Accounting for diagenesis overprint in carbonate reservoirs using parametrization technique and optimization workflow for production data matching
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Abstract
Diagenesis is rarely accounted for in the standard modeling workflows for carbonate reservoirs, although it has a huge impact on both porosity and permeability. This can be explained by at least two reasons: first, it is difficult to quantify the influence of diagenetic overprints on porosity and permeability; second, the integration of the diagenetic effects in carbonate reservoir models makes history matching much more difficult. Herein, a modeling methodology is proposed, in which the diagenetic imprints are included in the reservoir model and calibrated with dynamic data. The key point consists in defining a parametrization technique able to capture these diagenetic imprints. We assume that distinct regions of occurrence of a given diagenetic phase can be identified within the reservoir. Therefore, restricting our attention to a facies, we may distinguish regions characterized by low, medium or high proportions of the targeted diagenetic phase. The advantage of this parametrization technique is that the proportions of these regions can be easily driven by a reduced number of proportionality coefficients. Then, the overall modeling approach is integrated in an optimization workflow making it possible to vary the proportions of the region with a given occurrence for a given diagenetic phase, the variograms characterizing the spatial distribution of the regions, or even the way they are spatially distributed. The optimization process is run to adjust these various unknown parameters in order to match production history. The potential of the proposed methodology is finally investigated through the study of a twodimensional numerical example.
Keywords
Carbonates Diagenesis Historymatching OptimizationIntroduction
Carbonate reservoirs exhibit complex pore systems, which depend on their biological origin and great chemical reactivity (Flugel 2010; Moore 2001). The geometry and connectivity of these pores strongly control the dynamic flow of hydrocarbons in carbonate reservoirs. As more than 60% of the world’s oil and 40% of the world’s gas reserves are estimated to be held in such reservoirs, the oil and gas industry looks constantly to improve the representation of carbonate reservoirs through geological and simulation models that better capture and illustrate their heterogeneities. The standard reservoir modeling workflow consists in integrating seismic, geological and well data to create a static geological model. This one is usually populated by facies or lithofacies and their associated petrophysical properties like porosity, permeability and fluid saturations (Deutsch 2002). The static model is then inputted into a fluid flow simulator in order to understand fluid displacements and forecast production over reservoir lifetime.
In the current industry practice, diagenesis is generally integrated through the definition of lithofacies referring to textures and pore systems or rock types. However, this approach cannot be easily applied through history matching and field development optimization, because diagenesis imprint has not been modeled and thus cannot be modified according to production data. Moreover, even when taken into account, it is most of the time through a deterministic approach with a limited number of models. Due to the uncertainty in the description of the occurring diagenesis phases and their effects in the facies deposition model, a stochastic framework may be preferred. Several authors (Barbier et al. 2012; Doligez et al. 2011; Hamon et al. 2015; Labourdette et al. 2007) investigated and applied geostatistical methods to integrate depositional facies and diagenetic overprint models. The leading idea consists first in identifying the major diagenetic phases, which modify the petrophysical properties and second in qualitatively or semiquantitatively establishing the relationship linking the diagenetic phases to the depositional facies. The first identification step is usually performed from the analysis of thin sections and laboratory measurements of core samples. It is based upon the definition of the paragenetic sequence, which is a conceptual representation of the diagenetic events through time. The paragenetic sequence permits to schematically display the relationships between these diagenetic phases and the resulting porosity and permeability modifications (Moore 2001). The second step yields the relationships between the diagenetic phases and the depositional facies from petrographic observations. When this information is available, it can be integrated into reservoir models, provided that suitable stochastic simulation techniques are applied. For instance, Renard et al. (2008) extended the wellknown pluriGaussian simulation (PGS) technique (Le Loc’h et al. 1994) to get the bivariate pluriGaussian simulation (BiPGS) one. Such a method makes it possible to populate reservoir models with spatially varying facies and to account for diagenetic phases with spatially varying influences on facies. Even though data are provided through the analysis of thin section or laboratory measurements, they represent a very tiny part of the actual reservoir under consideration. Upscaling of diagenesis characteristics, but also of petrophysical properties like porosity and permeability, is still a major issue. Even when properties and relationships between facies and diagenesis have been clearly identified at the plug scale, the question remains about the corresponding properties to be provided to the grid cells of the reservoir model, which is associated with a much larger scale with macroporosity occurrences, locally baffling heterogeneities, fractures, etc. (Farooq et al. 2014; Nader et al. 2013). This is all the more true for diagenetized carbonate reservoir (Benyamin 2016).
The resulting static reservoir model is therefore very uncertain. The integration of other sources of information is required to better constrain the reservoir models used to represent the real one. This is why production data are also considered. They consist of all the data measured at wells during production: pressures, flow rates, gas/oil ratios, water cuts. The integration of production data into a static reservoir model is known as history matching (Jacquard and Jain 1965). This usually involves the definition of an objective function, which measures the leastsquare differences between the actual production data and the corresponding numerical responses simulated for the considered static model using a flow simulator. Then, an optimization process is run aiming at minimizing the objective function by successively adjusting some uncertain parameters of the static reservoir model or the associated flow model. The interested reader may refer, for instance, to Le Ravalec et al. (2014) for more details. This calibration process contributes to improve the reservoir model, i.e., to make it more reliable for predictions and to better describe fluid flows within the reservoir. The most valuable result is the capacity to predict reservoir production lifetime with a certain level of confidence and to evaluate its economic potential. One of the fundamental aspects of history matching is parameterization. A reservoir model and its associated flow features include a huge number of unknown or uncertain parameters. For instance, the flow characteristics depend on the definition of fault transmissivities, relative permeability and capillary pressure curves, coefficients describing the strength of the aquifers. On the other hand, the reservoir model itself consists of a grid that has to be populated by facies, porosity values, permeability values and initial saturations. The large number of uncertain parameters and the need for preserving geological consistency all along the historymatching process steered the development of specific parameterization techniques. The gradual deformation method (Hu 2000) and the probability perturbation method (Caers and Hoffman 2006) were then proposed to adjust the spatial distributions of facies and petrophysical properties from a reduced number of auxiliary parameters. In addition, they permit to preserve the spatial variability model inferred from the static data whatever the variations in the auxiliary deformation parameters. Other parameterization techniques were designed for handling facies proportions; PonsotJacquin et al. (2009) introduced a very simple and intuitive method that involves proportionality coefficients to drive the variations in the different facies proportions. A drawback of the method was the discontinuities occurring on the boundaries of the regions submitted to modifications. An improved krigingbased approach was then suggested by Tillier et al. (2010) to remove these undesired boundary effects. A preliminary tentative for integrating diagenesis effects into history matching was described by Pontiggia et al. (2010), but the approach was based upon screening, not on the iterative minimization of an objective function. There was no dedicated parameterization technique for diagenesis.
In this paper, we focus on the development of a simulation workflow making it possible to integrate diagenesis information in the generation of the static model and to consistently modify this model to also account for production data. In our case, “consistently” means that we want to be able to calibrate the model to the production data while preserving its geological characteristics derived from the analysis of static data, of which diagenesis. All of this actually calls for the definition of a new parameterization technique for driving diagenesis description. The one proposed hereafter is rooted in the approach developed by PonsotJacquin et al. (2009) for varying facies proportions. Section 2 describes the overall simulation workflow including the step especially added for diagenesis. In this section, the potential of the methodology is investigated on the basis of a synthetic case with three depositional facies. Finally, the third section focuses on sensitivity analysis and shows how the diagenetic modeling parameters influence the reservoir dynamic response.
Geological and reservoir simulation approach
To get the static reservoir model, we start generating facies, and then, we simulate the diagenesis phases given the facies before going through the distribution of the petrophysical properties. This process is described in the first part of this section. The key point consists in defining a simple and easy to use parametrization technique able to capture these diagenetic overprints. We assume that distinct regions can be identified within the reservoir depending on the occurrence of given diagenetic phases. For instance, we may distinguish regions within a facies, which are characterized by low, medium or high proportions of a targeted diagenetic phase.
Once the facies and the associated diagenesis imprint models are built, the facies is populated with porosity and permeability properties. Then, it is inputted into a fluid flow simulator to calculate dynamic flow responses that can be compared to the actual production data. The leastsquare differences between data and synthetic responses yield the objective function. The purpose of history matching consists in adjusting the unknown parameters everywhere to minimize the objective function: when generating facies (spatial distributions, variograms and proportions), when distributing the diagenetic phases (spatial distributions, variograms and proportions), when generating the petrophysical properties (spatial distributions, variograms and means) and when simulating fluid flow (aquifer strength, fault transmissivities, PVT, etc.). The optimization process applied in this paper is presented in the second part of this section. For simplicity, we focus on the parameters related to the simulation of the diagenetic phases. However, the generalization to any block of the simulation workflow is straightforward.
Diagenesis model definition
Quantitative relationships between facies A, B, C and diagenetic phases P1, P2, P3
Diagenesis  Facies  

A  B  C  
H (%)  M (%)  L (%)  H (%)  M (%)  L (%)  H (%)  M (%)  L (%)  
Phase 1  80  20  –  –  25  75  55  15  30 
Phase 2  10  50  40  50  25  25  35  25  40 
Phase 3  10  30  60  50  50  –  10  60  30 
Sedimentary facies modeling
Various geostatistical algorithms can be applied to randomly produce facies models. They can be split into three main groups with twopoint statistics pixelbased methods, multiplepoint statistics pixelbased ones and objectbased ones. In this paper, we focus on the truncated Gaussian simulation (TGS) technique, which belongs to the first group. It is used to generate a spatial categorical variable (Matheron et al. 1987) by simulating and truncating a Gaussian random function (GRF). This widely used approach exhibits a special feature that may be undesired in some cases: the generated facies respect a sequential ordering. However, this limitation is easily overcome with pluriGaussian simulation, which involves the truncation of at least two Gaussian random functions (Le Loc’h et al. 1994). For simplicity, the description below is restricted to TGS.
Diagenesis modeling
Petrophysical property modeling
As mentioned previously, the facies and diagenetic overprints are categorical variables that represent specific petrophysical rock characteristics. The last step of the static modeling workflow deals with the simulation of porosity, permeability and fluid saturation properties to populate the diagenetic levels. Distinct simulations algorithms can be applied to do so of which Sequential Gaussian Simulation (Goovaerts 1997) or Fast Fourier Transform Moving Average (Le Ravalec et al. 2000).
Reservoir modeling and history matching
When the static model is built, it can be inputted into a fluid flow simulator to get numerical production responses, such as pressures or fluid rates at wells. This last block of the overall simulation workflow is said dynamic since the production responses vary with time.
In such conditions, it can be shown that Y is also a Gaussian random function of mean m and covariance C whatever the value of the t deformation parameter. This property holds because the two Gaussian random functions are independent and because the sum of the squares of the coefficients applied to the two Gaussian random functions equals 1. Let us consider a realization y_{o} of Y_{o} and a realization u of U. y_{o} can be considered as the initial guess for the static model and u as a random complementary realization. For simplicity, the mean of the two Gaussian random functions is assumed to be 0. When the t deformation parameter is 0, the equation above yields a y realization identical to y_{o}. When t is 0.5, it leads to u. When t gradually varies from 0 to 0.5, the y realization is smoothly modified to evolve from y_{o} to u. This is why the deformation process is said gradual: it makes it possible to slightly and continuously change the y realization. The whole process is periodic with t varying in [− 1, 1]. When this parametrization technique is included into the minimization process, the purpose is to catch the y realization or equivalently the t deformation parameter that provides the smallest value for the objective function. Clearly, the chain of realizations built by varying t represents a very tiny part of the entire search space. Therefore, the complete gradual process consists in successively investigating different chains of realizations. The first one is produced from initial guess y_{o} and realization u. A first minimization process is run to explore this first chain and to determine the realization associated with the smallest objective function. This realization is said to be the first optimal one. Then, we replace y_{o} with the first optimal realization and randomly draw a new u realization. This provides a second chain of realizations. We can restart the minimization process to investigate this new chain and identify a new optimal realization, which permits to decrease further the objective function. The minimization loops are repeated until the data misfit is small enough.
A few authors (Roggero and Hu 1998; Thomas et al. 2005; Tillier et al. 2010) showed that the gradual deformation process can be also used to perturb the realizations of categorical variables and match well production.
Reference case
The methodology described above is applied to a synthetic case to evaluate its potential. First, we built a reference case populated with facies, diagenetic phases, and petrophysical property distributions.
The reference case is associated with a twodimensional grid including 200 × 200 × 1 blocks whose size is 5 × 5 × 5 m^{3} for each. The reservoir was assumed to consist of three sedimentary facies named facies A, facies B and facies C (Fig. 3, a). Their proportions were set to 0.5, 0.33 and 0.17, respectively. Their spatial variabilities were characterized by an anisotropic Gaussian variogram with a range of 600 m along the main axis, this one being defined by an azimuth of 45°. The range along the perpendicular axis was 120 m. The truncated Gaussian method was then used to generate the facies model.
Characteristics of diagenesis per facies: proportions, porosities and permeabilities
Facies  Diagenesis and levels  Proportion (fraction)  Porosity (fraction)  Permeability (mD)  

A  Silicification  Low  0.15  0.30  500 
Medium  0.21  0.15  100  
High  0.64  0.05  1  
B  Calcite cementation  Low  0.5  0.20  200 
Medium  0.5  0.10  50  
C  Silicification  High  1  0.05  2 
Diagenesis modeling
Parameterization of diagenetic proportions
As mentioned above, we aim at adjusting diagenesis from production data by varying the variograms considered for simulating the diagenetic phases, their spatial distributions, but also their proportions. The variograms involve scalar parameters such as the range that can be perturbed using any usual minimization algorithm. The variations in the spatial distributions can be driven by the gradual deformation method as explained in Sect. 2.5. The issue that still has to be solved is about the proportions of the various diagenetic phases, but also the proportions of the distinct levels of occurrence for a given diagenetic phase. All of these proportion values that have to be calibrated finally lead to a significant number of parameters. In this section, we introduce a new parameterization technique to vary all diagenetic proportions from a reduced number of proportionality coefficients. This will contribute to ease the ultimate optimization process.
This parameterization technique can be included into the optimization process described in Sect. 2.2. Then, a is viewed as an additional parameter to be determined from the minimization of the objective function. It is handled as any other scalar parameters. The methodology can become as complex as the diagenetic depositional conceptual model is. Several diagenetic phases affecting in different extents the carbonate reservoir can be considered. The interest of this new approach is the decrease in the number of parameters: the whole set of diagenetic proportions can be driven from one or a few proportionality a parameters.
Influence of variograms
Influence of the spatial distribution
Influence of level proportions
History matching: calibrating diagenesis from dynamic data
At this stage, the purpose is to see whether the proposed historymatching procedure makes it possible to calibrate diagenesis from the available dynamic or production data. For the numerical experiment under consideration, we now assume that everything about the static reference model is known except the parameters characterizing silicification in facies A. In other words, the spatial distribution of silicification heterogeneities as well as the proportions of the regions associated with the low, medium and high levels of occurrence is unknown. The issue to be investigated is about the possibility to calibrate this diagenetic phase from the reference water cut data provided in Fig. 6b. The matching process described in Fig. 1 is then run with unknown parameters in the diagenesis modeling block only. Two parameterization techniques are applied to drive the diagenesis model. The diagenesis proportion perturbation method proposed in this paper is considered to vary the proportions, while the gradual deformation method permits to change the spatial distribution of the diagenesis heterogeneities. This thus calls for the definition of two parameters: parameter a for the diagenesis level proportions and parameter t for the spatial distribution of diagenesis heterogeneities.
A given starting point, that is a given initial diagenesis model, is randomly generated (Fig. 4b). It is characterized by a seed that yields the heterogeneity distribution and a diagenesis proportion parameter a of 0.5. As in the example presented in Fig. 7, this a parameter gives at once the proportion of the region with a low occurrence level of silicification. This starting point is clearly different from the reference diagenesis model shown in Fig. 4, a: the value for the initial a parameter (0.5) is much larger than the one for the reference model (0.15).

Step 1 generate the facies model, define parameter a and generate Y_{0}, an initial Gaussian white noise (see Eq. 4),

Step 2 generate U, a complementary Gaussian white noise,
 Step 3 enter the gradual deformation loop that drives the t deformation parameter,
 a.
Combine Y_{o} and U to obtain Y.
 b.
Generate the diagenesis model.
 c.
Populate the resulting model with petrophysical properties.
 d.
Run the flow simulation.
 e.
Calculate the objective function.
 f.
Adjust parameters t and a.
 g.
Go back to 3(a) until convergence.
 a.

Step 4 update Y_{0} using the previous optimal Y and go back to step 2 until convergence.
Following this scheme, there are actually two loop levels. There is an inner loop that includes steps 3(a) to 3(g). At this level, U is fixed, while t and a vary. The purpose of this loop is to identify parameters t and a so as to minimize the objective function. As U is fixed, a tiny part only of the search space is investigated. This is why there is a secondlevel loop that groups steps 2, 3 and 4. This makes it possible to consider other U realizations, to build other chains of realizations and to explore further the search space.
The stopping criterion is defined as 10% of the initial objective function: when the objective function gets smaller than this value or when it is a value almost constant, the optimization process is stopped. The algorithm used for minimizing the objective function in step 3 is Simulated Annealing (Kirkpatrick et al. 1983). Its name comes from the annealing process in metallurgy, a technique with first heating and then cooling under a controlled temperature. The idea behind is to reach an equilibrium state corresponding to a global optimum. Given a starting point, the algorithm considers a random parameter perturbation and calculates the resulting change in the system energy (i.e., objective function). If this system energy decreases, the perturbation proposed is accepted. If it increases, the perturbation is accepted following a given probability. This prevents the algorithm from getting stuck in local minima. Perturbations are repeated until the system energy reaches a steady state or complies with the stopping criterion.
Optimization results for 10 different initial points: initial and final objective function, optimal a parameter
Cases  Initial fobj  Final fobj  Opt a 

1  0.3403  0.0060  0.2216 
2  0.3409  0.0126  0.1693 
3  0.4006  0.0137  0.3321 
4  0.2289  0.0095  0.2224 
5  0.0835  0.0065  0.2142 
6  0.1082  0.0099  0.2919 
7  0.4256  0.0400  0.1824 
8  0.2650  0.0133  0.2194 
9  0.42,449  0.0384  0.3584 
10  0.26,879  0.0207  0.2045 
Conclusions
We developed a methodology to calibrate diagenesis from production data. A diagenesis modeling step was added to classical modeling workflows, and a new, simple and intuitive parameterization technique was proposed to drive the proportions of the various levels for a given diagenetic phase. This made it possible to design a matching workflow aiming at calibrating the proportions, the spatial distribution and variability of the diagenetic phases on top of usual parameters.
A sensitivity study was then performed. It evidenced that the diagenesis modeling parameters such as the variogram range, the proportions and the spatial variability influence the simulated production responses to various extents. The proportions of the diagenesis level have the most prominent impact and then followed but the spatial distribution. This is related to the fact that these parameters control connectivity and hence fluid flow.
Last, a numerical experiment was developed to evaluate the ability of the proposed inversion methodology to constrain diagenesis parameters from production data. The proposed methodology was applied to a twodimensional synthetic case. We showed that the matching of the water cut profiles made it possible to properly capture the reference diagenetic model. The optimal a parameter was pretty close to its reference value, while the spatial distribution of the heterogeneities looks like the reference one.
A next step will consist in applying the methodology described in this paper to a real carbonate field with historical production. The challenge will be to consider more data, a complex configuration of diagenetic phases and the use of the pluriGaussian algorithm.
Notes
References
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