Uncertainty estimation in volumetrics for supporting hydrocarbon E&P decision making

Uncertainty estimation in volumetrics for supporting hydrocarbon E&P decision making Frans J.T. Floris 1 and Martin R.H.E. Peersmann Netherlands Institute of Applied Geoscience TNO P.O. Box 6012, 2600 JA, Delft email : floris@nitg.tno.nl, peersmann@nitg.tno.nl Abstract A methodology is presented for uncertainty estimation in volumetrics. Firstly, we stress the need for an open hierarchical methodology. This allows for a flexible work process in estimating uncertainty throughout the asset life-cycle, in which data of various scales and accuracy must be integrated. Secondly, a method is explained for the transfer of spatial uncertainty in structure and rock properties to uncertainty in hydrocarbon volume. Thirdly, a new technique is presented for calculating average water saturation. Application to a synthetic case study shows that scalar uncertainty calculation leads to an under-estimation of the uncertainty compared to a spatial approach. Spatial mapping of standard deviation of net hydrocarbon column indicates extra potential in the field. Incorporation of correlations in the field between, for example porosity, permeability and water saturation, increase the uncertainty range. Using extra wells in the uncertainty estimation reduces uncertainty. Now, the known volume lies within the estimated proven to probable range. Keywords : Spatial uncertainty, volumetric estimation, decision making, Monte Carlo Introduction Making decisions in reservoir management requires a method for quantifying uncertainty. In the early stages of the life cycle of a hydrocarbon asset, the estimation of hydrocarbon volume and associated uncertainty are of major importance. Later, when production history becomes available, focus shifts to asessing the uncertainty in the dynamic aspects of hydrocarbon recovery. This paper focuses on the volumetrics, ie. estimation of expectation and uncertainty of initial hydrocarbons in place. A review of current practise regarding probabilitistic volumetrics estimation in the oil industry can be found in Hefner and Thompson (1996). A formal framework for quantifying uncertainty in volumetrics has been presented in Berteig et al. (1988). The framework is applied to a simple case study, but in a later paper a realistic case study is performed (Lia et al. 1997). The resulting volumetric uncertainty estimates can be used as input for material balance calculation (see van Kruijsdijk, 1996) or for dynamic reservoir simulations which will provide a production forecast estimate with uncertainty. The production forecast in turn is used in economics or NPV calculations and further transformed into risk involved in decisions. In our approach we highlight three ingredients, process hierarchy, spatial uncertainty, water saturation estimation, that we believe are important in our approach. Process hierarchy During the asset life cycle more data is obtained in order to do technical analysis to provide input into the decision process. Initially, only very rough estimates of gross rock volume and hydrocarbon content can be made due to lack of data. When more data become available, for example from wells, the technical assessment study can become more detailed. Therefore, uncertainty estimation for supporting the decision making process must 1 Corresponding author

Uncertainty estimation in volumetrics for supporting... 2 allow for different levels of detail. A second problem is that correlations which hold in one field, may not be valid for another field. Therefore, we cannot completely fix the flow of processes that occur in the technical assessment. These two observations call for an open hierarchical system of calculation in which the user has enough control to specify the level of detail and process flow that he needs for the field under study. In order to accommodate the requirement of openness, we have developed an open Monte Carlo engine, in which the user can specify the calculations that need to be performed. In order to accommodate different levels of detail, the Monte Carlo engine can work with both scalar data for simple global estimates and arbitrary dimensional grid data, in order to allow for detailed calculations. Scalar and grid calculations can be mixed arbitrarily. The openness is realized by allowing the user to specify his own stochastic variables. The stochastic variables can be correlated through standard mathematical expressions. Moreover, a generic interface has been defined to couple any external calculation program in a transparent way. This external program may perform functions like interpolation, kriging or fluid flow simulation, in order to make more advanced correlations possible. Proprietary software may be plugged in this way. The Monte Carlo engine takes care of the calculation of probability distributions through the tree of stochastic variables. For certain fixed interpretation processes, templates can be used which pre-define the variables and calculations involved. Apart from being able to propagate probability distributions up through the processes, it is very important for the decision making process to be able to track the largest source of uncertainty down the tree. When the largest source of uncertainty has been found, data acquisition on this quantity will have the largest impact on the overall uncertainty. Figure 1 gives an example of a volumetrics calculation in which some of the components of the top level formula are expanded in more detail. We consider porosity as a scalar variable, but layer thickness is modelled in much more detail through velocity and reflection time grids derived from seismic. Note the use of spatial uncertainty grids, σ V and σ T which is the topic of the next section. Spatial uncertainty Some rock properties and rock property correlations of a reservoir are inherently spatial. Aggregating them into averages and estimating the expected hydrocarbon volume from these averages may lead to inaccurate and biased results, especially when the properties are correlated. Moreover, for estimating the uncertainty associated to the aggregate values their spatial distribution must be taken into account. Thus, in order to assess the impact of such rock property uncertainty and associated correlations on the uncertainty of total hydrocarbon volume, uncertainty should be included as a spatial characteristic. We note here that uncertainty tools such as the spreadsheet add-ons @RISK and CRISTAL BALL do not support this. Geostatistical methods can help to model spatial uncertainty. Here, we propose a method for generating possible images of the quantities occuring in spatial models, which each share the required geostatistical properties, mean, standard deviation and variogram. This method has been applied by Samson et al. (1996) in the context of gross rock volume estimation, but we use it in a more general context. Consider a quantity Y, e.g. the velocity or reflection time of the previous section or rock properties such as porosity or permeability. The mean of a quantity Y is defined as the map, Y µ, that would be obtained by a most likely interpretation of the raw data. A standard deviation map, Y µ, must be generated which reflects the uncertainty in the most likely interpretation of the data. Generally, uncertainty will be zero in the wells, if any are available. When the uncertainty map, Y σ, can be estimated in some other points, a smooth uncertainty map can be created by kriging. When considering seismic reflection time seismic resolution can be used to represent the uncertainty distant from the well locations.

Uncertainty estimation in volumetrics for supporting... 3 From the mean and standard deviation map, realizations can be constructed in the following way. A Gaussian random field, Z(x), is generated with a mean equal to zero, standard deviation equal to one and a given variogram (Deutsch & Journel, 1992). The variogram dictates the spatial structure in the Gaussian random field. A realization can then be constructed using Y(x) = Y µ (x) + Z(x) * Y σ (x) (1) Since the mean of Z equals zero, the mean of Y equals Y µ. Since the standard deviation of Z equals one, the standard deviation of the correction term equals Y µ. Thus, all realizations reflect the input mean and uncertainty map. Since Y σ is set to zero at the well locations, all maps will be conditioned to the values of Y µ at the wells. When spatial information of the uncertainty is not available, equation (1) can still be used. In such cases, a global (constant) estimate of the uncertainty can be used. Conditioning to the well data can now be done by conditioning the Gaussian random field, Z(x), to zero at the wells. In Figure 2 we show a mean and uncertainty map of the top of structure of an idealized anticline structure. The two white spots in the uncertainty grid (top right) indicate well locations, where uncertainty vanishes. The shown realizations were generated using a sequence of Gaussian random fields. Water saturation estimation Two approaches are in common use for calculating water saturation maps. The first approach makes maps of water saturation from well data in the same way as porosity or thickness may be mapped from well data. The second approach uses a saturation-height function to estimate the water saturation at any point above the oil water contact (OWC). Clearly, the first approach is not based on physical principles, but on a mapping algorithm. The second approach is based on physical principles. It must be combined with an integration technique to calculate in each point, the average water saturation over a complete layer. Usually, this is done by defining one or more integration points between the layers, calculating the water saturation in the integration points and doing the integration numerically. The selection of the location and number of integration points determines the accuracy of the integration process. This selection may be cumbersome in layers dipping into an oil-water contact, or in layers where the water saturation reaches connate saturation. In Appendix A, an analytical approach is explained, with which the integration can be done without the use of extra integration points and which is insensitive to the location of the layer with respect to fluid contacts or connate/residual saturations. The starting point is the capillary pressure law, which contains the Leverett-J function multiplied by an arbitrary rock property dependent function (traditionally (φ/k) is used here). The key of the method is to apply a preprocessing step in which the Leverett-J function is inverted and integrated. This needs to be done only once. The resulting function can be extended to incorporate the locations of the fluid contacts and connate/residual saturations. In a second step, the average water saturation for any location can be calculated simply but substituting the appropriate rock properties and layers elevations into the resulting expression for average water saturation. Although the method was developed to calculate average water saturation in complete geological layers, it can be used for any partitioning of the reservoir into homogeneous volumes, such as a 3D grid block model. Case study The synthetic case study we have chosen to demonstrate our approach is based on experience from the Northsea. In a synthetic case, the true reservoir is known. Its top structure, overburden velocity and two-way reflection time are given as 2D grids represented by maps (Figure 4). The STOIIP in the true reservoir equals 2.9 MM Sm 3. We

Uncertainty estimation in volumetrics for supporting... 4 note that in the uncertainty estimation average values and correlations are chosen to resemble as close as possible the true field. The top structure (Figure 4) shows that the reservoir comprises a simple elongated dome shape. However, the velocity map contains a high velocity anomaly caused by a set of stacked channels in the overburden. This high velocity anomaly distorts the structure of the reservoir in time (Figure 4C). When seismic processing and interpretation is focussed on mapping the top structure, the high velocity anomaly in the overburden may not be identified. The first well, indicated in Figure 4C, has been drilled in the crest of the seismic-defined structure. An oil/water contact is found at 2015 m. The next two wells are intended to constrain the structure. Since all three wells are in the center region of the high velocity anomaly (Figure 4B), the average velocities are all equal to the anomalous velocity. Thus, the interpretation using a constant overburden velocity seems justified. Thus, the two-way travel time map can be scaled to obtain the interpreted top of structure map. Obviously, we have run into a completely wrong interpretation, resulting in a far too small predicted reservoir structure. In a later phase, a water injector, well 4, is drilled in the supposed flank of our interpreted reservoir. This well, however, finds oil. In fact, it finds the highest top reservoir of all drilled wells (Figure 4A). Apparently, there is a large upside potential, and wells 5, 6 and 7 are drilled to assess it (Table 2). As shown in Figure 5, the reservoir has two layers above the oil/water contact. The layers dip eastward, just below 1 and are unconformly overlain by the reservoir seal. The thickness map of the true reservoir is given in Figure 5. The second layer is much thicker, thus it is unnecessary to consider any lower layers. The top layer has poorer reservoir quality than the bottom reservoir layer. Three wells; Scalar case We start off the uncertainty quantification with a scalar approach that uses only the data from wells 1 to 3. The basic equation for STOIIP calculation is (Figure 1) STOIIP = A*h*φ*NG*(1-S w ) /B o (2) The top reservoir seems fairly well constrained in the direction along the well trend. Perpendicular to the well trend, the flanks are fairly steep, thus suggesting that there is little uncertainty in the reservoir area. The same reasoning holds for the reservoir thickness. From the well data and background knowledge, the probability density functions (pdf s) for the rock and fluid parameters have been estimated (Table 3). Note that only pdf s for layer 1 are given, because in the current interpretation layer 2 is below the OWC and thus does not contribute to the HC volume. The resulting expectation curve for STOIIP obtained by applying Monte Carlo simulation, is given in Figure 6. The 85, 50 and 15 percentiles are equal to respectively 0.56, 0.87, 1.4 MM Sm 3. Due to the overconfidence of the interpretation, in particular the reservoir area and height, the expectation curve shows no upside potential and lies far below the true STOIIP. Three wells; Spatial uncertainty in structure Next, we go into more detail and apply the spatial uncertainty approach. The velocity field still is assumed to have a uniform average value of 2050 m/s. On top of that a standard deviation of 20 m/s is applied. One hundred Gaussian random fields are generated with this average and standard deviation, and conditioned to the well data (Table 4). For each possible velocity map the time-depth conversion is done. Note that for simplicity no uncertainty on the reflection time grid is used. The rock and fluid properties remain scalar at this stage. The average and error map for the net hydrocarbon column are given in Figure 7. The expectation curve is compared to the scalar case in Figure 8. The 85, 50 and 15 percentiles for STOIIP are now given by 2.0, 3.7, 7.7 MM Sm 3.

Uncertainty estimation in volumetrics for supporting... 5 The comparison shows that the main effect of the use of grids, is a shift of the expectation curve to higher values. Secondly, the uncertainty range is larger (1.4 orders of magnitude instead of 1 order of magnitude). The increase in uncertainty is due to a more open interpretation of the structure, in which the structure is not assumed to be as fixed as before. Because the structure is given more flexibility, the high porosity layer 2 occurs more often in the realizations (see Table 5), resulting in a shift to higher STOIIP. Note that the 15-percentile of the scalar estimation (1.4 MM Sm 3 ) lies below the 85-percentile of the grid estimation (2.0 MM Sm 3 ). The error map suggests that there is major uncertainty around the flanks of the reservoir in particular in the south west corner of the study area. This indicates that there may be more volume in the structure than anticipated. In a later phase, the location of well 7 was chosen in order to reduce uncertainty in this area. Three wells; Spatial uncertainty in structure and properties In the next phase, more detail is introduced regarding the rock properties based on the σ v = 20 m/s case. From the wells, some correlations have been extracted. Firstly, the porosity and net gross in layer 1 increase with respectively 5 %, and 7 % going from the bottom to the top. Laterally, the porosity doesn t show a strong trend. Secondly, there is a good linlog correlation between porosity and permeability. Finally, a Leverett-J curve is used to assess the capillary rise of water (data from Table 1 and Figure 3). The spatial distribution of porosity and net gross of layer 2 is fairly uniform. Again a good lin-log correlation is found between porosity and permeability. To model the lower residual water saturation in layer 2, the Leverett-J curve is scaled with respect to water saturation. The result of the STOIIP calculation, compared with the scalar and σ v =20 calculation is shown in Figure 9. The 85, 50 and 15 percentiles are given by 0.66, 1.76 and 4.2 MM Sm 3. At this stage, comparison can be made with the true STOIIP of 2.9 MM Sm 3, which corresponds to the 27 percentile of the expectation curve: it is in the range but on the high side. Because the grid calculation includes a capillary water rise calculation, the expectation curve shows a general shift to the left compared to the case where the structure is modelled with grids. More importantly, the total uncertainty has increased (from 1.4 to 1.8 orders of magnitude). The reason for this is the fact that we have now modelled the correlations existing between porosity, net to gross, permeability and water saturation. The product of a number of independent variables has less uncertainty than the product of a number of correlated variables, because the occurrence of multiple extremes is more likely when the variables are correlated. This means that when the rock properties are correlated with structure, as in our case, a split of the calculation for gross rock volume and fluid contents of the reservoir leads to an underestimation of uncertainty. Thus, the spatial uncertainty assessment should not be focussed at GRV as proposed by Samson et al. (1996), but should be integrated with spatial fluid content estimation. Seven wells; spatial uncertainty Next, the data from the other wells are included, and we investigate how the estimates change. The most important data arising from this investigation is the different well velocities for wells 4 to 6, which are around 2000 m/s. Clearly it is inappropriate to use a uniform expected velocity grid. Instead the expected velocity grid is generated by kriging the well velocities. The expected velocity field and corresponding top structure map are given in Figure 10. The new top structure map clearly indicates the presence of a larger volume of oil compared to the three well case.

Uncertainty estimation in volumetrics for supporting... 6 Now the full spatial uncertainty estimation is repeated. Because the structure is estimated to be much larger, the correlation length used in the generation of Gaussian random fields is doubled. Also, because the structure appears less elongate and slightly differently oriented, the anisotropy is set equal to 0.5 and the angle equal to 40. Apart from the base velocity grid, the changes in geostatistical parameters and the availability of conditioning data in four more wells, all parameter values are the same as those for the three well case. In the expectation curve for STOIIP compared to the three well case, the 85, 50 and 15 percentiles are now given by 1.9, 3.7 and 6.0 MM Sm 3 (Figure 11). The true STOIIP corresponds to the 32 percentile. The comparison of STOIIP (Figure 11) shows that 1. The uncertainty has reduced from 1.8 to 1.2 orders of magnitude; 2. The proven reserves estimate (P85) has drastically increased from 0.66 to 1.9 MM Sm 3 ; 3. The true reservoir dimensions are within the proven (P85) and probable (P50) range. These results quantify the added value that the well data have had on the reserves estimation. Conclusions In this paper, we have presented an integrated method for uncertainty estimation for static earth models at various scales of detail. The methodology allows for spatial correlations of variables as well as correlations through mathematical expressions or external simulators. From the application of our uncertainty estimation methodology to the synthetic case study, we conclude the following. 1. Scalar estimation gives conservative estimates for uncertainty in STOIIP compared to grid estimation. Our experience with other cases has shown that this conclusion is not limited to the current test case. The expectation of STOIIP using scalar estimation was too low for this particular case, due to a too narrow interpretation of the structure and layering in the reservoir. 2. Treating mutually correlated variables, for example porosity with permeability, as uncorrelated results in conservative uncertainty estimates for STOIIP. In particular GRV estimation and fluid content estimation should not be done separately because of their correlation. 3. Extra well data showed a reduction in uncertainty and a better estimate of the expected STOIIP which was know for our synthetic field (quantification of value of information, VOI). References BERTEIG, V., HALVORSEN, K.B., OMRE, H., Prediction of hydrocarbon pore volume with uncertainties, SPE 18325, presented at SPE Ann. Tech. Conf. & Ex., Houston, 2-5 Oct, 1988 DEUTSCH, C.V., JOURNEL, A., 1992, GSLIB Geostatistical software library and user s guide, Oxford U. Press HEFNER, J.M., THOMPSON, R.S., A comparison of probabilistic and deterministic reserve estimates : A case study, SPE RE, Feb 1996, p. 43-47 KRUIJSDIJK, C.P.J.W. VAN, Uncertainty analysis of reserve estimates, SPE 35593, presented at SPE Gas Technology Conf., Calgary, 28 April - 1 May 1996 LIA, O., OMRE, H., TJELMELAND, H., HOLDEN, L., EGELAND, T., Uncertainties in reservoir production forecasts, AAPG Bulletin, 81, no 5, May, 1997, p 775-802 SAMSON, P., DUBRULE, O., EULER, N., Quantifying the impact of structural uncertainties on gross-rock volume estimates, SPE 35535, presented at Europ. 3D Res. Mod. Conf., Stavanger, 16-17 April 1996

Appendix In this appendix, a new approach is presented for the calculation of average saturation maps based on a layered description of a hydrocarbon reservoir. The starting point is capillarygravity equilibrium ρ g h(s w ) = σ cosθ F(φ, k, NG) J(S w ) (3) Here ρ is the density difference between the reservoir fluids, g is the acceleration of gravity, h is height above the fluid/fluid contact, σ is the surface tension, θ is the contact angle, F is an arbitrary function of porosity φ, permeability k and net-to-gross ratio NG; and finally J is the Leverett-J function. A typical choice for F is φ k. Our goal is to calculate the average water saturation over a layer in an arbitrary x-y location S top 1, = Sw ( h) dh h w avg h hbottom where h equals h top -h bottom. In our current setting, we assume that the porosity, permeability and net-to-gross ratio are only a function of the areal coordinates, and not of height within the layer. Then, F may be considered as a constant in our integration. For notational convenience, introduce α = σ cos θ F(φ, k, NG) (5) ρg which, as argued before, is constant in our integration. Then, equation (4) can be rewritten using (3) and (5) as S w, avg α = h htop / α hbottom / α top h h S d w Sw J dj = α α α h ( ) h / α hbottom / α where the integration variable has switched from h to J. Denote the primitive of S w (J) by IntS w (J). Then, we finally obtain S w, avg = α h IntS w htop IntS α w h α When the Leverett-J function is given as a formula, which can be inverted and integrated, equation (7) can be evaluated analytically. However, generally, the Leverett-J function is derived in tabular form from a capillary pressure table. Figure 3 shows the steps to calculate IntS w (J). Figure 3a shows the Leverett-J function as a function of S w. Firstly, the axes must be exchanged to get S w (J) (Figure 3b). Secondly, this function must be integrated. Define IntS w (0) = 0. Then, using an arbitrary integration scheme, the function S w ( ) is integrated from 0 to J (Figure 3c). Figure 3b can be extended to the left with a saturation equal to one, and to the right with a saturation equal to the connate water saturation, S wc. This implied that IntS w (J) can be extended to the left with the linear function J and to the right with the linear function S wc J (arrows in Figure 3c). Through these linear extensions the calculation of averaged water saturation has become independent of the exact location of the layer with respect to the fluid contacts or residual/connate saturations. Consider the following example based on data from Table 1. Using this data in equation (5) with the standard F = φ k relationship gives α = 10. Thus, for a layer with a top at h top = 20 m above OWC and a bottom at h bottom = 10 m above OWC equation (7) gives bottom (4) (6) (7)

Uncertainty estimation in volumetrics for supporting... 8 Sw, avg = IntSw( 2) IntSw( 1) = 1. 3 0. 8 = 0. 5 (8) Next, we consider a case where the capillary transition zone is complete contained in the layer. For h top = 50 m above OWC and a bottom at h bottom = -10 m below OWC, the averaged water saturation is given by 1 2. 1 ( 1) Sw, avg = ( IntSw( 5) IntSw( 1) ) = = 0. 52 (9) 6 6 Application of the method to a 3D grid model is straightforward by using the grid block tops and bottoms in equation (7) instead of the layer tops and bottoms.

Uncertainty estimation in volumetrics for supporting... 9 List of tables Table 1 Data used for water saturation calculation Table 2 Well data for our case study Table 3 Probability density functions for the scalar case for layer 1. Table 4 Geostatistical data Table 5 Probability density functions for the scalar case for layer 2.

Uncertainty estimation in volumetrics for supporting... 10 List of figures Figure 1 Example of process hierarchy in volumetrics calculation. Data in three hierarchical levels are being processed into a final top level target : STOIIP (A = area, h = thickness, ΝΓ = Net to Gross ratio, φ = porosity, Sw = water saturation, Bo = oil volume formation factor,tos = Top of Structure, OWC = oil water contact, V = velocity, σv = velocity uncertainty, T = reflection time, σt = reflection time uncertainty). Figure 2 Based on a mean depth grid (top left) and an uncertainty grid (top right) a number of possible realizations of the depth map can be generated (bottom). Figure 3 Construction of integrated Sw-J curve: A) original J(Sw) curve, B) mirrored Sw(J) curve, C) IntSw(J) curve obtained using trapezoidal integration; arrows indicate extension of the curve outside initial J-range Figure 4 True reservoir with well locations indicated, A) Top Structure, B) Overburden velocity, C) Two-way travel time Figure 5 A) Cross-section of reservoir along wells 5, 4 and 1 and B) Thickness map of the top layer of the true reservoir. See table 2 for well data. Figure 6 Expectation curve of STOIIP for the scalar case with three wells. Figure 7 Average (left) and error (right) map of Net Hydrocarbon Column (darker indicates higher values). Figure 8 Comparison of expectation curve for scalar and grid case. Figure 9 Expectation curve for STOIIP using scalars, grids for structure or all grids. Figure 10 Velocity field obtained by kriging the well velocities and corresponding top of structure interpretation. Figure 11 Comparison of expectation curve for 7 well case with 3 well case.

Uncertainty estimation in volumetrics for supporting... 11 Table 1 Data used for water saturation calculation P cm = 0.02 N/m ρ = 200 kg/m 3 φ = 0.2 g = 10 m/s 2 k = 200 md S wc = 0.2 Table 2 Well data for our case study Well No X Y Top Depth Thickn Layer1 Poro Layer1 NG Layer1 Poro Layer2 NG Layer2 1 3650 2650 2005.7 21.5 15.8 70.5 31.0 87.0 2 4250 3550 2011.5 24.8 15.2 71.7 27.0 77.0 3 3050 1750 2009.6 8.6 13.6 73.0 32.0 85.0 4 2750 3050 2002.2 11.5 13.8 67.6 34.4 88.2 5 1850 3550 2015.8 0.0 0.0 0.0 27.1 80.6 6 1650 2050 2002.8 0.0 0.0 0.0 31.4 85.1 7 1650 650 2020.2 0.0 0.0 0.0 31.8 85.6 Table 3 Probability density functions for the scalar case for layer 1. Variable pdf-type layer 1 Area (10 6 m 2 ) TRIANGULAR 0.69 2.4 3.8 Height (m 2 ) TRIANGULAR 2.9 6.2 9.5 Porosity TRIANGULAR 0.10 0.15 0.20 Net to Gross TRIANGULAR 0.60 0.7 0.80 Water saturation TRIANGULAR 0.15 0.20 0.25 Oil Formation Factor (Rm 3 /Sm 3 ) CONSTANT 1.2 Table 4 Geostatistical data Anisotropy factor = 0.3 Angle w.r.t. north-south axis = 30 Variogram type = spherical Range = triangularly distributed with min=1000, mode=2500, max=10000 Table 5 Probability density functions for the scalar case for layer 2. Variable pdf-type layer 2 Porosity TRIANGULAR 0.20 0.30 0.35 Net to Gross TRIANGULAR 0.75 0.85 0.95 Water saturation TRIANGULAR 0.10 0.15 0.20 Oil Formation Factor Factor (Rm 3 /Sm 3 ) CONSTANT 1.2

Uncertainty estimation in volumetrics for supporting... 12 STOIIP A h NG φ S w B o Level 1 TOS OWC Level 2 V σ V T σ T Level 3 Figure 1 Example of process hierarchy in volumetrics calculation. Data in three hierarchical levels are being processed into a final top level target : STOIIP (A = area, h = thickness, NG = Net to Gross ratio, φ = porosity, S w = water saturation, B o = oil volume formation factor,tos = Top of Structure, OWC = oil water contact, V = velocity, σ V = velocity uncertainty, T = reflection time, σ T = reflection time uncertainty).

Uncertainty estimation in volumetrics for supporting... 13 Figure 2 Based on a mean depth grid (top left) and an uncertainty grid (top right) a number of possible realizations of the depth map can be generated (bottom).

Uncertainty estimation in volumetrics for supporting... 14 J 4 A S w 1.8 B 2 1.5 S wc =.2.4.6.8 1.5 1 2 J 4 S w.6.4.2 IntS w C 2.1 1.9 1.3.8.45-1 -1.5 1 2 4 J 5 Figure 3 Construction of integrated S w -J curve: A) original J(S w ) curve, B) mirrored S w (J) curve, C) IntS w (J) curve obtained using trapezoidal integration; arrows indicate extension of the curve outside initial J-range

Uncertainty estimation in volumetrics for supporting... 15 5 2 5 2 4 1 4 1 6 2010 2005 3 2000 6 3 2050 2015 2020 2025 A 7 2030 2040 2050 B 7 2000 2070 2060 5 2 2050 4 1 2030 2040 2025 C 6 2035 7 2020 3 2010 Figure 4 True reservoir with well locations indicated, A) Top Structure, B) Overburden velocity, C) Two-way travel time 2015 2005

Uncertainty estimation in volumetrics for supporting... 16 well 5 well 4 well 1 20 25 0 5 10 15 OWC A Figure 5 A) Cross-section of reservoir along wells 5, 4 and 1 and B) Thickness map of the top layer of the true reservoir. See Table 2 for well data B

Uncertainty estimation in volumetrics for supporting... 17 Figure 6 Expectation curve of STOIIP for the scalar case with three wells.

Uncertainty estimation in volumetrics for supporting... 18 Figure 7 Average (left) and error (right) map of Net Hydrocarbon Column (darker indicates higher values).

Uncertainty estimation in volumetrics for supporting... 19 all scalar grids for structure true STOIIP Figure 8 Comparison of expectation curve for scalar and grid case.

Uncertainty estimation in volumetrics for supporting... 20 all grids grids for structure all scalar true STOIIP Figure 9 Expectation curve for STOIIP using scalars, grids for structure or all grids.

Uncertainty estimation in volumetrics for supporting... 21 2090 2080 2070 2060 2050 2000 2000 2010 2020 2030 2040 2050 2040 2030 2020 2005 2010 2015 Figure 10 Velocity field obtained by kriging the well velocities and corresponding top of structure interpretation.

Uncertainty estimation in volumetrics for supporting... 22 3 wells 7 wells true STOIIP Figure 11 Comparison of expectation curve for 7 well case with 3 well case.