Carbonate Brazilian pre-salt fields have a large number of faults detected by seismic and well data. Nevertheless, because of limitations in seismic resolution, all existent faults cannot be identified. That is one of the main challenges for understanding related heterogeneities (vugs, karst) and the flow behavior. This paper deals with a fault analysis and modeling using an original approach and fault data of three pre-salt reservoirs.
One possible approach for characterizing and modeling the fault network (
The results presented on this article lead us to discuss the importance of how to choose the samples for modeling sub-seismic faults based on the ensemble of seismic faults available. This article answers the question about which available seismic faults we should use for estimating fractal dimension, should we use all available seismic faults near of the reservoir area or use only the faults inside the reservoir contour. After this short discussion on the fractal dimension choice from a spatial distribution point of view, the impact of this choice on flow was illustrated. The sub-seismic fault models were modeled using different fractal dimension. Subsequently, an upscaling step using analytical upscaling (
Characterizing sub-seismic faults has a major impact on the overall flow behavior of the field. The chosen methodology has been applied only on synthetic cases but never published using real data. This work will interest a practicing engineer. The fault network of these neighbor reservoirs allows us to illustrate the importance on the choice of fractal dimension for characterizing the fault network and its impact on the subseismic models and fluid displacement, consequently on production.
We have set up a general framework to obtain estimations of the geostatistical parameters (GP) such as the correlation length, lc and permeability variance, ln from well test data. Most often, in practical studies, the GP are estimated using geological and petrophysical data, but in a lot of cases, these data are too scarce to give high confidence results.
The method was tested using synthetic well-test data performed on some training images, and correct estimations of the underlying correlation length, lc and permeability variance, ln were recovered. Once the GP are estimated, other well established techniques can be used to get well-test matched reservoir images consistent with the geostatistical model. In practical applications, excellent well test data are needed, and the method should be improved using multiple well test data.
Hu, L.Y. (Inst. Français du Petrole and the HELIOS Reservoir Group) | Blanc, Georges (Inst. Français du Petrole and the HELIOS Reservoir Group) | Noetinger, Benoit (Inst. Français du Petrole and the HELIOS Reservoir Group)
A crucial step of the two commonly used geostatistical methods for modeling heterogeneous reservoirs, sequential indicator simulation and truncated Gaussian simulation, is the estimation of the lithofacies local proportion (or probability density) functions. Well-test-derived permeabilities show good correlation with lithofacies proportions around wells. Integrating well and well-test data in estimating lithofacies proportions could permit the building of more realistic models of reservoir heterogeneity. This integration is difficult, however, because of the different natures and measurement scales of these two types of data.
This paper presents a two-step approach to integrating well and well-test data into heterogeneous reservoir modeling. First, we estimate lithofacies proportions in well-test investigation areas with a new kriging algorithm called KISCA. KISCA consists of kriging jointly the proportions of all lithofacies in a well-test investigation area so that the corresponding well-test-derived permeability is respected through a weighted power-averaging of lithofacies permeabilities. For multiple well tests, an iterative process is used in KISCA to account for their interaction. After this, the estimated proportions are combined with lithofacies indicators at wells for estimating proportion (or probability density) functions over the entire reservoir field with a classical kriging method.
We considered some numerical examples to test the proposed method for estimating lithofacies proportions. In addition, we generated a synthetic lithofacies reservoir model and performed a well-test simulation. The comparison between the experimental and estimated proportions in the well-test investigation area demonstrates the validity of the proposed method.
Recent research on stochastic reservoir modeling constrained by well-test data has focused on the approach based on the Bayesian inversion theory and the Markov chain Monte Carlo methods.1-3 This approach is very attractive because it can deal directly with wellpressure data rather than well-test-derived permeability data and because it can be extended to history matching. However, it is limited to gross grid reservoir models in the context of continuous Gaussian-related variables (e.g., log-normal permeability field). In the case of a stabilized well-test, it is possible to define an effective permeability in the corresponding investigation area (well-test-derived permeability). A method based on simulated annealing has been used for conditioning permeability field to well-test derived permeabilities.4 Although the annealing process does not call for fluid-flow simulations, this method still can be very slow. This paper proposes an alternative approach for incorporating well-test-derived permeabilities into lithological reservoir models defined on fine grids.
Consider two commonly used geostatistical methods, truncated Gaussian simulation5 and sequential indicator simulation,6 for building reservoir lithological models. These methods consist of first estimating the local proportion functions [or probability density functions (PDF)] of lithofacies. Then, the lithofacies model is built by truncating a Gaussian random function with the proportion functions (or by randomly drawing lithofacies from the PDF). A realistic modeling of lithofacies distribution with these geostatistical methods depends greatly on the accuracy of the estimation of the lithofacies proportions functions (or PDF). In the case of few well data, the incorporation of other sources of information (including geological knowledge, seismic information, well-test data, and field production data) would improve the estimation of proportion functions (or PDF) significantly.
This paper covers the problem of incorporating well and well-test-derived permeability data into the estimation of lithofacies proportion functions (or PDF). We use a two-step approach: first the lithofacies proportions in well-test investigation areas are estimated with a new kriging algorithm called KISCA, then these estimates are combined with lithofacies indicators at wells for estimating lithofacies proportion functions (or PDF) over the entire reservoir field with a classical kriging method. Ref. 7 describes another method based on the cokriging technique for integrating well and well-test-derived permeability data. Also, there are existing methods for incorporating well and seismic data for estimating lithofacies proportion functions.8
Well and Well-Test Data
The data set is made of a lithofacies description at available wells and a number of well-test-derived permeability values. The continuous lithofacies description on the wells is regularly discretized with a fineness defined according to the lithofacies variability along wells. At each point xa of the well discretization, an indicator is defined for each lithofacies:
We consider that the covariance Cn(h) of each indicator can be inferred from the well data or other sources of information (e.g., analogous outcrop data).
For each well-test-derived permeability, kwt, the investigation area, V, is determined and a power-averaging formula is adopted to relate the well-test-derived permeability to the lithofacies permeabilities:
where kn stands for the permeability of lithofacies n and Pn(V) is its proportion in V. The averaging power ? is calibrated for each well-test9,10 and Pn(V) are to be estimated by the method described next.
Geostatistical models are useful to generate equiprobable realizations of oil reservoirs that allow the study uncertainties in the production forecasts. A key problem is the constraint of these models according to known production data, to reduce uncertainties. Due to the relative simplicity of well tests, numerous authors attempted to include thems in the geostatistical models, especially in a 2D context.
The goal of this work is to propose a simple analytical formula which relates the apparent permeability given by a well test interpretation and the original permeability map around the well for 3D heterogeneous reservoirs. The pressure derivative is used to interpret drawdowns. Assuming that this permeability is related to the original permeability map in the drainage volume by a simple power averaging formula, the best value of the averaging exponent is computed using perturbation methods. It depends on time, and on the vertical to horizontal permeability ratio kv/kh,, and geostatistical anisotropy. For sufficiently long times, when the apparent permeability is stabilized, we obtain an apparent permeability equal to the well-known steady-state equivalent permeability of the medium. This regime occurs when the investigation radius of the well test is larger than the correlation length. Numerical simulations of well tests performed in this regime are in good agreement with the calculation and demonstrate the validity and robustness of the proposed formula.
Progress in the power of computers and in reservoir characterization now makes it possible to represent detailled distributions of oil reservoir heterogeneities which leads to more realistic predictions of flow. Well data are generally scarce and must be complemented by indirect measurements such as seismic (references 1 and 2) and dynamic data. In particular, well-test data are important in the validation of stochastic models around wells. Ideally, starting from an initial geostatistical model, one should be able to generate reservoir realizations respecting all known data, particularly well tests constraints. Here, we suppose that petrophysical attributes of the different faciec are known, so we cannot adjust them to get a correct match (ref 3). In the present problem, the variability originates from the random character of the geostatistical images. The number of degree of freedom is on the order of magnitude of the number of grid blocks. This number is large such that "gradient methods" (see ref 3) cannot be used easily to get a practical solution. All proposed solutions (see ref 4 through 7) require a solution of the "direct problem" i.e. the well test pressure variations for a given permeability map.
This particular problem has already been investigated by many authors (references 3 and 4), mainly in 2 dimensions. Since no analytical expression exists for well tests in heterogeneous reservoirs, the first idea is to perform numerical simulations using a gridded reservoir simulator. As most methods to constrain geostatistical images to dynamic data use a great number of calls to the direct problem solver, simplifications are needed to get exploitable results. In this spirit, several empirical averaging formulas have been proposed and checked for their match against apparent permeabilities (references 3 through 5). These formulae are accurate only if the well test is stabilized, i.e. a single large scale permeability value K can be defined by a standard interpretation of the test. On the other hand, theoretical developments have been limited in the case of permeability heterogeneities of limited contrast, and no explicit solution is known to 3D cases.
Our present goal is to generalize these formulae for three dimensional reservoirs which advantages will be to mix the robustness of empirical formula with the limited permeability contrast results giving a predictive formula. In particular, we will check that the solution depends only on the reservoir petrophysical and geometrical anisotropies through the so called global anisotropy ratio: where is the vertical to horizontal permeability ratio, and Lh and Lv denote the horizontal and vertical correlation lengthscales.
The proposed formula has the structure of a power averaging of the small scale permeabilities over the investigation radius of the test:
Numerical simulation of pressure-transient tests in complex reservoirs is a growing interest topic in Reservoir Engineering. The aim of this paper is to build interpretation tools for synthetic well-tests in such geometries when classical analytical models are not applicable.
An approach based on moments is proposed. It consists in the evaluation of the time-variations of simple spatial global, properties of the simulated pressure-field. It provides at a negligible extra CPU cost a precise evaluation of the size of the drainage zone, as well as an alternative very simple determination of the reservoir equivalent porosity and permeability. It avoids a rather imprecise Log(t) interpretation, gives the main time- scales and length-scales of interest and the occurrence of reservoir- boundary effects. Such an approach is a first step toward an integrated interpretation of well-tests in complex reservoirs linked to a reservoir simulator. Simple mathematical elements, as well as numerical and practical examples of its use over geostatistically generated reservoirs are given.
Progresses in the power of computers and in reservoir characterization allow the simulation of flows in complex reservoirs generated either stochastically or from geological, geophysical interpretation (and references therein). In particular, well tests can be simulated with a sufficient accuracy and compared with actual real data to check the consistency of the chosen reservoir description.
Comparing field data to the simulation results requires new interpretation tools in heterogeneous geometries. Particularly, a complete interpretation of the whole pressure response curve can yield useful insights about the reservoir geometry and consequently can help to improve, e.g., a stochastic description by fixing some parameters of the model. Another important challenge is to incorporate well-test data in the geostatistical description, allowing to obtain images honouring not only the observed data coming from logs or seismic tools, but also from pressure transient behaviour.
The reasons for such a growing interest are:
-firstly, from a methodological point of view, well-test data must be treated as other data: there is no reason except technical difficulties to reject these data as conditioning ones.
- Secondly, pressure transient test investigates reservoir length-scales which can be smaller than the interwell spacing. Since in a geostatistical context, these length-scales cannot be accounted for as no data is a priori directly available and measurable on the field, it is of outermost importance to obtain informations about these length-scales. By now, only a rather qualitative geological knowledge e.g. an assumption about the correlation length can enter the description.
To explain further this important point, let us mention recent analytical works that have been performed by Oliver about pressure diffusion in slightly heterogeneous reservoirs (small permeability contrasts).
When performing fluid-flow simulations, up-scaling techniques must be used because of the required coarse grids. For this purpose, algebraic formula, combining arithmetic and harmonic averagings, extension of Cardwell and Parsons's results, derived from perturbation techniques are proposed taking into account the anistotropy, either natural or due to the reservoir geometry. These formula are valid for 2D or 3D problems and are very little computational time consuming compared to direct numerical methods.
Moreover they integrate rigorous mimima and maxima, thus showing the global uncertainty on equivalent permeability determination. They proved to work well for large permeability contrasts and anisotropy ratios either for simple log-normal or geostatiscally generated media.
In the first application case numerical calculations were carried out for the upscaling of isolated cells. In the other case well test simulations were performed either with the original fine grid or further coarser grids.
Progresses in the power of computers and in reservoir characterization allow the simulation of very high resolution images of complex reservoirs. But for the subsequent flow simulations, coarser grids are inevitable in most cases and one has to upscale the fluid flow parameters (for example permeability or porosity fields). The objective of the present work is to present a new algebraic formula for upscaling absolute permeability diagonal tensors. As opposed to more sophisticated numerical procedures, algebraic formulas are not intended to provide exact results in the general case, but just acceptable estimations for most of the practical applications. Their purpose is a good compromise between cost of calcul and precision. Several of them can be found in the literature and among them in the case of isotropic scalar permeabilities, those based on the bounds of effective permeability. But so far, to our knowledge, the influence of anisotropy was not considered in this context. This anisotropy comes from either sedimentation process or from reservoir geometry. Generally the vertical extent of reservoirs is at least one order of magnitude less than the horizontal one.
We have derived from a perturbation technique an extension of the estimator proposed by Le Loc'h and Guerillot:
Le Loc'h and Guerillot's estimator is only suitable for 2D and isotropic conditions. For this particular estimator K-, K+ are the absolute permeability minima and maxima defined by Cardwell and Parsons and explicated later in the text.
In a 2D case our derivation takes into account anisotropy with the following formula:
with the ponderation coefficient y :
where Ay is the global anisotropy coefficient :
where ky (resp. kx) is the Y (resp. X) direction permeability and x and y cells dimension.