Summary
This study investigates means for efficiently estimating reservoir performance characteristics of heterogeneous reservoir descriptions with reservoir connectivity parameters. We use simulated primary and waterflood performance for two-dimensional (2D) vertical, two- and three-phase, black oil reservoir systems to identify and quantify spatial characteristics that control well performance. The reservoir connectivity parameters were found to correlate strongly with secondary recovery efficiency and drainable hydrocarbon pore volume. We developed methods for estimating primary recovery and water breakthrough time for a waterflood. We can achieve this estimation with three to five orders of magnitude less computational time than required for comparable flow simulations.
Introduction
Several geostatistical methods have been developed over the past decade for generating fine-scale, heterogeneous reservoir descriptions. These methods have become popular because of their ability to model heterogeneities, quantify uncertainties, and integrate various data types. However, the quality of results obtained with these stochastic methods is strongly dependent on the underlying assumed model. Reservoir heterogeneities will not be modeled correctly if the appropriate scales of heterogeneities are not considered. Uncertainties in future reservoir performance will not be quantified if the entire range of critical spatial characteristics are not explored. Simulated reservoir performance will not match historical performance if the appropriate data constraints are not imposed. The likelihood of using an inappropriate model can be greatly reduced if production data is integrated into the reservoir description process. This is because production data is influenced by those heterogeneities that impact future rates and recoveries.
This paper investigates the applicability of using reservoir connectivity characteristics based on static reservoir properties as predictors of reservoir performance. We investigate two types of reservoir connectivity-based parameters. These connectivity parameters were developed to estimate secondary recovery efficiency and drainable hydrocarbon pore volume (HCPV). We use 2D vertical cross sections in the study. Previous work1–3 investigated the correlation of spatial reservoir parameters on reservoir performance for 2D areal reservoir descriptions. We first describe the general procedure. We then follow with definitions, more specific procedure details, and a discussion of the results for the two reservoir characteristics investigated.
General Method
We generated sets of permeability realizations, each set honoring at least the "conventional" geostatistical constraints (i.e., the univariate permeability distribution, the permeability variogram, and the wellblock permeabilities). We used simulated annealing4–6 to generate the permeability realizations and a linear porosity vs. log (permeability) relationship to obtain porosity values at each gridblock location.
Porosity and permeability were the only heterogeneous reservoir properties considered during the study; reservoir thickness was assumed to be a constant. We performed all the flow simulations at the same scale as the permeability conditional simulations. The two- and three-phase black oil flow simulations were run with Amoco's in-house flow simulator, GCOMP,7 on a Sun SPARC 10 workstation.8
We used flow simulation results and analytical calculations to determine water breakthrough time (tBt) and ultimate primary oil recovery. The results for each flow simulation were plotted vs. values of various spatial permeability and porosity-based parameters. We identified the spatial parameter having the strongest correlation with each simulated performance data type.
Recovery Efficiency
Definitions.
Secondary recovery efficiency is considered to be impacted by interwell reservoir connectivity characteristics. However, reservoir connectivity can be defined many different ways. A method has been reported that uses horizontal and vertical permeability thresholds to transform permeabilities to binary values.9 The least resistive paths are determined by finding the minimum distance required to move from one surface (i.e., a set of adjacent gridblocks) to another, for example, from an injector to a producer. We used a binary indicator approach to simplify the computations, thus resulting in an extremely fast connectivity algorithm. However, the success of the method is dependent on the applicability of the designated cutoff values. Such an approach would be most successful for systems comprised of two rock types (e.g., clean sand and shale), each having a small variance but significantly different means. The permeability distributions used in the present study do not fit in this category. Thus, attempts to correlate secondary recovery efficiency variables with the indicator-based connectivity parameters were unsuccessful. We concluded that a more sophisticated connectivity definition, accounting for actual permeability values, was needed to better quantify interwell reservoir connectivity.
As a result of further investigation, the following connectivity parameter was developed for 2D cross sections:
where IRe(i, k) is the secondary recovery efficiency "resistivity index" at gridblock (i, k), ?L is the distance between the centers of adjacent gridblocks, ka is the average absolute directional permeability between two adjacent gridblocks, krw(i) is the estimated relative permeability to water for the ith column, and A is the cross-sectional area perpendicular to the direction of movement. For a horizontal step, ?L/A=?Lx/?Lz, whereas for a vertical step, ?L/A=?Lz/?Lx .
The resistivity index parameter is derived from the analogy between Darcy's law for linear, single-phase fluid flow,
and Ohm's law for linear electric current
where I is the electrical current, ?E is the voltage drop, and R is the electrical resistance. Inspection of Eqs. 2 and 3 shows that the permeance of the fluid system, kA/µL, is analogous to the reciprocal of the electrical resistance. Eq. 1 is the multiphase flow equivalent of the reciprocal of the permeance, dropping the viscosity constant µ.
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