## Abstract

Analytical well-test solutions are mainly derived for simplified and idealized reservoir models and therefore cannot always honour the true complexity of real reservoir heterogeneities. Pressure transients in the reservoir average out heterogeneities, and therefore some interpretations may not be relevant and could be misleading. Geological well testing refers to the numerical simulation of transient tests by setting up detailed geological models, within which different scales of heterogeneity are present. The concept of geological well testing described in this paper assists in selecting from multiple equi-probable static models. This approach is used to understand which heterogeneities can influence the pressure transients. In this paper, a low-energy multi-facies fluvial reservoir is studied, for which data from a well test of exceptionally long duration are available. The pervasive low reservoir quality facies and restricted macro cross-flow between the reservoir layers give rise to an effective commingled system of flow into the wellbore (i.e. zero or very low vertical cross-flow between the reservoir units). In our model, facies transitions produce lateral cross-flow transients that result in a ‘double-ramp-effect’ signature in the test response. A sophisticated multi-point statistical (MPS) facies modelling approach is utilized to simulate complex geological heterogeneities and to represent facies spatial connectivity within a set of generated static models. The geological well-test model responses to a real well-testing cycle are then evaluated using dynamic simulation. The pressure match between simulated and recorded data is improved by generating multiple facies and petrophysical realizations, and by applying an engineering-based hybridization algorithm to combine different models that match particular portions of the real well-test response. In this example, the reservoir dynamics are controlled by subtle interaction between high-permeability channels and low-permeability floodplain deposits. Effective integration of geology and dynamic data using modern methods can lead to better reservoir characterization and modelling of such complex reservoir systems.

## Introduction

Classical transient well-test interpretation is based on the analytical solution of the diffusivity equation under various boundary conditions. These analytical solutions are mainly derived for draw-down cases and are used for the idealized reservoir characterization based on the estimation of the reservoir parameters. However, many of the analytical solutions are either very complex and computationally costly (Kuchuk 1996; Houzé *et al*. 2011) or may not exist for many geological and reservoir heterogeneities (Zheng *et al*. 1996; Zheng 1997; Kuchuk *et al*. 2010). Therefore, numerical techniques are implemented to solve the well-test diffusivity equation in complex reservoirs with their associated boundary conditions. Numerical well testing can be regarded as a ‘super type curve generator’ (Houzé *et al*. 2011). Numerical well-test simulations have been shown to give accurate results compared with analytical solutions and can be used to tackle non-linearity (e.g. non-Darcy flow, multi-phase flow and non-consolidated formation), complex well situations (e.g. multi-segment and slanted well) and combined complex reservoir heterogeneities (e.g. multi-layered, multi-facies, highly faulted and fractured).

The term ‘geological well testing’, where a greater emphasis is placed on geological modelling, is used in preference to the term ‘numerical well testing’. Geological well testing refers to the numerical simulation of transient tests by setting up the detailed geological models, within which different heterogeneity scales are honoured (Massonnat & Bandiziol 1991; Corbett *et al*. 1996; Zheng *et al*. 1996; Corbett *et al*. 2010). This process can be used to generate a family of geotype curves (Corbett *et al*. 2005) that investigate the limits of early, middle and late time well-test responses of particular reservoir heterogeneities with the acceptable ranges of uncertainty in structure, facies and petrophysical parameters (Massonnat *et al*. 1993; Corbett *et al*. 1996, 2005, 2010; Zheng *et al*. 1996; Corbett 1997, 2009; Zheng 1997; Sagawa *et al*. 2000; Ellabad *et al*. 2001; de Rooij *et al*. 2002; Robertson *et al*. 2002; Mijinyawa & Gringarten 2008). Therefore, the scope of well testing has evolved mainly from a parameter estimation technique to a more sophisticated discipline from which different levels of reservoir heterogeneity can be analysed (Du & Stewart 1994). The detailed analysis of the geotype curves provides a better perspective of transient performance of the reservoir model, which helps select appropriate reservoir models and constrains their response against the real well-test data. Therefore, the final matches to the recorded test data are obtained from numerical solutions of the geological testing rather than by forcing the simplified and average analytical solutions.

In this paper, the geological well-test transient pressure response of a commingled low-energy anastomosing fluvial environment is considered for a well test of exceptionally long duration. It is in these types of reservoir systems where detailed integration of geosciences and engineering is both challenging and potentially rewarding (Corbett 2012). The real test data show a ‘double-ramp-effect response’, where the pressure derivative curve increases with time and has two stabilization periods. The lateral cross-flow caused by the facies transitions is captured by employing multi-point statistical (MPS) facies modelling methods, which are anchored by the conditioning well data. Dynamic simulation of the conditioned MPS models enables us to reproduce a set of extended well-test responses and, hence, compare the dynamic behaviour of the models against that of the reservoir. In other words, the well-test data are effectively used to select appropriate reservoir models from a class of equally probable static models derived from an MPS approach. Therefore, any geological realization can be discarded if its dynamic simulation does not match the test response. Furthermore, successive evaluation of the simulated well-test responses resulted in the selection of two particular MPS models that were combined, using an engineering-based hybridization approach, to achieve a quality match with the real extended test data.

## Geological Modelling of the Ramp-Effect Well-Test Response in Commingled Fluvial Reservoirs

The transient well-test response of a single fluvial channel is commonly associated with a middle time linear-flow regime on the pressure derivative curve. This is a response of a simplified model of a pair of infinite parallel no-flow boundaries. This simplified geometry permits the derivation of an analytical solution for the linear-flow regime, and the estimation of the channel width. Although this estimated channel width seems to be a deterministic and engineering-based quantity, it can also be used in limited stochastic reservoir modelling workflows. However, clearly the lateral and vertical stacking of channels in fluvial environments impose further complexities, where the interlayer and/or intralayer cross-flows affect the well-test responses. Therefore, classical responses of commingled and cross-flow systems for isotropic and homogenous reservoirs (i.e. a zero-slope pressure derivative for the commingled systems, and the appearance of a V-shape signature in the derivative response (Bourdet 2002) for the cross-flow systems) are no longer valid in the presence of complex geology, and the simplified solutions cannot fully describe the dynamic behaviour of the system. Hence, it is beneficial to employ the geological interpretation from additional sources of data, such as outcrop and analogue studies.

Corbett *et al*. (2012) and Hamdi (2012) described three pressure transient extreme variants that are potentially present in fluvial systems. These well-test family members include the geoskin (Corbett *et al*. 1996), the geochoke (Corbett *et al*. 2005) and the ‘ramp-effect’ response (Corbett *et al*. 2012; Hamdi 2012, 2014; Hamdi *et al*. 2013). Of more interest, the ramp effect is a monotonic increase of the pressure derivative (with a more frequent half-slope trend) over at least one log cycle (Corbett *et al*. 2012). This response is present in variety of environments, where the lateral mobility or connectivity degrades away from the well. The ramp effect can be described either in terms of geostatistical parameters of the continuous petrophysical properties (i.e. permeability) – for example, the correlation length and covariance function (Hamdi 2012) – or transitional facies-contrast trends preserved in the fluvial environment with meandering (Corbett *et al*. 2012) or anastomosing channel deposits. This paper presents a special case of the ramp effect in a low-energy environment, where the lateral connectivity of the high-quality facies is drastically reducing away from the wellbore. The lateral cross-flow between ‘multiple’ facies of different quality occurs when the diffusion front reaches the distant areas. In this environment, the reservoir deposits are in the form of some elongated (and possibly isolated) sidebars with high-quality sands that are deposited in one side of the channel belt.

Hydrocarbon volumes in *partially* connected sandbodies impose particular challenges for geological modelling when fluid movement and production profiles are calculated (Henriquez *et al*. 1990). However, in this study it is the influence of the ‘cocooning’ low-permeable facies that strongly affects the response. Developing reliable geological descriptions of these fluvial environments is one of the key challenges. Sparse static conditioning data, limited spatial information, complex facies distribution within the three-dimensional (3D) channel network and non-unique solutions are among the main difficulties in producing reliable reservoir models that can preserve the spatial connectivity of the presented facies. In these situations, because the variogram-based two-point geostatistics cannot look for the spatial continuity between properties at more than two locations at a time, pixel-based algorithms give poor representations of the actual facies geometries (Strebelle & Journel 2001). The object-based models, however, have difficulty in reproducing the representative facies in the low-energy environment, such as meandering river deposits (Hu & Chugunova 2008). However, the more sophisticated MPS models provide more promising results in constructing the complex geological heterogeneities, which can better preserve the spatial shape and the connectivity of reservoir facies (Corbett *et al*. 2012). Such multiple-point statistics cannot be inferred from typically limited well data but could be read from training images depicting the expected subsurface heterogeneities (Strebelle & Journel 2001). The training image, which carries the spatial relationship of the reservoir facies in more than two points in the reservoir, could be a seismic map, an outcrop analogue, a hand-made conceptual model or even a satellite image.

## Geological Well Testing: Application to a Low Net-To-Gross Fluvial Environment

### Background

The sedimentary system of the anonymous real field (‘Field X’) corresponds to a low-energy anastomosing environment with expansive low-quality overbank deposits at a depth of around 4350 m true vertical depth (TVD). This corresponds to the low net-to-gross (NTG) fluvial deposits, which include channel and bar sandstones, alluvial splay siltstones, and floodplain shales. The main reservoir units correspond to the fluvial deposits within the side bars and point bars of the channel system. These individual sandbodies are small in size and are poorly connected. Furthermore, owing to predominant overbanking and limited channel erosion, the vertical connectivity of the sand patches remains extremely low.

#### Extended well-test programme

An extended well-test programme was carried out to investigate the reservoir volume and to evaluate the reservoir properties at areas distant from the tested well (i.e. Well X1). Therefore, six reservoir intervals with an overall thickness of 48 m (and a net interval of 32 m) within different reservoir units were selected for the perforation (Fig. 1: left). The test history is composed of many short-term draw-downs, with variable rates, and a series of build-ups for a total duration of 11.1 days that are followed by a very long build-up of 90 days. Figure 1 (right) shows the wellbore pressure and the imposed surface rate history of the extended well test.

The test data are of a good quality with minimal noise, and the pressure-rate synchronization has been accurately checked. Figure 2 shows the diagnostic log–log plot and the straight-line analysis approach for well-test interpretation of the last build-up period. This is an interesting long ‘ramp response’ example where the upward trend of the pressure derivative curves highlight the strong degradation in lateral connectivity away from the wellbore. In this particular case, there is a feature in build-up whereby the pressure drop (Δ*P)* and derivative (Δ*P*') curves cross each other at late times. It should be noted that, for multi-rate tests, the derivative is taken with respect to the superposition time, which is a complex function of rate changes and time (Bourdet 2002). The late time signature in this long build-up highlights the fact that the producing reservoir permeability declines gradually to a low value but does not actually become zero: that is, it is not a closed system (O. S. Fjaere pers. comm.). This response can be readily simulated using an analytical three-region radial composite model with the same rate history, where the mobility ratio of the inner to outermost regions is of the order of 100 to 1.

The input parameters for the analytical interpretation are listed in Table 1. By using the straight-line analysis approach, the early time stabilization of the pressure derivative curve provides an effective permeability of 30 mD and a skin factor of −1.9. This negative skin factor could be an indication of geoskin (Corbett *et al*. 1996), where some highly permeable streaks, with limited extent, are intersected by the wellbore. This early radial flow regime is followed by a half-slope trend and a secondary stabilization in the middle time region. The slope analysis of these flow regimes provides a channel width of 74 m (from the half-slope trend line) and an effective permeability of 4 mD with an associated skin factor of −5.4 (from the stabilization). It should be noted that, although the ‘channel’ interpretation looks valid in this fluvial environment, the half-slope derivative interpretation could be an immediate outcome of the composite regions with a mobility contrast of 7.5 (i.e. the outer region 4 mD to the inner region 30 mD). Moreover, the derivative response in the linear-flow regime during the ‘build-up’ usually has a slope of less than 0.5 on the log–log plot (Stewart 2011). Eventually, the derivative response follows a long unit-slope trend at the late times preceding a possible stabilization or turn-over after the times greater than 3000 h, which may provide an effective permeability of 0.03 mD. This unit-slope trend can either be interpreted as an indication of a high-mobility contrast or the reservoir compartmentalization, in the sense that there is a poor communication between the higher-permeable disconnected channel sandbodies.

This 2D analytical interpretation can be improved with a ‘multi-layered’ reservoir scenario that corresponds to the main productive perforation intervals. Although the layered model is more realistic, it introduces many more degrees of freedom, and selecting the appropriate analytical model for each reservoir layer would be challenging and lead to a highly non-unique system. Therefore, an alternate modelling approach is employed that aims to reduce the non-uniqueness by incorporating geological information. This is achieved by constructing a detailed static (geological) model using the MPS approach, which is then validated by the extended build-up data through dynamic simulation. This is an example of the power of integrating the multi-domain and multi-source information at the exploration/appraisal phase.

### MPS modelling

Limited well data were available for interpretation of geological settings and sedimentary modelling in this field. A well-to-well correlation was based on chemostratigraphic results, and detailed core and log interpretations were performed. Three-dimensional seismic data were also available to confirm the tilted and faulted structures of the system. Because of the low NTG and thin sands, the seismic sections did not supply useful stratigraphic information.

Regionally, there are strong variations in the sedimentary setting. However, all wells in Field X show the same low NTG behaviour in the siliciclastic series, which confirms that Field X is located in an area with small sand deposits. Ancient low NTG systems have been described in the field but, as with all field data, the 3D representation remains difficult.

The sedimentological interpretation of well data showed that the top geological units mainly correspond to thin anastomosing fluvial deposits with low-sinuosity distributary and shallow branching channels. In the lower parts, more laterally extensive channel belts are observed that correspond to lateral amalgamation during low accommodation prior to widespread lacustrine flooding. Figure 3 shows the wireline-log data and limited core data acquired for the subject test well, illustrating low NTG upper-reservoir units and higher NTG lower units.

Some modern analogues were researched and selected to represent the sedimentary systems in the various reservoir units, taking into account the palaeo-climate, sedimentary body geometries and sizes (i.e. thickness to width ratio databases), as well as the distributions of geological objects.. The closest present-day systems that were found are the lower Parana River flood plain areas (Argentina) as the analogue for the first two units (Fig. 4), and the Magdalena River bars (Colombia) as the analogue of the lower unit (Fig. 5). The photographs have been taken from Google Earth^{TM}.

The extracted satellite photographs were digitized to provide training images for the MPS facies-modelling process. Photographs give the plane view of the system, while the combination of images gives the vertical variability, bearing in mind that a few layers may represent the same environment of deposition. Non-uniformity of facies proportions and geobody shapes led us to use different layered training images for each unit. For example, in unit 3 the shale content in the training image decreases with decreasing depth. The vertical proportion of shale, which was derived from the well-log data, was used as an axillary variable for the MPS simulations. Figure 6 shows the constructed training images for each reservoir unit. The 2D training images attached to each layer have been constructed from various satellite images within the analogue area.

The 3D training images for each unit have 180 × 205 × 3 cells in the *x, y* and *z* directions, and each cell measures 10 × 10 × 1 m. A 3D search mask is implemented within each reservoir unit to construct the local probability distribution functions. The training image size is large enough to reproduce the facies shapes and interactions. The training images for this fluvial environment include four facies ranging from high-quality sandstones to low-quality shales. In Field X, the petrophysical quality of the rock, especially permeability, is highly related to the sedimentary facies. Therefore, it was considered easier to give the facies a name that reflects the petrophysical quality rather than using a rigorous sedimentological terminology.

Table 2 summarizes the colour codes and the various proportions of each individual facies within the training images. Facies description and facies names relate both to the facies observed at the well on cores and to the sedimentary bodies observed in the modern analogues. Table 2 shows that unit 3 possesses the highest proportions of the high-permeable facies (i.e. facies 3 and 4). However, the overall reservoir quality of the training image remains poor. This is then inherited by the MPS modelling, which results in a full-field facies model with a lower connectivity between the individual sand patches within dominant background shale.

The MPS modelling is performed on a fine-grid geological model (i.e. 290 × 320 × 106 cells in the *x, y* and *z* directions, and an average cell size of 11 × 11 × 2 m) to simulate five different facies realizations. The shape and spatial distribution of facies in the MPS model is read from the training image, which statistically describes the conceptual understanding of the subsurface geology. The resulting MPS facies modes are conditioned to the well data. Figure 7 shows the resulting facies images for the representative layers within the different reservoir units. Figure 8 shows a cross-section of a facies realization, which indicates how the MPS facies model honours the overall facies proportions at the well, while reproducing different cycle sequence orders that are consistent with the sedimentological understanding of the siliciclastic series in this field.

The conditional sequential Gaussian simulation (Deutsch 2002) algorithm is implemented to distribute the porosity field within the geological model, whereas the isotropic horizontal permeability values are estimated using three previously verified porosity–permeability relationships (Fig. 9). The statistical and geostatistical facies parameters required for the porosity distribution are listed in Table 3. The vertical connectivity of the system, however, is extremely low. This is largely reflected in the *k*_{V}*/k*_{H} ratio and the high proportion of the low-quality facies in the system. The *k*_{V}*/k*_{H} is equal to 0.001 for facies 1 and 2 (F1 and F2), and 0.1 for facies 3 and 4 (F3 and F4). These are inferred from core measurements and in-house log correlations, and are upscaled at the simulation model scale through a method similar to the one described by Massonnat & Wigniolle (2013). Moreover, because of the high proportion of shale in the model, the sparse producing layers are effectively separated by low-quality facies. This leads to production under commingled conditions, which results in ‘ramp response’ where the producing layers only communicate through the wellbore. This is confirmed by the production logging tool (PLT) and the detailed core studies.

## Results and Discussion

### Facies scenarios and matching procedure

A smaller sector of the fine model (i.e. 186 × 241 × 68 cells) is extracted to perform the numerical well-test simulation scenarios. The sector model size is selected large enough to ensure that transient flow occurs and to avoid any artificial boundary effect during the well-test time. Smaller models could result in a boundary-dominated flow, which leads to a rollover of the build-up derivative curve at late time. Figure 10 shows the sector model and the main reservoir units embedded within the reservoir structural framework.

A three-phase black oil simulator (i.e. Eclipse 100) is applied to perform the well-test simulations. The measured pressure–volume–temperature (PVT) properties for live oil and dry gas have been used to create black oil PVT tables to simulate the well test. The initial reservoir pressure is 823.9 bars measured at a depth of 4431 m TVD. The saturation pressure is 324.7 bars and in all simulations the pressure stayed well above the saturation pressure. The total oil volume in the sector model is around 8×10^{6} m^{3}, of which 58% is contained in the high-permeability sandstones, 34% in the low-permeability sandstones, 4.4% in the alluvial siltstones and 3.6% in the alluvial shales. The perforation interval and the rate history are the same as the real test operation.

Figure 11 shows the log–log diagnostic plot for five facies realizations. Although all of the realizations seem to follow the same upward trend, the overall qualities of matches are not acceptable. However, a careful look at the diagnostic plots for all other realizations (Fig. 11) reveals that two of those (realizations 4 and 5) depict a better trend than the other realizations (Fig. 12). Despite a vertical shift, which indicates a much lower estimated lateral connectivity than reality, the well-test response of realization 5 follows the same trend as the real build-up response at early times. Clearly, there is a major issue in the late time trend, where the pressure derivative and pressure drop curves fail to reproduce the real test response. Figure 12 also shows that facies realization 4 reproduces the same late time signature as the real build-up, while the early time trend is not correctly captured. In particular, the middle time stabilization does not appear in the simulated responses and the duration of the linear-flow regime (1/2 slope trend in the derivative curve) is overestimated.

Having found the proper facies realizations, a few matching procedures, which are explained in detail in the next sections, were selected. These procedures are listed as follows:

uniform upgrading of the petrophysical properties;

multiple petrophysical realizations;

facies hybridization and petrophysical realizations.

Amongst these approaches, the ‘facies hybridization’ could provide the most reasonable match to satisfy the geological constraints. Figure 13 shows the general workflow used in this paper for matching the well-test data.

#### Uniform upgrading of the system properties

The vertical shift observed on the simulated responses of facies realizations 5 and 4 suggests that the overall connectivity of the system needs an improvement. Therefore, the overall system permeability (*k*_{H} and *k*_{V}) was improved by a factor of 2.3. This factor, which is the amount of pressure derivative shift upwards, is equal to the ratio of the pressure derivative for the facies realization 5 (or realization 4 but at the late times) to the real test derivative value. Figures 14 and 15 (grey lines) show the well-test simulation results for this uniform upgrading of permeability in facies realizations 5 and 4. The simulated response curves have shifted downwards and are closer to the measured build-up curves. However, there is still an obvious mismatch in the late times. It should be noted that by increasing the permeability values not only is there a vertical downward shift in the derivative response curve but also there is a shift towards the left. In other words, the spatial heterogeneities are felt earlier in time with the ‘permeability upgrading’.

To overcome this shortcoming, the porosity values might also be improved by the same factor. Figures 14 and 15 (black curves) show the well-test response curves for facies realizations 4 and 5, where the porosity distribution is ‘also’ upgraded by the same factor of 2.3. As expected, the overall match is improved. However, there are still some clear mismatching features. The facies realization 5 (Fig. 14: black curve) cannot reproduce the expected late time feature, and the overall well-test match obtained from facies realization 4 (Fig. 15: black curves) is not convincing (in particular, the middle time stabilization does not match). This approach with porosity upgrading should remain as a simple sensitivity study and cannot be considered as a correct matching procedure. This is because porosity upgrading with a factor 2.3 violates the confirmed range of porosity variations from the geological studies of Field X.

#### Petrophysical realizations

In this subsection, the effects of different petrophysical realizations are examined. The objective is to model the effect that varying petrophysical properties has on the early and late time signature of the well-test response in facies realization 5, and to find out whether the these variations can lead to a reasonable well-test match. In this context, two hypotheses (Hyp1 and Hyp2) are considered and five different realizations are simulated for each realization. The Hyp1 uses the same geostatistical parameters as previous simulation studies, whereas, in the Hyp2, the horizontal and vertical correlation lengths of porosity (i.e. λ_{x}, λ_{y} and λ_{v}) are changed. These changes are based on the geological understanding of the porosity and permeability variations in Field X, and are within the accepted ranges of uncertainties. Table 4 summarizes the input parameters required for the sequential Gaussian simulation algorithm. Because the porosity and the permeability are correlated, this automatically changes the permeability field within the individual facies.

Figure 16 shows the build-up diagnostic plot for simulated petrophysical realizations. The figure indicates that increasing the petrophysical correlation length tends towards a flattening of the derivative curve. This, for instance, can be observed in realization 6 (of Hyp2), where the middle time stabilization disappears. It is also noted that the late time convergence of the real data curves is almost reproduced (Fig. 17). Meanwhile, the preceding steep derivative (with a unit-slope trend) has been flattened and, more importantly, the overall connectivity increase is not remarkable. However, this is common for all of the simulated petrophysical realizations. Furthermore, Figure 16 reveals that the pressure derivative curves for different realizations cross each other at some points. This is attributed to the different spatial permeability distributions that have been created by each petrophysical realization.

This exercise shows that overall connectivity of sand patches remains an issue and highlights the fact that the facies distribution may be considered a stronger controlling parameter than the geologically constrained petrophysical variations for this general sand–shale reservoir. This consideration is more consistent with previous studies (Corbett *et al*. 2012), where the low-permeability facies exert a strong control on effective properties.

#### Facies hybridization

Several facies realizations (most of which are not documented in this paper) were generated. Only two of those realizations (i.e. realizations 4 and 5) showed trends that closely match the real data. Facies realization 5 closely matches the early time, while facies realization 4 matches the late time cross-over of Δ*P* and Δ*P*′ (Fig. 18). An engineering hybridization algorithm is utilized to take advantage of both geological models, and to combine parts of the spatial facies structures in realizations 4 and 5. This is to ensure that the early time signature of facies realization 5 and the late time signature of facies realization 4 are both preserved in the resulting well test response of a hybrid facies model. The hybridization algorithm appears functionally similar to the gradual deformation method (Roggero & Hu 1998; Hu 2000), where two realizations are geostatistically combined in Gaussian space and realizations are created that change smoothly while preserving the global statistical features of the model (Gallo & Ravalec-Dupin 2000). However, in the hybridization algorithm used here, two engineering approaches were implemented to combine the facies models:

(1) Effective Volume Cropping;

(2) Box Cropping.

These methods aim at finding the cells responsible for a well-test response over a particular time window in the heterogeneous reservoir models. This may be seen as analogous to the use of the radius of investigation in 2D homogenous models.

(1) **Effective Volume Cropping**: In order to define an effective volume around the wellbore in facies realization 5, a single draw-down with an average production rate is simulated. This is to give a clearer view on the pressure diffusion process and to reduce the unnecessary simulation time. With a cumulative production of 6375 m^{3} during 11.18 day (just before the start of the main build-up period), an average oil rate of 570 m^{3} per day is obtained.

Figure 18 shows the hybridization concept where the merging time of 35 h is selected based on the derivative signatures. This time is used to define the effective volume in which the pressure drop in any location of the reservoir model is greater than a small value (*ϵ*). This is mathematically shown as follows:

in which **EV** is the effective volume and Ω(**x**, *t*_{m}) is the volume defined by the pressure drop after *t*_{m} = 35 h of production. In the presence of numerical artefacts, defining a meaningful ‘*ϵ*’ may not be easy, and needs a trial-and-error approach. However, several numerical experiments (under commingled conditions) suggest that selecting different *ϵ* values for the reservoir layers provides adequate results. In this context, the volume is defined so that the spatial pressure drop in each layer of the reservoir is greater than 1–5% of the maximum pressure drop at each individual layer connection to the wellbore. Figure 19 (left) shows the 3D effective volume for the facies realization 5 obtained by this procedure. The figure also highlights the non-homogeneous nature of pressure diffusion in heterogeneous layered reservoirs. Obviously, the layers within unit 3 (the lower layers of the model) have a higher petrophysical quality and, therefore, pressure diffusion is faster in these layers.

(2) **Box Cropping:** In this approach, a cuboid is defined around the tested well in facies realization 4, which is then replaced by the one cropped from facies realization 5. The motivation for the Box Cropping was to test a way for a future development to automate the hybridization using a box with gradually increasing sides, and to check the transition effect on the resulting facies connectivity. In this study, the box is simply defined by a cuboid that circumscribes the effective volume which was obtained in the effective draw-down procedure. Figure 19 (right) shows a cuboid that is obtained by this approach. This volume, which is obtained from facies realization 5, replaces the equivalent portion of the model in facies realization 4.

Figure 20 shows the build-up responses obtained by these two different hybridization approaches. Although both of the approaches preserved the required features of the facies realizations 4 and 5, the Box Cropping approach still requires a permeability upgrading. This is in contrast to Effective Volume Cropping, which gives satisfactory results without permeability upgrading. The resulting well-test signature shows the expected features of facies realizations 4 and 5, as well as an important connectivity upgrading. This is because the spatial feature of realization 5, within the effective volume, has been spatially superimposed on the spatial facies structure of facies realization 4 and the model gets richer in facies 4 (high permeable facies). This point is clearly illustrated in Figure 21. This figure illustrates the facies hybridization workflow within one layer of the reservoir model (layer 33) and shows that the lateral connectivity has been improved in near wellbore areas. Therefore, the pressure derivative response shifts downwards and results in a proper derivative matching. It is important to note that the model obtained by the Effective Volume Cropping is geologically consistent and follows the common facies structures generated by the MPS facies modelling. This has been checked layer-by-layer to ensure that the general trend of facies distribution is adequately preserved in the final model and no visual abrupt changes in the facies distribution occurs. Figure 21 shows an example for layer 33 of the model.

The hybrid facies realization obtained by the hybridization algorithm provides the necessary framework for the conditional petrophysical distribution. Ten different petrophysical realizations under two hypothetical scenarios are simulated using the sequential Gaussian simulation. The geostatistical distribution parameters and the probability distribution function of porosity for both scenarios follow the same values as presented in Tables 3 and 4. The only difference is that the first five realizations use cross-plot 1 (Fig. 22) to estimate the permeability field, while cross-plot 2 (Fig. 22) is used for the second set of five realizations. Cross-plot 2 uses a different porosity–permeability relationship for facies 4 (F4: highly permeable facies), where its estimated permeability values are twice the values predicted by cross-plot 1 (Fig. 9).

Figure 23 shows the well-test response for all 10 petrophysical realizations. As expected, the derivative curves for the ‘second’ five realizations are somewhat shifted down relative to the real derivative curve and fail to match the middle time stabilization. The late time behaviour has also been slightly affected; hitherto, the curve cross-over has been reproduced in all of the realizations.

Of all the simulated realizations, realization 1 provides a better well-test match. Figure 24 shows the build-up response for this realization.

Although the overall real test signature is adequately captured in the hybrid model transient response, the early time mismatch (Fig. 24, for *t* < 0.1 h) shows the inability of the model to represent the near-wellbore geological complexities. There is also a vertical displacement of the pressure drop curve, which may be due to skin effect. More detailed near-wellbore modelling (Chandra *et al*. 2013) can be employed for better representation of the near-wellbore geology from log and core data. However, in this study the detailed well data were not provided and high-resolution near-wellbore geological description could not be constructed. Therefore, it was decided to upgrade the permeability of few cells in the immediate vicinity of well. This is equivalent to the concept of geoskin (Corbett *et al*. 1996), where the highly permeable streaks straddle the wellbore. It is acknowledged that the uniform upgrading is unlikely to represent the near-wellbore geology. Nevertheless, for matching purposes, the candidate cells for permeability upgrading are defined by employing another ‘Effective Volume Cropping’ after 0.1 h of production that corresponds to a distance of approximately 30 m from the well in the most permeable layers. This time corresponds to when the simulated and the real well-test data merge (Fig. 24). The permeability values within this volume are upgraded by multiplying with a constant value of 1.8 that corresponds to the derivative mismatch. It should be noted that this volume is very sparse along the wellbore and only few cells in the highly permeable layers are affected. Figure 25 shows the superimposition of the effective volumes.

Defining a secondary effective volume and upgrading the permeability within that volume can significantly improve the well-test match. Figure 26 shows the main build-up response where the pressure drop and the pressure derivative curves are vertically displaced, and could match the real test response.

The quality of the match can also be checked against the semi-log plot (Fig. 27) and the history plot (Fig. 28). These figures show that, although there are very small mismatches in the early times, the overall quality of match is very good.

## Conclusions

The geological well-test matching of a low-energy commingled fluvial reservoir was performed and the transient test response of the final model was matched against the exceptionally long well-test data. The well-test data showed a prolonged ramp-effect response, indicating a drastic reduction of lateral connectivity in the model. Systems with complex interactions and geometries between high-permeability channels and cocooning low-permeability facies develop difficulties in interpreting the associated well-test responses, where the idealized analytical well-test models cannot fully represent the extent of spatial heterogeneity in the system. Therefore, the geological well-test method with the MPS facies statistics approach was used to model the complex channel network and to preserve the lateral (dis-)connectivity of the facies in the model.

In this work, the MPS approach was successfully implemented to read the key patterns from a 3D training image and to generate the geologically realistic features in a stochastic geological model. Several sensitivity studies were then performed to simulate different facies and petrophysical realizations. The results showed that the facies modelling, rather than the petrophysical modelling, was the main controlling parameter in the simulated transient test response of this shale–sand reservoir. Although, none of the models could reproduce the overall well-test response, some facies models could partly follow the build-up derivative shape, although with some vertical displacements evident (i.e. the connectivity–mobility issue). A hybridization algorithm was successfully employed to merge two facies models to match the early and the late time well-test responses. This workflow has resulted in a model with an improved lateral connectivity, which could reasonably match the entire well pressure history.

As a result, the operator of Field X generated a dynamically calibrated reservoir model in which the key geological features of the depositional setting, as well as sedimentological knowledge of the reservoir, were better captured. It is worth noting that as with any other inverse problem, this approach also suffers from a lack of uniqueness in the solution. The geological well testing is a multi-domain platform, which effectively integrates the available static and dynamic data, and tends to *reduce* the non-uniqueness in solutions. However, although the final hybrid model is constrained by the well-test data, it still remains a single facies realization that has been validated against static and dynamic data.

Despite the simplifications used in certain steps – single porosity–permeability relationships for each facies, and the use of a modern analogue without considering preservation of the complex geometries – the results of this work are considered a step forward in the integration of static geological data and dynamic engineering data.

## Acknowledgments

H. Hamdi would like to thank Total Geoscience Research Centre in the UK for the financial support of his PhD study and the Interactive Reservoir Modeling, Visualization and Analytics Research Group of the University of Calgary for the financial support of his Postdoctoral fellowship. The authors thank Schlumberger for use of Petrel and Eclipse, Kappa for Saphir, and Weatherford for Pansystem.

- © 2014 EAGE/The Geological Society of London