Facility of the Lawrence Berkeley National Laboratory (Berkeley, CA, USA) and was subsequently released in an aqueous sodium bicarbonate solution ([11C]NaHCO₃, 55 mM) (Vandehey & O’Neil Reference Vandehey and O’Neil2014). The latter constitutes the radioactive solution to be injected during the pulse-tracer experiments. An equivalent (non-labelled) NaHCO₃ aqueous solution was used to saturate the core sample and as the background carrier fluid throughout the experimental campaign.

2.2 Rock sample and permeability heterogeneity

The properties of the Berea sandstone (Cleveland Quarries, Ohio, USA) core used in this study are summarised in table 1. A key feature of this specific sample is that its subcore-scale permeability distribution is precisely known at a resolution of a few mm³; the latter was obtained in a previous study (Pini & Benson Reference Pini and Benson2013) and is again shown in figure 1(a) together with the histogram plot of the corresponding permeability values. Notably, the distribution of permeability values at the subcore scale is relatively wide (200–500 mD, measured average permeability $k_{m}=330$  mD) and is bimodal, with approximately 25 % of the voxels belonging to fine-textured (and less permeable) strata that can be readily recognised in the three-dimensional (3-D) permeability map. Although Berea sandstone is the archetype of a well sorted and homogeneous rock, the presence of these characteristic layers has been reported in earlier studies (Brooks & Corey Reference Brooks and Corey1964). The latter are not readily noticeable in the dry or fully saturated state, but introduce fine-scale (mm–cm) variations in the permeability, whose effects have been shown to greatly influence fluid flow for both single (Walsh & Withjack Reference Walsh and Withjack1994) and multiphase (e.g. gas–water) systems (Perrin & Benson Reference Perrin and Benson2010; Krause et al. Reference Krause, Perrin and Benson2011).

Figure 1.

Multidimensional reconstructions of the Berea sandstone core used in this study in terms of permeability maps together with their corresponding histograms. (a,e) ‘Original’ reconstruction obtained from a laboratory characterisation study (Pini & Benson Reference Pini and Benson2013). In the histogram, the curves are normal distributions representing three subfamilies of voxels (dashed lines) and their contribution to the whole population of values (solid line). Three additional maps have been derived from (a,e), namely (b,f) ‘streamtubes’, (c,g) ‘tuned streamtubes’ and (d,h) ‘two layer’ (2P-ML). Voxel size: for all maps, ( $x\times y$ )  $=$  (2 $\times$ 2) mm², while $z=3$  mm for (a) and 83.5 mm for (b–d). Additional details are given in § 2.2.

An empirical semivariogram was calculated to investigate the correlation structure of the voxel permeability values in the $x,y,z$ directions by using the following equation:

(2.1) $$\begin{eqnarray}\hat{{\it\gamma}}_{i}(h)=\frac{1}{2N(h)}\mathop{\sum }_{N(h)}(w_{j}-w_{l})^{2},\end{eqnarray}$$

where the subscript $i$ refers to the direction used for calculating the variogram ( $i=x$ , $y$ or $z$ ), $h$ is the lag (i.e. the distance separating two voxels), $N$ is the set of all pairwise distances ( $j-l=h$ ) and $w_{j}$ and $w_{l}$ are data values at spatial locations $j$ and $l$ , respectively. Note that (2.1) was applied on the log-transformed data, $w=\log _{10}(k)$ , because the latter follow a normal distribution with location and scale parameter given by ${\it\mu}=2.52$ and ${\it\sigma}=0.0715$ , respectively, as shown in figure 2(a). The computed variograms are plotted in figure 2(b), together with predictions from an exponential model that was used to match the experimental data (Sarma Reference Sarma2009),

(2.2) $$\begin{eqnarray}{\it\gamma}_{i}(h)=(C_{i}-{\it\gamma}_{i}^{0})\left[1-\text{exp}(-3h/a_{i})\right]+{\it\gamma}_{i}^{0},\end{eqnarray}$$

where $C={\it\gamma}(h\rightarrow \infty )$ is the sill (i.e. the limiting value of the variogram), $a$ is the range (i.e. the distance at which the sill is reached and the data are no longer autocorrelated) and ${\it\gamma}^{0}={\it\gamma}(h=0)$ represents the so-called nugget effect. It can be seen from the plot that the correlation length in the $z$ -direction (parallel to flow) is much larger than either value obtained in the $x/y$ -direction, with the value of the range being just below the sample length ( $a_{xy}\approx 1.5~\text{cm}<a_{z}\approx 7~\text{cm}~\approx L$ ). The sill in the $x/y$ -direction is also the largest (and close to the variance of the whole distribution), suggesting that permeability heterogeneity in this sample may be primarily attributed to variations on this plane. These results again confirm the presence of those textural features that are characteristic of Berea sandstone and that are readily visible in figure 1(a,e) as layers running parallel to the direction of flow. As explained below, these observations are used to justify the simplification of the original voxel permeability map into an ensemble of streamtubes with constant permeability.

Figure 2.

Statistics of the voxel permeability map of the Berea sandstone sample used in this study and shown in figure 1(a,e). (a) Histogram plot of the log-transformed voxel permeability values together with a normal distribution probability density function (PDF) curve ( ${\it\mu}=2.52$ , ${\it\sigma}=0.0715$ ). (b) Computed empirical semivariograms (symbols) in the $x$ , $y$ and $z$ directions. The variograms are computed using (2.1) and normalised with the variance of the log-transformed permeability distribution. The curves are best matches obtained from an exponential variogram model (2.2); the values of the fitted parameters are as follows: $C_{z}=0.0037$ , $C_{y}=0.0047$ , $a_{z}=68.8$  mm,  $a_{y}=15.8$  mm,  ${\it\gamma}_{z}^{0}=0.0012$ , ${\it\gamma}_{y}^{0}=0$ .

Figure 3.

Schematic of the experimental apparatus used for the pulse-tracer experiments with simultaneous PET imaging.

With the intent of investigating the role of permeability heterogeneity on miscible displacements, four permeability maps are considered in this study that are derived from the original (experimental) map. These include a single-permeability (homogeneous) case, together with other three scenarios shown in figure 1(b–d, f–h) as 3-D reconstructions of the rock sample. Histograms are shown on top of each map to represent the corresponding distribution of permeability values. Map (b,f) reproduces the permeability field with parallel ‘streamtubes’ that have been obtained by taking the harmonic average of the original voxel permeability values in (a) along the longitudinal axis of the sample. For map (c,g), ‘tuned streamtubes’ have been obtained from (b,f) by further multiplying by a factor of 0.91 and 1.015 the streamtubes’ permeability values that are found below and above a threshold value of 285 mD, respectively. The values of these parameters were selected so as to attain a best match to the tracer effluent curve measured in one tracer test and were then used throughout the study. This is somewhat analogous to the procedure adopted in the stochastic advective–dispersive streamtube approaches (Cirpka & Kitanidis Reference Cirpka and Kitanidis2000; Ginn Reference Ginn2001) to obtain a distribution of streamtube travel times (i.e. velocities), with the difference that in our case their spatial arrangement is not free, but determined by the experimental permeability map. The major effect of such ‘tuning’ is that the range of streamtube permeability values is slightly stretched and reproduces more closely the one observed in the original voxel permeability map (figure 1 a,e), when the two main families of voxels are considered (90 % of the entire population). Finally, map (d,h) reduces the complexity of map (c) by using only two representative permeability values, while maintaining a similar spatial pattern as in (a,e). We refer to this scenario as ‘2P-ML’ to emphasise that two different values of permeability appear in multiple parallel layers. It is worth noting that each map possess the same average permeability of 330 mD that was determined from laboratory experiments.

2.3 Apparatus

The pulse-tracer experiments were performed using the experimental set-up depicted in figure 3. The core holder (Larson Engineering, Boulder, CO, USA) can accommodate a cylindrical sample of 5 cm in diameter and of variable length. The sample is positioned between two Delrin^® end-caps, which have circular grooves to distribute the fluid over its entire cross-sectional area. Compressed air provides a confining annular pressure between the jacketed sample and an outer 1.5 mm-thick aluminium tube (internal diameter (ID) 7 cm), so as to prevent bypass flow during the experiments. In addition, four all-thread rods secure the assembly and ensure good contact between the sample and the edges of the end caps. The core holder is placed horizontally on the bed of the scanning instrument (Siemens Biograph 6 PET/CT scanner with a 80 cm ring diameter and 4 mm detector element width) and is connected to two shielded detectors (miniature Si PIN diode detectors, Carroll-Ramsey and Associates, USA) that monitor the activity of the fluid entering and leaving the sample at 5 s intervals. The cumulative void volume between the sample and the detectors is approximately 3 ml and its exact value has been found by matching the mean residence time of the measured effluent tracer profile to the expected average residence time with the given flow rate and rock’s pore volume (see table 1). A displacement pump (Waters, Model 590 HPLC) is used for continuous injection of the carrier fluid through the sample at controlled flow rates (1.4 or $3.4~\text{ml}~\text{min}^{-1}$ in this study). A loop consisting of four syringes is used to inject the tracer and allows diverting the flow of the background fluid during injection of the tracer, so as to maintain a constant total volumetric flow rate through the sample. This generates a radiotracer input function with the shape of an isosceles triangle that can be easily reproduced for modelling purposes. The various components the system are connected by means of PEEK and PTFE tubing (with OD 1/16”–1/8”).

2.4 Experimental procedure and PET imaging

All experiments have been carried out at room temperature and ambient pressure conditions. To ensure a tight annular seal around the sample, the latter was wrapped with a 3 mm thick sheet of soft rubber gasket material that was fixed with PTFE thread-sealer tape; three o-rings were additionally placed along the length of the wrapped core and held by means of an outer heat-shrinkable Teflon layer. Care was taken to maintain the same orientation of the sample as for the 3-D permeability map shown in figure 1. After mounting the jacketed sample in the core holder and placing it on the bed of the scanner, the confining pressure on the sample was increased and set to a value of approximately 40 psia. The system was then purged with gaseous CO₂, followed by the injection of the aqueous background solution that was circulated for various pore volumes, so as to dissolve any remnant CO₂. After that, a X-ray scan of the entire core sample was taken to register its position. Each pulse-tracer experiment consisted of the injection of a bolus ( ${\sim}$ 3.5 ml) of the radioactive solution ([11C]NaHCO₃ at 55 mM) with an activity concentration of approximately 1 mCi (37 MBq), while keeping a constant volumetric flow through the sample. While the term ‘activity concentration’ represents the correct notation for measurements with radionuclides (in $\text{mCi}~\text{ml}^{-1}$ or $\text{Bq}~\text{ml}^{-1}$ ), we are going to interpose the terms ‘activity’ and ‘concentration’ when describing the exact same property. For the sake of better clarity, the symbols $a$ (or $A$ ) are used when referring to experimental results and $c$ (or $C$ ) to model predictions.

For the entire duration of the experiment (30–60 min depending on the flow rate), the activity $a^{D}$ of the fluid entering and leaving the sample is measured and corrected to the radioactivity at the injection time $a_{0}^{D}$ by accounting for radioactive decay, i.e. $a_{0}^{D}(t)=a^{D}(t)\text{e}^{{\it\lambda}t}$ with ${\it\lambda}=\text{ln}(2)/t_{1/2}$ . Simultaneously, the whole axial field of view of the PET scanner ( ${\sim}$ 22 cm) is imaged and the corresponding tracer activity in the rock sample is detected. This information is converted into a 3-D image with a voxel dimension of $0.509~\text{mm}~\text{pixel}^{-1}$ by using the algorithm provided as part of the instrument’s reconstruction software package. For the PET scanner used in this study, the effective spatial resolution is expected to be ${\sim}$ 2.5 mm at FWHM (Levy & Hoffman Reference Levin and Hoffman1999), and the reconstructed images will be analysed accordingly. The preparation of the PET data for subsequent analysis is completed by (i) resampling the data set into an (arbitrary) number of constant time frames ${\rm\Delta}t$ and by (ii) correcting the measured tracer concentration for radioactive decay. Mathematically, step (i) is expressed as follows:

(2.3) $$\begin{eqnarray}A_{j}^{PET}(t_{i})=\frac{1}{{\rm\Delta}t}\displaystyle \int _{t_{1}}^{t_{2}}a_{j}^{PET}(t)\,\text{d}t,\end{eqnarray}$$

where $A_{j}^{PET}(t_{i})$ is the average tracer activity in any given voxel $j$ over a time frame ${\rm\Delta}t$ , while $t_{1}=t-{\rm\Delta}t/2$ , $t_{2}=t+{\rm\Delta}t/2$ and $t_{i}=(t_{1}+t_{2})/2$ . Step (ii) requires accounting for the amount of decay during the selected image frame, i.e.

(2.4) $$\begin{eqnarray}\displaystyle DF(t_{i})=\frac{1}{{\rm\Delta}t}\int _{t_{1}}^{t_{2}}\text{e}^{-{\it\lambda}t}\,\text{d}t=\text{e}^{-{\it\lambda}t_{i}}\left(\frac{1-\text{e}^{-{\it\lambda}{\rm\Delta}t}}{{\it\lambda}{\rm\Delta}t}\right),\end{eqnarray}$$

with $DF(t_{i})=A_{j}^{PET}(t_{i})/A_{0j}^{PET}(t_{i})$ representing the decay factor. In deriving (2.4) we assume that radioactive decay is the sole mechanism affecting the tracer concentration during the time frame ${\rm\Delta}t$ . We note, however, that in this study ${\rm\Delta}t=1$  min and the term in brackets takes a value near unity ( $0.983\approx 1$ ); accordingly, the previous assumption can be relaxed and the decay-corrected tracer average activity in any given voxel $j$ is simply calculated as:

(2.5) $$\begin{eqnarray}A_{0j}^{PET}(t_{i})=\text{e}^{{\it\lambda}t_{i}}A_{j}^{PET}(t_{i}),\end{eqnarray}$$

where $A_{0j}^{PET}(t_{i})$ refers to a concentration measured at time $t_{i}$ and lumps together all events taking place over the time frame ${\rm\Delta}t=(t+{\rm\Delta}t/2)-(t-{\rm\Delta}t/2)$ . We also note that the selection of 1 min intervals represents a trade off between a temporal resolution that is useful for analysis and good image quality at the spatial resolution that is required in this study. The latter is significantly reduced when a small amount of radioactivity is injected, thus requiring sufficiently long ‘exposure’ when reconstructing the image frames. The obtained voxel activity values are reported in radioactivity concentration per unit voxel volume ( $\text{mCi}~\text{ml}^{-1}$ or $\text{Bq}~\text{ml}^{-1}$ , as noted throughout the manuscript with $1~\text{mCi}=2.7027\times 10^{-8}~\text{Bq}$ ).

3.1 Solution procedure

The partial differential equation (3.1) is discretised in space using the finite-difference method with 164 grid points corresponding to a constant width ${\rm\Delta}x=0.05$  cm. To this aim, the space derivatives are approximated using the central difference operator for each internal node and the backward difference operator for the first and last node. This produces a system of 164 ordinary differential equations (ODEs), which are solved simultaneously in time using the ode15s solver in MATLAB. The solver uses an implicit method for the numerical integration of the ODEs based on backward differences with a variable time step. The latter is chosen so that the local error $e_{i}$ in $c_{i}$ at each time step satisfies $|e_{i}|\leqslant r|c_{i}|+a_{i}$ . In this study, the relative ( $r$ ) and absolute ( $a_{i}$ ) error tolerances have been set to 0.01 % and $1\times 10^{-4}~\text{mCi}~\text{ml}^{-1}$ , respectively, while the maximum allowed time step was limited to 0.01 min. The input parameters used for the model are the sample properties listed in table 1. These are common to each experiment and modelling scenario considered, while the flow rate, $q$ , is adjusted depending on the experimental conditions. The longitudinal dispersivity ${\it\alpha}_{L}=D_{L}/v$ represents the sole fitting parameter, which was estimated for each experiment by reproducing the experimentally obtained tracer effluent profiles (i.e. the detector’s data) with the model described above, while minimising the following objective function:

(3.7) $$\begin{eqnarray}{\it\Phi}^{2}=\frac{1}{N_{P}}\displaystyle \mathop{\sum }_{j=1}^{N_{P}}\left(\frac{c_{j}^{out}-a_{0j}^{D}}{a_{0j}^{D}}\right)^{2},\end{eqnarray}$$

where $c^{out}$ is the flux-averaged effluent concentration predicted by the model, while $a_{0}^{D}$ is the decay-corrected activity concentration measured with the detector. $N_{P}$ is the number of experimental data points in the tracer effluent curve. The parameters of the input pulse function ( $t_{inj}$ and $c_{F}^{0}$ ) are summarized on the right of table 2 together with the corresponding amount of activity injected. A grid convergence analysis was carried out to quantify the error affecting the numerical solution of (3.1) by estimating the pulse breakthrough time, $t_{B}$ . The latter has been estimated as the time at which the derivative of the BTC takes a value of $2\times 10^{-3}~\text{mCi}~\text{ml}^{-1}~\text{min}^{-1}$ . The error is computed as ${\it\varepsilon}=(f_{160}-f_{i})/f_{160}$ , where $f_{160}$ represents the value of $t_{B}$ computed on a grid with 160 points and $f_{i}$ is the solution obtained when the grid is varied between 20 and 320 points. Results from this exercise show that the error takes a value of ${\it\varepsilon}=8.9$ , 0.6 and 0.2 % for $f_{i}=20$ , 40 and 120, and stabilises at a negligible level beyond that. Other measures, such as the first moment of the BTC, give a very similar trend, while a negligible error ( ${<}0.1$  %) is observed in the material balance irrespectively of the grid refinement, thus confirming the accuracy of the simulations results presented below on a grid with 164 nodes.

Table 2.

Overview of the experiments conducted in this study. Reported variables are: volumetric flow rate ( $q$ ), average residence time ( ${\it\tau}={\it\phi}AL/q$ ), dose amount ( $M_{exp}^{out}$ with corresponding error, ${\it\varepsilon}=1-M_{exp}^{out}/M_{exp}^{inj}$ ) and mean residence time ( $t_{R,exp}^{\ast }$ ). Model parameters are: the tip ( $c_{F}^{0}$ ) and base ( $t_{inj}$ ) of the triangular tracer input function, and the simulated dose amount ( $M^{out}$ with corresponding error, ${\it\varepsilon}$ ).

3.2 Model scenarios

As anticipated above, three modelling approaches are considered in this study that differ in the level of characterisation of the rock sample in terms of permeability, namely the 1-D (homogeneous), the ‘streamtube’ and the ‘2P-ML’ model. For these last two cases, (3.1) is solved for each streamtube ( $N_{K}=431$ ) or layer ( $N_{K}=2$ ) and the average effluent tracer concentration is calculated as $c^{out}=\sum _{i=1}^{N_{K}}c_{i}q_{i}/q$ , with $c_{i}$ being the effluent concentration from streamtube (or layer) $i$ and $q_{i}$ its volumetric flow rate. The latter is related to the average flow rate $q$ through the relationship that describes Darcy’s flow in a system of parallel horizontal layers without cross-flow, i.e. $q_{i}/q=(S_{i}k_{i})/(Sk_{m})$ , with $S_{i}$ and $k_{i}$ being the cross-section and permeability of layer $i$ , respectively. For the ‘streamtube’ model the values of $k_{i}$ are known and $S_{i}$ is set by the scanner resolution ( $S_{i}=4~\text{mm}^{2}$ in this study), while for the ‘2P-ML’ model $S_{i}$ is estimated based on the selected pairs of permeability values, i.e. $S_{1}/S=(k_{m}-k_{2})/(k_{1}-k_{2})$ and $S_{2}=S-S_{1}$ . It should also be noted that the total volumetric flow rate used in the streamlines scenario has been corrected to account for a cross-section of the sample ( $S=431\times 0.04=17.24~\text{cm}^{2}$ ) that is slightly smaller than the physical value reported in table 1.

A few remarks are warranted with respect to the application of the models that are based on streamtubes (or layers). First, in adopting the streamtube geometry we are assuming that the correlation length of the permeability in any direction perpendicular to flow is much smaller than the sample diameter; accordingly, the correlation length in the direction of bulk flow is at least as large as the sample length and the permeability within a streamtube is constant. As discussed in § 2.2 and shown in figure 2 this assumption is justified, as it reflects the characteristic structural features of the sample considered in this study. As to the underlying transport processes, the adopted formulation assumes that the solute in each streamtube undergoes longitudinal mixing and instantaneous lateral mixing within the streamtube, while mixing between streamtubes (e.g. transverse dispersion) is neglected. For the flow regime investigated in this study ( $Pe\approx 10{-}20$ ), the ratio between the longitudinal and transverse dispersion coefficient takes a value of $D_{L}/D_{T}\approx 35$ , as estimated from results reported in literature for Berea sandstone, i.e. $D_{L}/\mathscr{D}\approx 100{-}200$ (Hulin & Plona Reference Hulin and Plona1989; Gist et al. Reference Gist, Thompson, Katz and Higgins1990) and $D_{T}/\mathscr{D}\approx 3{-}5$ (Bijeljic & Blunt Reference Bijeljic and Blunt2007), and in agreement with common observations for unconsolidated porous systems (Perkins & Johnston Reference Perkins and Johnston1963). We believe that these figures, together with the small scale of our experiment, support the approximation of negligible transverse dispersion, while the streamtube formulation allows resolving of the important mechanism of macrodispersion, i.e. the spreading of the tracer plume along distinct convective paths within the rock sample. Finally, irrespectively of the number of streamtubes, each scenario uses a single value for the dispersivity, ${\it\alpha}_{L}$ , while the dispersion coefficient in each streamtube changes according to its permeability, because $D_{L}=v{\it\alpha}_{L}$ . This approach ensures that the obtained results are consistent with observations from the literature that support this linear relationship for experiments where diffusion can be neglected (advection–dispersion control) (Perkins & Johnston Reference Perkins and Johnston1963; Illangasekare et al. Reference Illangasekare, Frippiat, Fucik, Thibodeaux and Mackay2011).

3.3 Residence time, mass balance and concentration profiles

The concept of residence-time distribution (RTD) (Fogler Reference Fogler1999) is used to evaluate the results from the pulse-tracer tests. When applied to the effluent tracer profile predicted by the model, the RTD distribution function $E(t)$ is defined as

(3.8) $$\begin{eqnarray}E(t)=\frac{c^{out}(t)}{\displaystyle \int _{0}^{\infty }c^{out}(t)\,\text{d}t}.\end{eqnarray}$$

Accordingly, the denominator in (3.8) is used to compute the total amount of tracer produced, i.e.

(3.9) $$\begin{eqnarray}M^{out}=q\displaystyle \int _{0}^{\infty }c^{out}(t)\,\text{d}t\end{eqnarray}$$

while the first moment of the RTD function provides an estimate for the mean residence time $t_{R}$ , i.e.

(3.10) $$\begin{eqnarray}t_{R}=\displaystyle \int _{0}^{\infty }tE(t)\,\text{d}t.\end{eqnarray}$$

The integrals in (3.8)–(3.10) are computed via the trapezoidal method and are truncated at $t=60$  min for this study. The value of $t_{R}$ is corrected to account for the mean residence time of the tracer input function, i.e. $t_{R}^{\ast }=t_{R}-0.5t_{inj}$ , in order to be compared with the expected average residence time of the rock sample, ${\it\tau}={\it\phi}AL/q$ . Application of (3.9) to the injected tracer profile allows for a verification of the material balance by quantifying the error, ${\it\varepsilon}=1-M^{out}/M^{inj}$ . The exact same exercise is repeated for evaluating the experimental results and the corresponding parameters are calculated, namely $M_{exp}^{inj}$ , $M_{exp}^{out}$ and $t_{R,exp}^{\ast }$ . Note that in the latter case the computed mean residence time is additionally corrected for the amount of time ( $t_{d}=V_{d}/q$ ) that the tracer has spent in the tubings between the rock sample and the detectors. Finally, tracer concentrations computed within the rock are resampled over the time interval ${\rm\Delta}t=1$  min to allow for a direct comparison with the dynamic image frames generated from the PET scanner (see § 2.4), i.e.

(3.11) $$\begin{eqnarray}C(t_{i})=\frac{1}{{\rm\Delta}t}\displaystyle \int _{t_{1}}^{t_{2}}c(t)\,\text{d}t\end{eqnarray}$$

with $t_{2}-t_{1}={\rm\Delta}t$ . Note that when compared to those measured by PET ( $A_{0j}^{PET}$ ), the concentration values obtained from (3.11) need to be additionally multiplied by the porosity ${\it\phi}$ , as the former are reported per unit voxel volume.