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# Spe 59781 leo

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• Good morning. My Name is Leonardo Vega. I come from Texas A&M University. My presentation is entitled: “Determining OGIP and Aquifer Performance with No Prior Knowledge of Aquifer Properties and Geometry.”
• I have developed a new approach for the simultaneous determination of OGIP and Aquifer performance.in a water-drive gas reservoir. With my approach, we don’t need to know, in advance, neither the geometry nor the properties of the aquifer. We only need performance data in the form of production rate history and some reservoir pressure measurements.
• The main advantages of the new method over previous methods are that: 1. Previous methods require a lot of information about the aquifer. 2. Previous methods require that the aquifer be homogeneous. 3. My new approach is very practical. 4. Other methods are very idealistic
• Now let me make an overview of the rest of my presentation. First, I will give you an introduction, followed a description of previous methods. Then, I will present my new approach and the results that I used to finally arrive at the conclusions.
• For a volumetric or depletion type gas reservoir, the Material Balance Equation can be expressed as----- In which it can be seen that a plot of p/Z versus Gp would be a straight line--- With intercepts at p/Z at the initial conditions when Gp=0---- And at Gp=OGIP when p/Z=0.----- However, in the presence of an aquifer, the equation is not linear any more--- With different levels of pressure support depending on the strength of the aquifer. In this equation, the term Crw, accounts for the compressibility of the rock and the water in the reservoir, and W e is the cumulative water.
• This plot shows that the performance of water-drive gas reservoirs is rate-dependent. The blue line shows the typical gas production rate schedule of many gas reservoirs in which there is periods of high production rates and periods of low production rates due to curtailment. Notice that during the periods of curtailment, the pressure tends to increase. This plot was created by assuming an OGIP equal to 800 Bscf and monthly pressure measurements
• In practice, however, pressure measurements are only made on a yearly basis. In that case, the plot shown in the previous slide would look like this. This line represents the behavior if the reservoir were volumetric. The diamonds show its actual behavior. Notice that, even though this is a water-drive gas reservoir, the p/Z plot follows a fairly linear. The common is to attribute this non-linearity to gauge errors. Thus, if we had used the p/Z technique, we would have overestimated the OGIP by 122%.
• Now let me make a review of previous methods..
• In 1949, van Everdingen and Hurst presented a solution to the diffusivity equation that satisfied certain simplifying assumptions such as aquifer homogeneity and elementary reservoir-aquifer geometries
• In 1964 and 1965 Havlenah and Odeh presented a technique for aquifer fitting. Using this technique it was possible to determine the OGIP by expressing the material balance equation in the form of a straight line. This is a plot of the underground withdrawal term versus the cumulative water influx term in the material balance equation. Basically, their technique consisted of assuming the properties and the shape of the aquifer. Then, in an iterative manner, several aquifer sizes were assumed. If the resulting curve was concave upward, it was because the assumed aquifer was too small. This procedure is repeated several times until a 45º straight line is obtained. The intercept with the y axis is the actual value of the OGIP.
• However the Havelnah-Odeh lacks uniqueness As I mentioned in the previous slide, the properties and shape of the aquifer have to be assumed. This information is hardly ever known with any degree of certainty since an operator would not be willing to drill wells into an aquifr just to determine what its properties are. Thus, if a set of aquifer properties and shape are assumed, a value of the OGIP would be obtained, whereas with a different set of conditions, a different value of the OGIP would be obtained.
• To better understand the new approach, let me give you a definition of what an Aquifer Influence Function is. An AIF is “the pressure drop due to a unit water influx rate at the original GWC.. They are like type curves, unique to each aquifer. Mathematically they can be expressed like this, In which pi stands for the initial pressure in the aquifer, p(t) is the pressure at the original GWC at any time t, ew=1 specifies the fact that the pressure drop is for a unit water influx, and F(t) is the AIF at any time t.
• In 1964, Coats, et al presented an aquifer model. (pause) They proposed the use of AIF (click) for a system of arbitrary geometry and homogeneity.
• Coats and his coworkers found that the exact solution to the diffusivity equation for this kind of system could be expressed like this: In which the there is a linear terms which represents the stabilized flow, And an infinite sum of exponential terms which accounts the transient flow.
• In 1988, Gajdica proposed a new method. He calculated the OGIP and aquifer performance from production performance data. He used the linear programming technique. along with 32 field data sets to validate his results.
• Gajdica’s technique used the term Relative Error, which he defined as the sum of the ratios between the difference of the observed and calculated pressure drops and the assumed values of the OGIP. Since a minimum could be observed in the Relative Error function when it was plotted vs. the assumed values of the OGIP, Gajdica heuristically claimed that this minimum corresponded to the optimum OGIP.
• However, Gajdica’s technique had problems. Sometimes, it was determined that the optimum OGIP was smaller than the cumulative volume of gas which had already been produced.
• Two questions may then arise about the use of Gajdica’s technique: 1. Is the technique presented by Gajdica valid at all? 2. Is the anomaly the previous slide due to errors in some of his field data?
• Let us consider the first question: Is Gajdica’s technique valid at all As I mentioned earlier, Gajdica’s technique is based on term that he defined as Relative Error. However, this term lacks a sound statistical meaning.
• Now, let me present my new approach.
• For my new analysis, I decided to use the absolute error function. The absolute error function, A, is defined as the sum of the absolute values of the difference between the calculated and the observed pressure drops. Unlike the Relative Error, the Absolute Error function has a sound statistical meaning, since it basically represents the distance between a continuous function and a set of discrete points. So that I could make data sets comparable, I normalized the A by dividing by the total number of observed data points.
• Thus, the methodology to analyze the performance behavior included the following two ideas: 1. To analyze the behavior of the normalized absolute error instead of the relative error. 2. To use synthetic data to isolate the true nature of the problem
• In the process of using the new approach, the following procedure was followed to determine the OGIP and the AIF. 1. Assume several values of OGIP. 2. For each assumed OGIP, minimize the the A N . 3. Plot the minimized A N versus the assumed OGIP 4. Report the AIF for each assumed OGIP. During the optimization process , the AIF must meet certain constraints. Those constraints have to do with the shape of the AIF.
• In other words, the AIF must meet certain smoothness constraints. First, for a positive water influx rate, the AIF must be positive or zero. Mathematically, it can be expressed like this: F(t) must be greater than or equal to zero.
• Second, for a positive water influx rate, the must increase or remain constant. Mathematically, it can be expressed like this: The first derivative of the AIF with respect to time must be greater than or equal to zero.
• And third, for a positive water influx rate, the AIF must be concave downward or be a straight line. Mathematically, it can be expressed like this: The second derivative of the AIF with respect to time must be less than or equal to zero.
• This table shows the reservoir data that I assumed to generate the synthetic performance. Please notice that in this particular case I assumed an OGIP equal to 700 Bscf.
• To better understand the behavior of the A N , I will first illustrate its behavior in a volumetric depletion reservoir. This is a plot of an arbitrary flow rate versus the cumulative gas produced. Then, using values on this plot, the reservoir data that I showed you in the previous slide, and the material balance equation, I calculated the corresponding values of p/Z. Now I have all of the performance data that I need as input for my approach, namely, production rate and pressure as a function of time.
• It could be observed that the AN curve displayed a typical behavior in volumetric depletion reservoirs. It was divided into regions: 1. From OGIP equal to zero to the actual OGIP, the A N was characterized for having very small values and following an almost flat trend. 2. From point A to point B, there is a sudden increase in the values of the curve. 3. From point B onwards, the A N curve tends to be flat. This pattern can be explained by taking a look at the behavior of the calculated water influx rate from the material balance equation for a volumetric reservoir.
• This table shows values of e w calculated from the Material Balance Equation. Notice that when the OGIP is equal to the actual one (700 Bscf), the water influx rate is equal to zero, since the data set was constructed assuming a volumetric depletion gas reservoir. However, for values of the OGIP greater than the actual one, the values of the water influx rates are all positive and increase with time. As a result, the changes in the water influx rate will always be positive. On the other hand, for values of the OGIP larger than the actual one, the water influx rates are all negative and decrease with time. As a result, the changes in the water influx rate will always be negative.
• Let’s analize the behavior for G&lt;Gactual as shown previously. 3. The AIF is constrained to be non-negative Therefore, it will always be possible to obtain a feasible solution. Effectively, there was a different AIF for each assumed value of the OGIP, all of which satisfied the three smoothness constraints shown on the left.
• Surprisingly, all of them matched the observed pressures equally well, at least to the naked eye.
• As a result of these almost equally good pressure matches, the A N function displays very small values and follows an almost flat trend in the Low Region.
• Lets now see what happens when G>Gactual. As I showed you before, the changes in the water influx rate will be always negative. The pressure drop term is always non-negative. Therefore, the only feasible solution would be when F(t) is equal to zero due to the first smoothness constraint. Thus, when the assumed OGIP turns out to be in this range, the F(t) curve turns out to be a straight line equal to zero. In conclusion, it can be stated that in this region, F(t) is always equal to zero and the calculated pressure drop is also equal to zero..
• Since  p cal is equal to zero, the normalized absolute error becomes the sum of the ratios of the calculated pressure drops to the total number of observations. Which is a constant. That’s the reason for the flat appearance in this upper region.
• It was observed that when the straight line representing the intermediate region was extrapolated to the abscissa, the point of intersection corresponded to the actual value of the OGIP.
• And this was not just a heuristic observation. This is a zoomed view. Notice that a minimum can be observed.
• Now let’s analyze the behavior of the A N in a water-drive gas reservoir. This table shows the aquifer data that I assumed to generate the synthetic performance data. I will use the same gas reservoir data that I used for the volumetric reservoir example.
• To generate the performance data in this water-drive gas reservoir, I have used a constant gas production rate of 50 MMscf/D, as indicated by the horizontal green line. Then, I used the material balance equation along with the analytical solution to the diffusivity equation for a linear system to generate the p/Z data, indicated by the black X&apos;s. Remember that the value of the OGIP that I assumed was 700 Bscf. Notice that the p/Z vs. Gp follow an almost linear trend. As a result, an engineer would be tempted to assume that the reservoir is of the volumetric depletion type. If he did so and extrapolated the resulting straight line to the abscissa to estimate the value of OGIP, he would over estimate it by 39%.
• Let’s now use the new approach. This plot shows the behavior of the A N in this particular water-drive gas reservoir Notice that as in the case of the volumetric reservoir, the plot is divided into three regions. The only difference is that the middle region is not a vertical line. However, the point of interception between the low region and the middle region still coincides with the actual value of the OGIP.
• Again this is not just a heuristic observation. This plot shows a zoomed view of A N for this particular water-drive gas reservoir. Notice that there is a minimum at the actual value of OGIP. Then, I repeated the same experiment with the same reservoir and aquifer properties, but changing the gas production rate schedules. I used other constant qg schedules, increasing q g schedules and decreasing q g schedules.
• Just as I expected, I always obtained the same AIF at the optimum OGIP, no matter what the gas production rate schedule was. This illustrates the fact that the AIF is like a fingerprint of the aquifer and is not affected by the production rate schedule like the p/Z plot.
• Let me now show you some results..
• To illustrate the practical applicability of my new approach, I used data from a field in the Gulf of Mexico. For convenience, I will call it Field “A”. This graph shows the performance data of Field “A”.
• Notice that the p/Z vs. Gp data appear fairly linear. Thus we would normally be tempted to use the p/Z Technique. If we do, we would calculate a value of OGIP of approximately 270 Bscf.
• Let me now show you the conclusions of my research:
• Thank you very much. I’d be glad to answer any questions you may have.
• Gas research can be divided into two categories: First, those in which the HCPV does not change significantly as a result of production, namely volumetric or depletion type gas reservoirs. The reason for this is that there is a complete lack or very little water encroachment from a neighboring aquifer.

## Spe 59781 leoPresentation Transcript

• Determining OGIP and Aquifer Performance With No Prior Knowledge of Aquifer Properties and Geometry Leonardo Vega Texas A&M University Masters’ Division SPE International Student Paper Contest October 5, 1999
• New Approach
• OGIP
• Aquifer Performance
• No Prior Knowledge of Aquifer Properties or Geometry are Required
• Only Production Performance Data
• Previous methods require a lot of information about the aquifer
• Previous methods require a homogeneous aquifer
• New method is very practical
• Other methods are very idealistic
• Overview of Presentation
• Introduction
• Previous Methods
• New Approach
• Results
• Conclusions
• Overview of Presentation
• Introduction
• Previous Methods
• New Approach
• Results
• Conclusions
• Material Balance Water drive Strong Moderate 0 p i /z i 0 G p =G p/Z G p Depletion
• Performance of Water-Drive Gas Reservoirs Is Rate-Dependent 2,500 2,700 2,900 3,100 3,300 3,500 3,700 0 50,000 100,000 150,000 200,000 250,000 300,000 350,000 400,000 G p G p , MMscf p/z © Leonardo Vega 0 100 200 300 400 500 600 Gas Rate, MMscf/Day
• p/Z Plot of Water-Drive Gas Reservoir May Look Linear p 2 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 OGIP G p , Bscf p/z psia
• Overview of Presentation
• Introduction
• Previous Methods
• New Approach
• Results
• Conclusions
• In 1949, van Everdingen-Hurst Presented Solution to Diffusivity Equation
• Aquifer Homogeneity
• Elementary Reservoir-Aquifer Geometries
Aquifer Gas Reservoir
• In 1964-65 Havlena-Odeh presented Technique for Aquifer Fitting © Leonardo Vega F/E g (Bscf) W e B w /E g (Bscf) Aquifer too small Aquifer too large Correct Match 45 o G
• Technique of Havlena-Odeh Lacks Uniqueness F/E g (Bscf) W e B w /E g (Bscf) © Leonardo Vega Aquifer Description 2 Aquifer Description 1 G 1 G 2
• Definition of Aquifer Influence Functions (AIF)
• Pressure drop due to a unit water influx rate at the original GWC
• like type curves
• unique to each aquifer
• In 1964, Coats et al. Presented an Aquifer Model
• Systems of Arbitrary Geometry and Heterogeneity
• They Proposed The Use AIF
Aquifer Gas Reservoir © Leonardo Vega
• Coats et al. ’s Exact Solution © Leonardo Vega
• In 1988, Gajdica Proposed New Method
• Calculated OGIP and aquifer performance from production performance data
• Used linear programming technique
• Used 32 field data sets to validate results
• Gajdica’s Technique
• Gajdica’s Technique Had Problems OGIP, Bscf Relative Error, psi/Bscf © Leonardo Vega G p max G opt 0 200 400 600 800 1000 1200 1400 0 2 4 6 8 10 12
• Two Questions Arise About Use of Gajdica’s Technique
• Is the technique presented by Gajdica valid at all?
• Is the anomaly due to errors in some of his field data?
• Is Gajdica’s Technique Valid At All?
• The Relative Error Function Lacks any Statistical Meaning
• Overview of Presentation
• Introduction
• Previous Methods
• New Approach
• Results
• Conclusions
• Absolute Error Function Has Sound Statistical Meaning © Leonardo Vega
• Methodology to Analyze Performance Behavior
• Analyze the behavior of the Normalized Absolute Error, A N , instead of the Relative Error
• Use synthetic data to isolate the nature of the problem.
• Procedure To Determine OGIP and AIF
• Assume several values of OGIP.
• Optimize the A N .
• For each assumed OGIP, report the optimized A N , and corresponding AIF.
• Plot the optimized A N versus the assumed OGIP.
• AIF Must Meet Certain Smoothness Constraints © Leonardo Vega
• The AIF must be positive or zero
Aquifer Influence Function 0.0000 6.0000 0 30 t F(t) t
• AIF Must Meet Certain Smoothness Constraints © Leonardo Vega
• The AIF must increase or remain constant
Aquifer Influence Function 0.0000 6.0000 0 30 t F(t) t
• AIF Must Meet Certain Smoothness Constraints © Leonardo Vega
• The AIF must be concave downward or be a straight line
Aquifer Influence Function 0.0000 6.0000 0 30 t F(t) t
• Reservoir Data Assumed to Generate Synthetic Performance © Leonardo Vega
• Volumetric Reservoir Flow Rate and p/Z Performance p 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 0 20,000 40,000 60,000 80,000 100,000 0 10 20 30 40 50 60 70 80 90 100 © Leonardo Vega G p , MMscf
• A N Displayed Typical Behavior In Volumetric Depletion Reservoirs 0 50 100 150 200 250 300 350 0 200 400 600 800 1,000 1,200 0-A : Lower Region A-B : Middle Region Larger than B: Upper Region B A © Leonardo Vega A N , psi/point OGIP, Bscf
• e w Calculated from Material Balance Equation Volumetric Depletion Reservoir For G<G actual  e w >0 For G>G actual  e w <0 © Leonardo Vega
• For G<G actual (Volumetric Depletion Reservoir)
• Pressure Match When G<G actual (Volumetric Depletion Reservoir) © Leonardo Vega
• A N Behavior for G<G actual (Volumetric Depletion Reservoir) 0 50 100 150 200 250 300 350 0 200 400 600 800 1,000 1,200 A N , psi/point OGIP, Bscf
• For G>G actual (Volumetric Depletion Reservoir) F(t)=0  p cal =0
• For G>G actual (Volumetric Depletion Reservoir) Since  p cal =0
• Middle Region (Volumetric Depletion Reservoir) © Leonardo Vega
• Zoomed View of Lower Region (Volumetric Depletion Reservoir) © Leonardo Vega
• Water-Drive Gas Reservoir Aquifer Data Assumed © Leonardo Vega
• Water Drive Gas Reservoir q g =50 MMscf/D 0 10 20 30 40 50 60 200 400 600 800 1,000 G p , MMscf 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Gas Rate Observed Pressure OGIP overestimated by 39% © Leonardo Vega OGIP actual =700 Bscf
• Behavior of A N Water Drive Gas Reservoir ( q g =50 MMscf/D) © Leonardo Vega
• Zoomed View of A N Water Drive Gas Reservoir ( q g =50 MMscf/D) © Leonardo Vega
• Obtained The Same AIF For All Production Schedules 0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02 6.0E-02 0 100 200 300 400 500 600 Time, days © Leonardo Vega
• Overview of Presentation
• Introduction
• Previous Methods
• New Approach
• Results
• Conclusions
• Performance Data of Field “A” © Leonardo Vega Rate and Pressure Behavior as a function of Gp 0 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 30 35 G pa , BScf 5,200 5,300 5,400 5,500 5,600 5,700 5,800 5,900 6,000 6,100 Gas Rate Observed p/Z q g , MMscf p/Z, psia
• p/Z Technique 0 10 20 30 40 50 60 70 80 90 100 50 100 150 200 250 300 G pa , BScf 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 Gas Rate Observed q g , MMscf/d p/Z, psia
• OGIP Determined with New Approach © Leonardo Vega 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0 100 200 300 A N , psi/point OGIP, Bscf
• Overview of Presentation
• Introduction
• Previous Methods
• New Approach
• Results
• Conclusions
• Conclusions Unlike the p/z plot, the shape of the A N permits the recognition of the reservoir drive mechanism. The A N allows the determination of the OGIP in a water-drive reservoir. No prior knowledge or assumptions about the aquifer properties and geometry are required . © Leonardo Vega
• Conclusions The optimum OGIP is located where the middle and the lower regions coincide. The drive mechanism and the optimum OGIP can be easily recognized, even when few production-pressure data are available. © Leonardo Vega
• Conclusions Even though the risk of a non-unique solution exists, its occurrence has been diminished. Unlike the Havlena-Odeh method, when used along with the the van Everdingen­Hurst exact solution, this method does not need a continuous re-evaluation of the aquifer. © Leonardo Vega
• Determining OGIP and Aquifer Performance With No Prior Knowledge of Aquifer Properties and Geometry Leonardo Vega Texas A&M University Masters’ Division SPE International Student Paper Contest October 5, 1999
• Volumetric Gas Reservoirs No Water Encroachment © Leonardo Vega
• How to Generate Synthetic Data
• Assume aquifer geometry.
• Assume k,  ,  , c t , A, and x e in aquifer.
• Assume reservoir properties T,  g , c f , c w , G and p i ..
• Calculate z, B g , B w , and V p .
• Assume q g (t) and calculate G p .
• Calculate p D (t D ) from exact solution of Diffusivity Equation.
• Calculate p(t) using the Superposition Principle.
• Use t, p(t) and DG p (t) as input to the AIF program (Reservoir Performance Data).
How to Generate Synthetic Data
• Water Drive Gas Reservoir q g =100 MMscf/D © Leonardo Vega
• A N Behavior Water Drive Gas Reservoir ( q g =100 MMscf/D) © Leonardo Vega
• Water Drive Gas Reservoir q g =150 MMscf/D © Leonardo Vega
• A N Behavior Water Drive Gas Reservoir (q g =150 MMscf/D) © Leonardo Vega
• Water Drive Gas Reservoir Variable Production Rate OGIP overestimated by 32% © Leonardo Vega
• A N Behavior Water Drive Gas Reservoir (Variable Production Rate) © Leonardo Vega
• A N Behavior (Zoomed View) Water Drive Gas Reservoir (Variable Production Rate) © Leonardo Vega