Original Oil In Place (OOIP) calculations based on material balance methods are strongly influenced by data uncertainty. Although some research is available in literature, usually the effects of data uncertainty on material balance calculations are rarely considered and quantified in most studies. This work presents an approach to properly quantify and account for the impact of reservoir pressure and PVT data uncertainty on material balance calculations under different drive mechanisms and using different material balance methods. This study allows reservoir engineers properly select the most suitable material balance method when uncertainty on reservoir pressure and PVT data is significant.
In this work, two different methodologies are proposed. First, a sensitivity analysis was conducted using generated realizations of reservoir pressure and PVT data to evaluate their effect on material balance calculations. Second, a more robust approach was proposed using experimental design and analysis of variance to systematically evaluate the influence of reservoir pressure and PVT data on material balance calculations in an optimal and integrated fashion. In both methodologies, different material balance methods were used and computed OOIP were compared to reference values from a conceptual reservoir model with known PVT data and simulated reservoir pressure. A MATLAB-based program, with graphical user interface, was coded for this purpose1.
Application of the proposed methodologies allowed to determine and quantify the most significant parameters that influence material balance calculations. Interestingly, the most important parameter was the selected material balance method used to compute the OOIP. More accurate results were obtained using the traditional graphical method (F-We vs. Et) for volumetric oil reservoirs with minimal pressure and PVT data uncertainty. In those cases with moderate to significant water influx and gas cap, and some uncertainty on pressure and PVT data, less accurate original oil in place was obtained when graphical methods were used. Reservoir pressure uncertainty was the most significant parameter on the material balance calculations. Gas-oil ratio uncertainty was also significant. Oil and gas gravity, and reservoir temperature were less significant.
Material balance is a simple and one of the most important reservoir engineering tools. Calculations require production/pressure data, PVT data, and aquifer parameters, so that original oil in place and drive mechanisms can be quantified. Data quality is an important issue in material balance calculations. Uncertainty due to data errors can be found in field production data, measured PVT properties, and average reservoir pressure.
Usually, it is expected that oil and gas production are measured with confidence since industry revenues are based on oil and gas sales, and consequently error in production data can be considered minimal. However, reservoir pressure is uncertain since limited well measurements are usually available and averaging procedures might introduce some uncertainty in the computed reservoir pressure history. PVT data can be also uncertain since some reservoirs have no representative fluid samples for a complete PVT analysis and correlations are used instead for material balance calculations.
Quite a number of problems that cannot be bandied economically with the methods usually applied in reservoir engineering can be solved by the theory of integral equations. The main purpose of this paper is to draw attention once more to the usefulness of integral equations. Their benefit is shown by considering a typical example; namely, the behavior of a horizontal multilayered system of oil reservoirs bounded by aquifers.
A number of reservoir engineering problems involving the flow of compressible fluids in porous media are so complex that they cannot be solved analytically, yet the usual numerical techniques require excessive computer time. An example of such a problem is simultaneous production from a horizontal multilayered system of production from a horizontal multilayered system of separate oil reservoirs, each reservoir being bounded by its own aquifer but sharing production wells with the other reservoirs. The difficulties in obtaining an analytical solution to this problem are insuperable; consequently, Tempelaar-Lietz and Lefkovits et al have restricted their studies to particular aspects under simplified conditions. The modern approach is to simulate the flow problem numerically. This implies that the differential equations that describe the problem are approximated by finite-difference equations, problem are approximated by finite-difference equations, which are subsequently solved by a computer. As both the aquifers and the reservoirs must be simulated, the numerical calculations require considerable computer time and storage capacity. In this paper it is shown that problems involving the influx of fluids (e.g., fluid influx from pay zones into wellbores, or water influx from aquifers into reservoirs) can often be described by a system of intergral equations that require much less computer time for solution. These integral equations can be derived only if the motion of the inflowing fluids can be described by linear and homogeneous differential equations, so that the superposition principle applies. The fluid influx can then be principle applies. The fluid influx can then be calculated from integral expressions that contain standard functions, This obviates the need for numerical simulation of those parts of the reservoir from which the influx originates. The standard functions are the cumulative influx as a function of time for a unit pressure drop at the influx boundary, and the pressure drop at the influx boundary as a function of time for a unit influx rate. These functions are interrelated so that if one is known the other can be calculated. For homogeneous and isotropic flow regions of linear or radial geometry, the standard functions have been evaluated in dimensionless form by van Everdingen and Hurst. For other flow regions, they can be evaluated with an auxiliary computer program. The integral equations presented here are by no means new. They are special cases of Duhamel's theorem (see also Carslaw and Jaeger, p. 30), and one of them is explicitly formulated by Samara in his analysis of pressure decline. The simplest form of our integral equations is
t y (t) . y (t) = (t) + K(t,0) . y (0) . d0, 0
. . . . . . . . . . (1)
where the functions v(t), (t) and K (t, 0) are known, and y(t) must be determined. The equation is termed "integral" because y(t), the function to be determined, also appears under the integral sign. The known function K(t, 0), which depends not only on the current time variable t, but also on the auxiliary variable 0 in the interval O 0 t, is known as the kernel of the integral equation. In many of our problems it is a function of t - 0 only; i.e., of the Faltung type. Moreover, it often becomes infinite if 0 approaches t. However, the improper integrals appearing in this paper can be made proper by transformation with a Dirichlet formula, thus avoiding the evaluation of improper integrals.
When fluids are withdrawn from a petroleum reservoir, the space left behind is filled partly by the expansion of the remaining fluids and rock and partly by the influx of water from a contiguous aquifer, if it exists. The Volumetric Balance Equation (VBE) is an expression Of this same statement. Its simplified form is:
Z = Ax + BY (1)
Z = Total Fluid Withdrawals in MMRB
A = Original Active Oil-in-Place in MMRB
X = Unit Expansion
B = Water Influx Constant
Y = Water Influx Function
When sufficient historical data on X and Z are available, various functions for Y can be tried, and by the technique of least squares a set of values can be calculated for A and B.
It has been convenient to write the equation in the following form:
Z/X = A + B (Y/X) (5)
The advantage of this form is that firstly, it has only two variables, and least square calculations are easier for it; and secondly, the values of Y/X and Z/X can be graphically plotted, so that a linear trend can be visually examined. The disadvantage is that the equation has a low resolving power, and can produce erroneous answers. Nevertheless, in a literature survey carried out it was found that most authors used this form of Eq(5).
A paper by Hurst and van Everdingen in 1949 led to the practical solution of many nonsteady-state flow problems. Subsequently, applications of this material have been discussed by several authors. The purpose of this note is to outline a simplified direct procedure for water-influx pressure predictions for pressures above the bubble point. The proposed procedure is unique in that after the "time-conversion factor" and the "influx constant" are obtained, predictions are made without trial and error. The elimination of trial and error is made possible by expressing certain PVT characteristics as explicit functions of pressure. The procedure is applicable to fields producing at pressures above the bubble point and under radial water drive from an infinite aquifer. In the usual applications it is required to predict (1) withdrawals given future pressure history or (2) reservoir pressure, given future withdrawals or production.
The theory, assumptions and conditions under which the nonsteady-state solutions hold are covered in the references and are not repeated. The simplified direct procedure outlined here has been developed from published solutions for the so-called "pressure case". A similar direct procedure for the same special conditions has been developed for the "rate case"; however, in this instance the procedure is lengthy and offers no advantage over the trial-and-error so1ution. Consequently, this paper covers only the procedure based on the pressure case.
Under the pressure case prediction of withdrawals from a waterdrive field, given the future pressure history, involves the use of a nonsteady-state water-influx term together with the material balance equation. The solution of a particular problem requires the following steps: (1) division of past pressure and production data into equal time intervals; (2) computation of cumulative water influx vs pressure drop for each time interval of past history using the material balance equation; (3) determination of the best values for the "time-conversion factor" and the "influx constant" by equating the nonsteady-state influx term with the influx computed in Step 2 (this requires a trial-and-error procedure); (4) prediction of influx for a given pressure drop using the influx term and the computed factors from Step 3; and (5) calculation of fluids produced using the predicted influx in the material balance equation. A forecast of the water-oil ratio or water production rate is required to complete the calculation.
The material balance equation has been used for many years by engineers to determine reservoir performance. The use of this equation in general has been twofold: first, to determine the oil-in-place in a reservoir; and second, to predict the future performance. The Schilthuis form of the material balance equation, or one equivalent to it, has been used by many of the engineers. The use of the Schilthuis form in predicting future performance has proved to be laborious in that one generally must make several estimates at each step of the trial and error calculation before arriving at a check of the oil-in-place.
A method is presented for expediting the calculations. The Schilthuis form of the material balance has been rearranged into a more useable form. In using the material balance to predict the future performance of a reservoir, it has become common practice to estimate the incremental oil production for each step of the calculation which corresponds to a given pressure reduction. Instead of estimating the incremental oil production, the method outlined in this paper better lends itself to estimating the instantaneous gas-oil ratio. A check of oil-in-place is more easily accomplished by estimating gas-oil ratio since it is less sensitive to small inaccuracies. This method incorporates the same self-checking feature inherent in the Schilthius form.
Development of Simplified Equation
In the development of the simplified equation, the Schilthuis form of the material balance equation will be used.
The present paper contains a method which combines the material balance equation with the water influx equation to obtain reliable values for the active oil originally in place and a quantitative evaluation of the cumulative water influx. The method is illustrated by an application to a reservoir without original gas cap. In the absence of an original gas cap, results may be obtained using only field production and pressure data, PVT analyses, and a minimum of subsurface information.
Characteristics of Reservoir and Reservoir Fluids
Production in the field under review is obtained from the top of the Wilcox formation of Eocene age, at a depth of approximately 8,100 ft subsea. The structural map, Fig. 1, shows that the accumulation, half elliptical in shape with the long axis in east-west direction, is trapped to the north and on the upthrown side of a normal fault. A number of east-west faults with additional minor faulting in random directions are found in the general area, and seismic information and production performance indicate in this particular case the existence of a second fault a short distance downdip, with a strike more or less parallel to the fault controlling the accumulation. It is estimated that 1,830 acres were originally underlain by oil and that the maximum thickness of the original oil column, about 37 ft, was only slightly greater than the maximum thickness of the oil-bearing sand. Maximum net sand thickness was estimated at approximately 26 ft. From logging and subsurface information the gross sand volume has been placed at 37,400 acre-ft, which reduces to about 27,500 acre-ft after probable non-productive intervals have been deducted. The following table presents laboratory analyses data for the Wilcox sand cores.