Multi-point Modeling of Clay Lenses and its Impact on Aquifer Vulnerability

Transcription

1 Multi-point Modeling of Clay Lenses and its Impact on Aquifer Vulnerability Whitney Trainor and Jef Caers Earth, Energy, and Environmental Sciences Stanford Center for Reservoir Forecasting April 30 & May 1, 2009 Abstract Effective groundwater management requires hydrogeologic models built from various data sources. While multi-point geostatistical algorithms have been widely applied in petroleum reservoir characterization, applications to hydrogeology are still few. In this paper we show how the multi-point algorithm snesim is employed to characterize facies distributions, using a dataset of a groundwater system from Århus, Denmark. Geological heterogeneity of this area is the result of glacial depositional and erosional processes. This geologic scenario has created some clay lens features on the surface and within the buried valleys. The clay coverage and lens features determine whether a flow path exists between a surface contaminant and an extraction well. Bayesian classification of lithology from resistivity was performed such that the more extensive resistivity soundings could be used as soft data (representing probability of clay) in the snesim simulations. These facies realizations are then used for flow simulations with tracers placed exhaustively at the surface. The geobody idea was recycled to define the aquifer vulnerability measure: number of tracer concentration bodies intersecting with extraction wells. How to represent the changing support volume of the resistivity soundings is one of several outstanding issues. Preliminary results suggest another data set with more near-surface information will be more consequential to vulnerability results. 1 Introduction & Motivation Much of the world s drinking water is supplied from groundwater sources. Over the past several decades, many aquifers have been compromised by surface-born contaminants due to urban growth and farming activities. Further contamination will continue to be a threat until critical surface recharge locations are zoned as groundwater protection areas. This can only be successfully achieved if the hydraulically complex connections between the contaminant sources at the surface and the underlying aquifers are understood. 1

2 Denmark is one example of this type of scenario. Since 1995, in an effort to identify crucial recharge zones, extensive geophysical datasets were collected over 37% of the Danish countryside the areas designated as particularly valuable due to their high water extraction. The work presented in this report utilizes the geophysical data and flow information available in an area north of the city of Århus. As with the rest of the country, the data was collected with the intention of making more informed decisions regarding the designation of recharge proctection zones. The magnitude of these decisions is considerable, as it could involve the relocation of farms and large compensations for these explusions. Consequently, incorrectly identifying an area as hydraulically connected to important aquifers or vice versa, can lead to a costly error. Although spatial information has an important role and long tradition in decision making for petroleum applications, this has not been the case for the groundwater field. The long-term goals of this research are to develop a methodology that can quantify the value of information for spatial decision-making for groundwater management. An important distinction should be made clear here: although the Århus North data and geology are used in this study, these are mainly used to create a realistic synthetic case. Therefore, no conclusions should be drawn to a particular Danish case from the results in this study. This report will describe the uncertainty in the subsurface heterogeneity, the geophysical information and how these uncertainties were taken into account during the subsurface modeling. More importantly, flow simulation is performed on these models. These results were used to define a measure of vulnerability the critical measure for the Danish groundwater decision. 1.1 Geological Scenario: Buried Valleys Buried Valleys are considered the informal term for Pleistocene (Quaternary) subglacial channels. They have also been described as the result of waxing and wanning of Pleistocene ice sheets [BurVal Working Group, 2006]. The primary method valleys are formed is by subglacial meltwater erosion (sudden outbursts of meltwater released by glacial lakes). Thus, the valley formation is directly related to the morphology and erodability of the geological strata. The secondary method is through direct glacial erosion by ice sheets [BurVal Working Group, 2006]. Several of the processes that create and fill buried valleys are important for understanding the complexity of the Danish aquifer systems and their vulnerability to surface-born pollutants. In Denmark, the superposition of 3 different generations of glaciation have been observed. Thus, multi-generation glacial valleys cross-cut each other and can also appear to abruptly end (as seen in Figure 1). The existence and location of these glacial valleys can be thought of as the primary level of Denmark s aquifer system structure. If largely filled with sand, the buried valley has potential for being a high volume aquifer (reservoir). However, these buried valleys can be re-used as revealed by the observed cutand-fill structures. This describes the secondary level of uncertainty of heterogeneity in the Danish aquifer systems. Most cut-and-fill structures are narrower than the overall buried val- 2

3 Figure 1: Network of Buried Valleys; darker to lighter representing older to younger generations ley, but in some places very wide structures that span the entire valley width can be seen. The complex internal structure can be observed in seismic surveys, electromagnetic surveys and occasionally in borehole data. [Sandersen & Jørgensen, 2003] The possible combinations of heterogeneity are indeterminable, on account of the erosion and deposition processes that create both the valleys themselves and the structures inside them. Figure 2 shows a few different possible internal heterogeneities and varying extent of overlying strata, which deems the valley as actually buried. Figure 2: Possible Eroded Substrate & Internal Structures of Buried Valleys modified from [Sandersen & Jørgensen, 2003] 1.2 Hydrogeology: Aquifer Vulnerability Due to the generally complex internal structure of the valleys, potentially protective clay layers above the aquifers are likely to be discontinuous...the aquifers inside the valley will thus have a varying degree of natural protection. Even if laterally extensive clay layers are present, the protective effect will only have local importance if the surrounding sediments are sand-dominated....the valleys may therefore create short-circuits between the aquifers in the valley and the aquifers in the surrounding strata. [BurVal Working Group, 2006] 3

4 Experience has led Danish hydrologists to believe that aquifers with a clay cap thickness of less than 15m are more vulnerable to anthropogenic contamination, specifically the leaching of nitrate [Thomsen, et. al, 2004]. Thus, these clay features, whether they are interior structures (as in scenario H in Figure 2) or the continuous cap (as in scenario I in Figure 2) are the determining factor whether surface contaminants reach the underlying or neighboring aquifers. 2 Geophysical Dataset: Time-Domain Electromagnetic Soundings Favorable electrical conductivity contrasts exist between the flow-barrier clay facies and the high permeability sand facies; clay generally has an electrical resistivity less than 30 ohm-meters, whereas sand is usually greater than 80 ohm-meters. Hence, most of the geophysical surveys in Denmark have been either electrical or electromagnetic (EM). The Time-domain EM method (TDEM or TEM) works with a transmitter loop that turns on and off a direct current to induce currents and fields into the subsurface [Christiansen, 2003]. How the TEM method works and measures the earth s resistivity structure can be summarized into six steps with the appropriate Maxwell equation 1 : 1. The constant current I wire in the transmitter produces a primary magnetic field H p : H p = I wire 2. Current suddenly terminates; the changing magnetic field induces a secondary electric field E s in the earth that attempts to oppose the change (where µ is magnetic permeability of the earth): E s = (µh p) t 3. The induced electric field produces an image current, which in turn produces a secondary magnetic field H s (where σ is electrical conductivity of earth): I s = σe s H s = I s 4. Current diffuses outward and downward over time (t). The diffusion rate: is proportional to 1/sqrt(t) depends on earth conductivity (σ) 5. The diffusing, time varying current produces time varying secondary magnetic field. ( H s ) t = I s dt 1 modified from lecture notes of D. Alumbaugh, University of California, Berkeley 4

5 6. This decaying magnetic field H s produces time-varying voltage V in the receiver coil s (where ˆn is the unit normal vector through the receiver coil): V = (µhs ) ˆnds t As depicted in Figure 3, the induced fields are the result of the current being turned off in the smaller transmitter coil (T x) which is inside the larger receiver loop (Rx). The receiver loop measures the changing magnetic field (H) from the induced currents (I t ). In short, one can think of the TEM method as sampling the earth as described in step 4 outward and downward. At later time gates (measurement times which range from 10 6 to 10 3 seconds), the measurement swath is wider, but the signal is weaker and perhaps less reliable. The TEM measurement presents a challenge, as this changing volume support is not easily represented on regular grids, commonly used to create the lithological models. This is discussed further in the Section 2.1. As the Maxwell Equations suggest, the geophysical inversion to obtain resistivity models from TEM data is computationally demanding. The state of the science is developing feasible 3D inversions. In view of that fact, it is understandable that the more than 3,000 TEM soundings in the Århus North area were inverted using 1D inversion codes. The resulting 1D electrical resistivity models, representing the X- and Y-locations of the transmitter loop, are parameterized into vertical layers, each with an electrical resistivity (ρ) and thickness value (h). The geophysical inversion also provides an uncertainty measure on both of these parameters (also described in Section 2.1). Figure 3: Left: Tx & Rx loops on surface of conductive, layered earth; Center: Induced, diffusing currents I t ; Right: Secondary H field 2.1 Calibration of TEM to lithology The real goal is to relate these TEM models of electrical resistivity to the property of consequence: lithological facies that have different flow properties and affect aquifer vulnerability. Several challenges exist in relating or calibrating the TEM models to lithology. The first is silt. Silt has the electrical properties of sand, and the flow properties of clay. Second, again as seen in Figure 3, is the changing TEM volume of support. The geophysical inversion partially accounts for this through the simulation of the physics. However, the simulations assume homogeneous halfspace (where resistivity can only change vertically). And lastly, the lithological 5

6 information available (as what is common in the hydrogeologic community) are drillers logs. These data are subjective, lithological observations made while wellcuttings are being excavated. The drillers log classifications for the Århus North area were reduced from more than 20 to two: either clay or non-clay (gravel, sand, silt, limestone, etc). Two general approaches are common when calibrating the larger support data (usually with more exhaustive coverage) to the property of interest at a smaller scale. The first technique is to calibrate the larger support data to represent a probability of the high resolution property existing at a certain location. This is useful for use with multiple-point algorithms which use Bayes law to combine probabilities (such as a probability of a repeating geologic pattern). The second is to utilize the volume support data as a proportion measurement. This approach was considered in attempt to downscale TEM models to clay proportions. However, results were unworkable, limited by the poor point-scale variogram from the driller s logs. Along with the 3,000 TEM models, Århus North has 750 drillers logs. However, only 7 TEM-drillers log pairs are within 25 meters map distance (XY location). Perfect collocation most likely doesn t exist since the TEM measurement can be ruined by metal in the well head or casing. On average, the TEM models reach 300 meters depth while the drillers logs are usually 100 meters deep. Both the TEM models and the drillers logs were re-sampled at 1 meter vertical spacing, such that some TEM layers resistivities were repeated to account for layer thickness >> 1m (this may account for the high modes seen in Figure 4). Generally Figure 4 shows a favorable resistivity separation between sand and clay. However, it is important to note that the effects of re-sampling, the different support size of the two measurements, and the subjective nature of the drillers logs have not been taken into account. Also included in Figure 4 is the error estimates from the TEM inversion. The confidence interval for each parameter is based on a linear approximation to the covariance of the estimation error C est : C est = ( G T C G) 1 (1) where C is the covariance matrix of the estimated data error and G is the Jacobian, ie the partial derivatives of the data vector to the model parameters of layer resisitivity (ρ) and thickness (h) [Tarantola & Valette, 1982]. Standard deviations on model parameters are calculated as the square root of the diagonal elements in C est. The analysis gives a standard deviation factor (STDF) on the parameter p s (representing ρ or h) [Christiansen, 2003]. ST DF (p s ) = exp( C est,ss ) (2) The inversion is executed in the logarithmic space. Therefore, the quotient and product with the model parameter p s and the resulting STDF factor can be used to define a confidence interval of two standard deviations (2σ): [ ] p s 2σ = ST DF (p s ), p s ST DF (p s ) 6 (3)

7 Figure 4: Resistivity (Ohm-m) and Lithology of 7 collocated TEM models and Drillers Logs These confidence intervals give a measure of the inaccuracy of the TEM models, both the resistivity value and its vertical location, due to data error. This can lead to misclassification of lithology at the 7 collocations with the drillers logs. To account for this, the variability is modeled using realizations of each of the original 7 collocated TEM models. Each realization is modified according to the confidence intervals of the original. Thirty realizations of the 1D TEM model were made by drawing from a multivariate normal distribution. This distribution s mean is defined by the original TEM model parameters, and the diagonal of the covariance is defined by the uncertainty values from the TEM parameters. Since resistivity and thickness are negatively correlated in the TEM response, the off-diagonal terms were assigned a -0.6 value. This value is typical for statistical rock physics simulations involving attributes that are known to be physically negatively correlated. The calibration provides the likelihood of the electrical resistivity ρ given the lithology being clay or non-clay. Using Bayes rule (Equation 4), the posteriors or soft probabilities can be calculated: given the TEM model resisitivities what is the probability of the lithology being clay or non-clay? The marginal (p(ρ)) and prior (p(litho)) are scaling factors and were assigned a value of 1. p(litho ρ) = p(litho)p(ρ litho) p(ρ) (4) As is apparent in Figure 4, the resistivity values discriminate well between clay and non-clay. Figure 5, shows the 3D view of the locations of TEM models, with their original posterior values. 3 Geostatistical Simulation As mentioned in Section 1.1, two levels of subsurface heterogeneity exist: the location of the buried valleys (the reservoir) and the interior structure of the valleys. Although the locations of the valleys themselves could be represented stochastically using the patterns of Figure 1, current work focuses on the interior structures. Thus, 7

8 Figure 5: Clay posterior; 3D view from above Figure 6: Deterministic Model of Buried Valley (Red) and Non-valley (Blue) Locations. Vertical Exaggeration x25. View from South valley and non-valley are modeled deterministically from the TEM models (shown in Figure 6). Very little information exists in the geological literature on the dimension of these internal structures. To represent the uncertainty on clay lens dimension and spatial patterns, several binary clay lens training images (TI representation of repeating patterns of geologic system) were created (seen in Figure 7) using the GSLIB Ellipsim program [Deutsch, et. al, 1998]. Blue represents the clay (flow-barrier) facies, and red is the non-clay (permeable) facies. These TI s are used with the algorithm snesim within Stanford s Geostatistical Earth Modeling Software (SGeMS) [Remy, et. al, 2002]. Single Normal Equation Simulation (snesim) is a multiple-point geostatistical algorithm, which generates a stochastic facies realization using the TI [Strebelle, 2002]. How snesim accomplishes this can be roughly summarized into seven steps 8

9 Figure 7: 5 different Clay Lens Training Images On the Training Image: 1. snesim scans the TI for all independent patterns using a template 2. The template captures different data patterns 3. The Search tree is used to organize and store different data events or unique patterns from the TI On the simulation grid: 4. Randomly select a location 5. Define a data event within the template 6. Use the search tree to find similar patterns and define the probabilities on the possible discrete values for the center of the template 7. Draw from these probabilities to determine the center value. The region feature was used to only simulate clay features within the buried valley locations. Additionally, to test the influence of the posterior on both the snesim realization and eventually the flow simulations, a synthetic, less discriminating data set of resistivity and lithology was created (Figure 8) as an alternative to the one provided in Figures 4 and 5. However, to make either the data-calibrated posterior or the less-discriminating posterior compatible with the TI s of Figure 7 and the snesim algorithm, smoothed cubes of the posteriors were created. In order to avoid artifacts due to the vertical support of the TEM models (ie Figure 5), the vertical sampling of the TEMposteriors was decimated such that only every 3rd cell (or 12m) was retained. Using 9

10 this thinned set of posteriors as hard data in an SGSIM realization, the final posteriors were obtained. Figure 9 shows the same cross-section location for both the synthetic and original posteriors. 4 Flow Simulation: Defining a Vulnerability Measure All the binary snesim realizations were then utilized in flow simulation. The clay facies was assigned a permeability of 0.1 milli-darcy (md) and non-clay 1000 md. Specifically of interest is how surface-born contaminants will reach extraction (pumping) wells. To observe how different flow property heterogeneity will influence the surface-to-well hydraulic connections, a conservative tracer was placed exhaustively on the surface. The sources (boundary recharge and rainfall) and sinks (extraction wells) are simulated for 20 years. A conventional metric to compare surface-to-well connectivity is the volume of tracer pumped out. However, we propose a different method extending the geobody idea to concentration bodies. The geobody concept was developed to identify connected bodies (cluster identification) in binary facies models [Hoshen & Kopelman, 1976], and later used to modified the snesim algorithm to condition to tracer tests (hydraulic connection data) [Renard & Caers, 2008]. To utilize the geobody idea, a concentration threshold is defined for the simulation grid at the last time step. For this study, a threshold of lbs/stb (pounds per standard barrel field units in an oil reservoir simulator). Defining a threshold allows for the continuous concentration cube to be transformed to a binary representation of connected concentration bodies (Figure 10). A connection evaluation can then be made about how many independent concentration bodies intersect the 61 pumping wells, or the total length of these concentration body-well intersections can be evaluated. This discrete measure is advantageous compared to the more abstract metric of the volume of tracer pumped out. The concentration bodies metric provides a more concrete notion of the number of connections from Figure 8: Resistivity (Ohm-meter) vs Lithology: Example of less discriminating TEM 10

11 Figure 9: Cross-Section at X= Top: X-Section of Synthetic Posterior. Bottom: X-section of Original Posterior. Red color means higher probability of clay at that location. Grey indicates non-valley location. Figure 10: Top: Log Concentrations. Bottom: Binary representation with threshold= Black lines represent locations of pumping wells. 11

12 the surface to extraction wells, and is therefore considered a suitable indicator of aquifer vulnerability. Figure 11: Number of concentration body intersects from flow simulations on realizations from original posterior; Clay percentage for SNESIM realizations: Left plot 15%, right plot 25%. Shapes depict different random seeds In Figures 11 and 12 the x-axis represents the area of the largest clay lens featured in the TI. Above each string of models is the TI number that was used to generate them (shown in Figure 7). The plot of the left represents the realizations with 15% clay, and on the right, those with 25% clay are plotted. Each group has three models generated with different random seeds. Specifically, the plots of Figure 11 show two trends. First, as the size of the largest clay lens increases, the number of concentration bodies-to-well connections decreases. And second, the realizations with 25% clay (right) show fewer connections than those with 15% (left). Figure 12 displays the total length of all the concentration body-well connections. Again, the two trends (decrease in length with increase in clay lens area and clay percentage) are generally seen except for the suite of models from TI 5. The length of the intersections is longer for TI 5. Unlike the other two training images, TI 5 has more clay features of different sizes and shapes. These complex patterns may restrict the possible preferential flow paths from the surface to well, which may explain why the intersection lengths are longer. The cross-sections in Figures 13 and 14 give insight into how the clay influences the tracer. Figure 13 demonstrates the log tracer concentration and permeability for the small clay lenses of TI 1, while Figure 14 is from a model created with TI 5, with distribution of smaller and larger clay lenses. Figures 15 and 16 show two results from models created with TI 3 and TI 4 respectively. The smaller clay features of Figure 13 result in more fingering of the concentration bodies. In Figure 14, the left side demonstrates an absence of clay (which is consistent with the clay posterior), and consequently the average tracer depth is 10 cells. The right side has more clay and more limited tracer penetration (average penetration 5 cells). Therefore, information with more near-surface details will have increased consequences for the vulnerability measurement. Figures 17 and 18 are the results from the realizations that used the soft probability from the synthetic, less-discriminating posterior. These figures demonstrate 12

13 Figure 12: Length of concentration body-well intersection from flow simulations on realizations from original posterior Figure 13: Cross-Section at X= for model from TI1 15% clay. Top: Log Tracer Concentration. Bottom: Permeability (Pink=Clay) Figure 14: Cross-Section at X= for model from TI3 15% clay. Top: Log Tracer Concentration. Bottom: Permeability (Pink=Clay) the same trends of Figures 11 and 12. The flow simulations on the models created from TI 5 (with 3 different clay lens sizes) show the same tendency to have longer concentration body connections with the wells. Also similar to the simulations with the original posterior, the models with the greatest clay lens area and 15% clay demonstrate greater concentration body-well connections. This could be because the marginal (clay percentage) and the length of the clay lens of TI given to the snesim algorithm are incompatible, causing the actual clay lenses in the realizations to be not as large. 13

14 Figure 15: Cross-Section at X= for model from TI2 15% clay. Top: Log Tracer Concentration. Bottom: Permeability (Pink=Clay) Figure 16: Cross-Section at X= for model from TI2 15% clay Scale 2. Top: Log Tracer Concentration. Bottom: Permeability (Pink=Clay) Figure 17: Number of Intersections from realizations using synthetic posteriors. Left plot: models with 15% Clay. Right plot: models with 25% 5 Summary The work presented here represents the first developments of a workflow which includes geological information, geostatistical modeling and flow simulation. The modeling is performed with the objective of resolving the groundwater decision: which locations are vulnerable to surface contaminants and therefore should be protected? Currently, modeling of the buried valleys is done deterministically using the 3,000 TEM models. It was these resistivity models that were also used to determine the likelihood of the presence of interior clay structures. After generating snesim realizations conditioned to these soft probabilities and different clay lens training images, flow simulation was performed. A measure of vulnerability was established recycling the geobody concept for tracer concentration bodies. However, it was observed from the cross-sections of tracer concentrations that the heterogeneity in the upper 5 cells (20m) really determines the preferential flow paths of the tracer. Therefore, a different data source should be considered to extract more near surface information. Electrical profiling, where current and potential electrodes are inserted into the 14

15 Figure 18: Intersection Lengths from realizations using synthetic posteriors. Left plot: models with 15% Clay. Right plot: models with 25% ground at different spacings, gives good resolution of near-surface conductive features such as clay. However, because of the need for galvanic contact with the subsurface (insertion electrodes) and the particular spacing between electrodes needed for the inversion algorithms (moving of electrodes), this method was too labor intensive and hence expensive for extensive coverage. The method was improved by mounting the instruments on a vehicle that could be pulled; it is now known as the PACES method (pulled array continuous electrical sounding) [Sørensen, K. I., 1996]. Secondly, the design provided more electrode separations. Having more current electrodes, the new method provides better possibilities for interpreting multiple near-surface layers, and, hence, the presence or absence of protective clay covers. Besides utilizing PACES, other future work may include analyzing: the uncertainty of drillers logs, how decimation may have affected the near-surface information in the posterior, how to speed up flow simulations and more simulations (results) for more resolute conclusions. We saw that the critical depths for aquifer vulnerability were in the top five grid cells. However, in order to make a viable soft probability cube, only every 3rd TEM-posterior was retained. Also, the changing (diffusing) TEM volume support was not truly accounted for. Each one of the 20 year simulation, requires 1 hour on 8 nodes using the parallel Eclipse flow simulator. The mass conservation computation (necessary for the tracer) significantly increases the CPU time. For future work, it maybe worthwhile to consider streamline simulation in place of flow simulations. Further simulations with varying clay lens sizes are needed to define in which situations, higher quality data is more useful in detecting the consequential subsurface heterogeneity. 15

16 References [BurVal Working Group, 2006] Groundwater Resources in Buried Valleys: A Challenge for Geosciences Leibniz Institute for Applied Geosciences (GGA-Institut), Stilleweg 2, Hannover, Germany. [Christiansen, 2003] Christiansen, A.V., Application of airborne TEM methods in Denmark and layered 2D inversion of resistivity data. PhD Thesis, Department of Earth Sciences, University of Århus, Denmark. [Deutsch, et. al, 1998] Deutsch, C., and Journel, A., GSLIB Geostatistical Software Library and User s Guide, Oxford University Press: New York. [Hoshen & Kopelman, 1976] Hoshen, J., and Kopelman, R Percolation and Cluster Distribution. I. Cluster Multiple Labeling Technique and Critical Concentration Algorithm. Physical Review B 14: [Huuse & Lykke-Andersen, 2000] Huuse, M., Lykke-Andersen, H. Overdeepened Quaternary Valleys in the Eastern Danish North Sea: Morphology and Origin, Quaternary Science Reviews, 19, [Sandersen & Jørgensen, 2003] Sandersen, F. and Jørgensen, Buried Quaternary Valleys in Western Denmark Occurrence and Inferred Implications for Groundwater Resources and Vulnerability, Journal of Applied Geophysics, 53, [Sørensen, K. I., 1996] Sørensen, K. I., Pulled array continuous electrical profiling. First break 14, [Strebelle, 2002] Strebelle, S., Conditional Simulation of Complex Geological Structures using Multiple-point Statistics. Mathematical Geology, V. 34, No. 1, [Thomsen, et. al, 2004] Thomsen, R., Søndergaard, V.H., and Sørensen, K.I., Hydrogeological Mapping as a Basis for Establishing Site-specific Groundwater Protection Zones in Denmark. Hydrogeology Journal, V. 12, [Remy, et. al, 2002] Remy, N., and Journel, A., J.GsTL: A Geostatistical Template Library in C++, Computers & Geosciences; Proceedings of the IAMG2001, Annual Conference of the International Association for Mathematical Geology, 28, 8, [Renard & Caers, 2008] Renard, P., and Caers, J., Conditioning Facies Simulations with Connectivity Data, SCRF Annual Reports [Tarantola & Valette, 1982] Generalized Non-linear Inverse Problems Solved Using the Least Squares Criterion. Reviews of Geophysics and Space Physics 20(2),