The Effect of Skin and Damage in Each Layer on Flow Performance
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Analysis of Layered Gas Reservoir Performance Using a Quasi-Analytical Solution for Rate and Pressure Behavior
Research Proposal I Nengah Suabdi
Chair of Advisory Committee: Dr. W. J. Lee Co-Chair of Advisory Committee: Dr. Thomas A. Blasingame Objectives The overall objectives of this research are: 1. To provide a quasi-analytical solution for the depletion (i.e., boundary-dominated flow) performance of wells produced at a common production pressure in a layered gas reservoir. This scenario is that of a "layered, no crossflow" case, and as such, we duplicate the numerical simulations given by Fetkovich, et al.1 using our quasi- analytical approach as a validation.
2. To use the analytical liquid flow solutions and the quasi-analytical gas flow solutions as mechanisms for characterizing the performance of layered gas reservoirs—in parti- cular, cases of commingled production (no crossflow in the reservoir). The methods derived from these solutions will be used to estimate: The total original gas-in-place (G). The moveable (or recoverable) reserves in each layer (EURj), The permeability ratio (2-layer case), The total flow capacity (kh product), and The productivity index for individual layers (Jgj). 3. To investigate the sensitivity of individual layer properties on the depletion perfor- mance of layered reservoirs. The reservoir properties to be investigated include: The permeability ratio (2-layer case), Skin factors for individual layers, Reservoir layer volumes, and The effect of drawdown (i.e., the magnitude of the wellbore flowing pressure) 2
______This proposal follows the style and format of the SPE Journal.
Deliverables
The expected deliverables of this work are: 1. Presentation of a single-layer, quasi-analytical solution for the depletion performance of gas wells produced at constant bottomhole pressure. This solution will be used to develop rate and cumulative production relations for wells in a layered gas reservoirs (with no crossflow in the reservoir).
2. Investigation of the sensitivity of individual layer properties on the depletion performance behavior for a layered reservoir system. This study includes: The permeability ratio (2-layer case), Skin factors for individual layers, Reservoir layer volumes, and
The effect of drawdown (i.e., pwf). 3. Development of analysis techniques for commingled production data from layered gas reservoirs—in particular, the following techniques are provided:
a. Modified p/z versus Gp analysis plot for layered (no crossflow) systems, b. Type curve analysis approach for the analysis of rate-time data from layered (no crossflow) gas reservoir systems (this includes the generation of new production performance type curves for layered gas reservoirs), c. Direct extrapolation technique for the estimated ultimate recovery (EUR). 4. Application/validation of these new analysis techniques using both field and synthetic data (the synthetic cases are generated using a commercial reservoir simulation program).
Present Status of the Question
Decline Curve Analysis for Single Layer Reservoir Systems:
Decline curves and decline type curves are the primary mechanisms for the analysis and prediction of oil and gas production in the petroleum industry. This is not an overstatement of the value of decline curve methodologies — however, we must recognize that decline curve methods have evolved from simple (basically empirical) extrapolation techniques into very 3 comprehensive (and hence, complex) reservoir models. The pioneering work in this area was provided by Arps,2 who presented empirical exponential and hyperbolic models for the purpose of rate extrapolation. Interestingly, the exponential model is actually the rigorous solution for a well producing a slightly compressible fluid from a closed reservoir (boundary-dominated flow prevails). In a similar fashion, the "hyperbolic" relations proposed by Arps have been shown to be approximate solutions for both the dry gas and solution gas drive reservoir cases. This is not a trivial issue — the fact that the "empirical" Arps relations have an analytical (or semi- analytical) basis means that these relations hold promise in providing representative predictions of reservoir performance.
In 1973 Fetkovich3 presented the classic work on the topic of decline type curve analysis. Fetkovich showed that the Arps exponential decline relation is the rigorous (exact) solution for a well producing at a constant bottomhole pressure at pseudosteady-state flow conditions. The exponential decline equation was used as a correlating function to establish the "dimensionless decline variables" (tDd and qDd), where these variables were then used to develop the so-called "Fetkovich" decline type curves. The "Fetkovich" type curve is unique in that it can be used for both analysis and prediction. This type curve permits us to forecast well performance, but it also allow us to estimate reservoir properties (i.e. flow capacity, kh, skin factor, s, fracture half-length, etc), as well as the oil or gas in place. This classic work by Fetkovich is the foundation for all modern work in production data analysis. The primary issue related to the present work is the extension to multilayer reservoir cases. Fetkovich, et al.1 provide an approach for the analysis/interpretation of multilayer gas reservoir performance, but we note that neither the Fetkovich, et al. work (nor any subsequent work) has effectively addressed the use of decline type curves for multilayer reservoir systems.
In 1985 Carter4 presented a new set of type curves developed exclusively for the analysis of gas reservoir performance data. Carter used a finite-difference numerical model to generate a sequence of decline type curves for gas wells produced at a constant bottomhole pressure. The most significant contribution of this work was that Carter demonstrated that the constant pressure behavior of a gas well is path-dependent (as we might expect), but that this path is uniquely defined by the level of drawdown (i.e., the bottomhole pressure). 4
In short, Carter showed that a correlating parameter () depends only on the reservoir fluid properties and presented a sequence of rate solutions for various cases of a constant wellbore pressure. Carter's type curves provide a consistent (and rigorous) approach for the analysis of production data from a gas well produced at a constant bottomhole pressure. However, these type curves have several limitations — the approach is only valid for wells produced at a constant bottomhole pressure, and we would like to have a more generalized approach where the gas flowrate-time behavior can be correlated with the flowrate-time behavior predicted by a liquid theory.
In 1987 Fraim and Wattenbarger5 developed a correlation of the gas and liquid flow solutions using a "pseudotime" (or normalized time) function to account for changes in gas properties with pressure and time. This significantly improved the use of the Fetkovich type curve for the analysis and interpretation of gas well performance — and essentially made the use of Carter's type curves unnecessary. The Carter type curves remain popular because this approach is simple and straightforward (comparatively) — however, the Fraim-Wattenbarger approach is preferred because of its ability to "correlate" gas and liquid solutions (a much more general approach).
The only significant issue related to the Fraim and Wattenbarger5 technique is the requirement of an accurate estimate of gas-in-place (where G is used to "seed" the pseudotime calculation). This most often results in an iterative procedure (based on G), but this is easily resolved in one or two steps, and the iterative calculations are easily solved with modern computational tools. As a practical note, it is not recommended that the iterative calculation procedure be performed automatically, but rather, manual intervention is recommended (successive hand analyses will yield a convergent solution, but an automated regression solution may not yield the most representative estimate of gas-in-place).
In 1986, Aminian, et. al.6 also developed a set of type curves base on a semi-analytical model for gas flow behavior in a closed reservoir. These type curves are formulated in a "dimensionless" format like the Fetkovich-style type curves, although the dimensionless variables are defined in terms of gas reservoir variables. This work is similar to that proposed by Carter,6 but presented in a less rigorous form (these decline type curves can be applied in a straightforward fashion, but will not be as rigorous as the Fraim and Wattenbarger5 approach). 5
In 1991 Blasingame, et al.7 introduced constant pressure analog time and constant rate analog time functions as mechanisms to analyze production data exhibiting variable-rate/variable pressure drop behavior. This work provides a rigorous, straightforward approach for the analysis of liquid or gas production data, and offers the option of using analogs for either constant rate of constant pressure (Fetkovich decline type curve format). As noted, these analog time functions are formulated in terms of a pseudotime for the case of a gas reservoir, though the approach still requires an iterative procedure for the gas-in-place. Blasingame, et al. provide verification of this approach using simulated and field performance data for the oil (liquid) and gas well cases. The approach was not specifically tied to the use of a decline type curve for the constant rate analog case.
In 1993 Palacio and Blasingame8 presented a new, unified method for the analysis of production data (single phase oil or gas flow) using an appropriate pseudopressure-pseudotime trans- formation and type curves analysis technique. This method is completely general, and should be suitable for the analysis of all production data (excluding shut-ins and recompletions). The approach uses a modified time function (which addresses both the variable-rate/variable pressure issue, as well as the variation of gas properties with time).
Palacio and Blasingame also presented a "Fetkovich-Carter" type curve, which combines the constant pressure gas flow solution and the Arps' decline curve stems on a single type curve. The "rate-integral: and "rate integral-derivative" functions have been added to the "Fetkovich- Carter" type curve, which improves the overall utility of this approach for scenarios of constant pressure production. Although this work represents an improvement in the interpretation and analysis of well performance data from gas reservoir systems, it remains limited to the single- layer case. An extension of the "Fetkovich-Carter" type curve to the case of a multilayered gas reservoir would be useful.
Knowles9 (1999) presented a new approach for developing an approximate solution for boundary-dominated gas flow in single-layer reservoirs. His approach was to use a straight line approximation for the behavior of the ct product with the p/z profile (i.e., the "first-order" poly- nomial approximation). This approach yields a p/z-squared (i.e., (p/z)2) form of the stabilized flow equation, which in turn provides the proper algebraic form for coupling the stabilized flow 6 equation directly with the gas material balance equation in order to yield an explicit analytic rate- time gas equation.
Ansah, et al.10 (2000) presented a suite of semi-analytical relations for boundary-dominated gas flow in single-layer reservoir systems that can be used for rate-time, rate-cumulative and p/z (gas material balance) predictions. Similar to Knowles, they also employed the "first-order" poly- nomial approximation assumption for the behavior of the ct product with the p/z function. This relation is (generally) limited to pressures below 5000-6000 psia, but the simplicity of the relation gives rise to a number of potential applications (reserves prediction, production extrapo- lations, etc.). Ansah, et al. present these solutions in terms of dimensionless variables and in the form of type curve plots. Field-variable solutions can easily be derived from the dimensionless solutions.
Doublet, et al 11 presented a variety of oil field case analyses using the Fetkovich/McCray type curve to estimate oil-in-place and moveable oil reservoir volumes, as well as the flow charac- teristics of the reservoir. In this decline type curve analysis they use the "rate integral" and "rate integral derivative" functions to analyze and interpret production rate and pressure data from oil reservoir systems. This work provides a strong validation of the approach as well as the utility of using decline type curve analysis for single-layer reservoir systems. An extension of this approach for multilayer gas reservoir systems is needed and should be developed.
In the proposed work, we present a new, quasi-analytical flow solution for the depletion performance of gas wells produced at constant bottomhole pressure in a layered reservoir. This approach is quite simple, and borrows from the Knowles9 and Ansah, et al.10 work (the single- layer approximate solution for gas systems is used for the multilayer reservoir case). We provide new type curves in terms of rate-time, pressure-cumulative-production, and cumulative produc- tion-time performance for layered gas reservoir. The type curves are applied and verified using both field and synthetic well performance data.
Behavior of a Well Producing from a Layered Reservoir System: (No Crossflow)
Templaar-Lietz12 (1959) provided one of the earliest studies of layered reservoir behavior, where they studied the effect of oil production rate on well performance in a two-layer reservoir (equal layer thickness and steady-state flow). Lefkovits, et al.13 (1961) then modified the Templaar- 7
Lietz depletion equation to model a two-layer reservoir with unequal thickness. Lefkovits, et al. presented an analytical solution for a two-layer system (without crossflow) — their solution was developed using the Laplace transform and they obtained an analytical inversion result. Lefkovits, et al. found that differential depletion between layers exists only during transient flow, when the more permeable layer is depleted faster than the less permeable layer. It is relevant to note that, for steady-state flow, both layers contribute equally to production.
Raghavan14 (1989) discussed the behavior of commingled wells in layered system, and suggested methods to determine layer properties and well performance especially for reservoirs with significant contrasts in reservoir properties. The primary conclusion of this study is that during the transient flow period the rates from different reservoir layers are not constant, and, in short, the rates are controlled by the kh-products of individual layers, the total flowrate, and the skin factors for individual layers. At later times, when pseudosteady-state flow becomes established, the layers contribute to production proportional to the pore volumes (Vp) of the individual layers. Both Lefkovits et al.13 and Raghavan considered layered behavior from the perspective of well- test analysis, rather than from the standpoint of long-term production performance. In addition, neither Lefkovits et al, nor Raghavan considered the behavior of layered gas reservoir systems.
In 1991, Johnston and Lee15 presented new guidelines for analyzing production and pressure transient test data from either multilayer reservoir systems. They used analytical solutions to illustrate a variety of well performance cases for various multilayer well completions. Johnston and Lee focused on identifying patterns of characteristic behavior and gave guidelines for inter- preting well performance behavior in multilayer reservoir systems.
Gao and Lee16 (1993) presented a new approach which used pseudopressure and pseudotime formulations to model the performance of commingled reservoirs with pressure dependent fluid properties in each layer. They developed an analytical model (which used liquid flow solutions) which was then compared to the results from a numerical model. Gao and Lee focused their attention on the "short time" analysis of well test data, and did not consider long-term pro- duction performance. We considered extending the Gao and Lee approach for the analysis/ modelling of long-term well performance, we be have instead elected to consider the develop- ment of multilayer gas flow solutions using quasi-analytical gas flow solutions. We believe that this approach will be more straightforward and robust. 8
Depletion Performance of a Layered Gas Reservoir System: Fetkovich, et al.1, applied a fully implicit, radial numerical model to generate rate/time and pressure/cumulative production data for the case of a single gas well produced at a constant wellbore pressure. All of their work pertained to single-phase gas flow in a two-layer gas reservoir — without crossflow (i.e., commingling of fluids is only permitted in the wellbore, there is no crossflow in the reservoir).
For comparison, Fetkovich, et al also investigated the case of two non-communicating layers where each layer contained a single gas phase (this configuration was used to model annual 72- hour shut-ins and 48-hour deliverability tests as well as long-term production). In addition, this model was used to verify the results obtained from the proposed analysis approach (i.e., a p/z
1 versus Gp approach for multilayered reservoir systems). Fetkovich, et al. noted that Arps depletion-decline exponents ("b"-values) between 0.5 and 1 can be obtained with using the "layered no crossflow" model. In contrast, for the case of a single homogeneous layer, the maximum decline exponent (i.e., "b"-value) is 0.5 (ref. 3). The fundamental work by Fetkovich (ref. 3) stated that when two liquid flow solutions are commingled (i.e., added), the resulting rate decline behavior is hyperbolic, (i.e., b 0 — specifically for the case presented by Fetkovich,1 the resulting "b"-value was 0.2).
It is relevant to note that Fetkovich, et al.1 proved that the permeability contrast and layer volume ratio can be inferred (or estimated) from the value of the decline exponent (b).1 It was observed that if a gas well exhibits b>0.5, this behavior can indicate the presence of a layered reservoir system, and in such systems the b-value can approach 1.0. This appears to be a unique characteristic of a layered gas reservoir system. Fetkovich, et al.1 also showed that the shape of the p/z-factor versus cumulative gas production plot is governed by the permeability contrast and layer volume ratio, and that this behavior can (and should) be used to detect layering.
In summary, Fetkovich, et al.1 attempted to demonstrate the theory behind decline type curve analysis for the single layer system as applied to the case of multiple layers, commingled at the wellbore (the so-called "layered-no crossflow" case). Fetkovich1 provided a general result for 9 the case of production at a constant wellbore pressure that was derived as a combination of the material balance equation and the pseudosteady-state flow equations (for gases and liquids).
The second work (i.e., Fetkovich, et al.1) investigated the behavior of both rate-time and p/z- cumulative production performance for a single well in a 2-layer gas reservoir, specifically for the case of production at a constant bottomhole flowing pressure. As a comment, our proposed work will integrate these investigations using a new semi-analytical solution for a single layer gas reservoir to develop a solution for a multilayer gas reservoir.
McCoy, et al.17 continued the work of Fetkovich et al,1 and they included the effect of a poorly stimulated layer on the overall depletion performance of a two-layer gas reservoir. The goal of the work presented by McCoy, et al. was initially to reproduce the results of the Fetkovich et. al.1 study (ref. 1), then to investigate the effect of changing the skin on the lower permeability layer. In a practical sense this situation would be presented by a "stimulation" treatment where the skin factor in the low permeability layer is changed. McCoy, et al. found the controlling parameters to be the layer volume ratio and the skin factor in the low permeability layer — in particular, these parameters control the overall recovery and the time dependency of recovery. The conclusions of the McCoy, et al. study were focused on the dependency of recovery in the low permeability layer, in particular on the skin factor for the low permeability layer. Their goal was to demonstrate the value (i.e., increased recovery) that could (or should) be achieved when the low permeability layer is stimulated.
Another investigation of layered gas reservoir performance was performed by Kuppe, et al.18 In this work they developed a simple spreadsheet model to estimate original gas-in-place, the productivity indices for each layer, and the recoverable reserves. The analysis approach is based on a combined technique which use the gas material balance equation and the pseudosteady-state relation for gas flow. This methodology was demonstrated for wells in the Cooper Basin (Australia), where the production from multilayer gas reservoirs is commingled. Kuppe, et al matched field performance data using a spreadsheet-based program which averages the calculated p/z curves for the high and low permeability layers using an appropriate set of "weighting" factors based on the productivity index. The purpose of the "weighting" factor is to "adjust" the 2 computed p/z trends into a single "average" p/z trend, which is then compared to the measured/observed p/z trend. 10
Procedure
The primary task of this work is to demonstrate the use of our proposed quasi-analytical solution as part of a performance-based reservoir characterization. In particular, our goal is to provide a representative analysis of a typical multilayer gas reservoir sequence (in our case, we will study the Hugoton Field in Kansas (USA)). We will use production data, wellhead pressure data, and annual shut-in pressure data obtained for individual wells. To achieve the objectives of this research, we propose the following tasks:
1. Development of the quasi-analytical model for a layered gas reservoir system produced at constant bottomhole pressure. The production is "commingled" at the wellbore, there is no "crossflow" in the reservoir.
The pressure and flowrate models are given in dimensionless format as follows: (example plots are also provided to illustrate characteristic behavior)
Dimensionless Wellbore Pressure for an Individual Layer: (pDj) 1 pDj ( pwD 0) ...... (1) (1 0.5 tDdj )
(1 pwD ) (1 pwD ) exp(- pwD tDdj ) pD pwD ( pwD 0) ...... (2) (1 pwD ) (1 pwD ) exp(- pwD tDdj )
( p/z) j pDj ...... (3) ( pi /zi )
Dimensionless Wellbore Pressure for the Total System: (pD) 1 pD ( pwD 0) ...... (4) (1 0.5 tDd )
(1 pwD ) (1 pwD ) exp(- pwD tDd ) pD pwD ( pwD 0) ...... (5) (1 pwD ) (1 pwD ) exp(- pwD tDd )
( p/z) pD ...... (6) ( pi /zi ) 11
p/z versus Total Cumulative Gas Production: (pwD = 0.1 case)
Fig. 1 - p/z versus total cumulative gas production for a commingled gas well produced at a constant wellbore pressure (pwD=(pwf/zw)/ (pi/zi) = 0.1), -3 3 with k1/k2 varying from 1x10 to 1x10 .
Total Cumulative Gas Production versus time: (pwD = 0.1 case) 12
Fig. 2 - Total cumulative gas production versus production time for a commingled gas well produced at a constant wellbore pressure -3 3 (pwD=(pwf/zw)/ (pi/zi) = 0.1), with k1/k2 varying from 1x10 to 1x10 .
Dimensionless p/z-difference function versus Dimensionless Total Cumulative Gas Production: (pwD = 0.1 to 0.5)
Fig. 3 - p/z-function versus total cumulative gas production function for a commingled gas well produced at a constant wellbore pressure -3 3 (pwD=(pwf/zw)/ (pi/zi) = 0.1-0.5), with k1/k2 varying from 1x10 to 1x10 .
Dimensionless Decline Rate for an Individual Layer: (qDdj) 1 q ( p 0) Ddj 2 wD ...... (7) (1 0.5 tDdj ) 2 4 pwD exp(- pwD tDdj ) q ( p 0) Ddj 2 wD ...... (8) (1 pwD ) (1 pwD ) exp(- pwD tDdj )
Dimensionless Decline Rate for the Total System: (qDd) n 1 q ( p 0) Dd 2 wD ...... (9) j 1 (1 0.5 tDdj ) 13
n 2 4 pwD exp(- pwD tDdj ) q ( p 0) Dd 2 wD (10) j 1 (1 pwD ) (1 pwD ) exp(- pwD tDdj )
Type Curve for a Layered ( No Crossflow ) Gas Reservoir: (pwD = 0.1 to 0.5)
Fig. 4- Dimensionless decline gas rate versus dimensionless decline time for a commingled well produced at various constant wellbore pressures (pwD = (pwf/zwf)/ (pi/zi) = 0.1 to 0 3 0.5), with k1/k2 varying from 1x10 to 1x10 .
Dimensionless Cumulative Production for an Individual Layer: (GpDj) 0.5 tDdj G pDj ( pwD 0) ...... (11) (1 0.5 tDdj ) 2 (1 - pwD ) 1- exp(- pwD tDdj ) G pDj ( pwD 0) ...... (12) (1 pwD ) (1 pwD ) exp(- pwD tDdj )
Dimensionless Cumulative Production for the Total System: (GpD) n 0.5 tDdj G ( p 0) ...... (13) pD (1 0.5 t ) wD j 1 Ddj 14
n 2 (1 - pwD ) 1- exp(- pwD tDdj ) G pD ( pwD 0) ..(14) j 1 (1 pwD ) (1 pwD ) exp(- pwD tDdj )
Dimensionless Cumulative Production Versus Dimensionless Time: (pwD = 0.1)
Fig. 5 - Dimensionless cumulative gas production versus dimensionless decline time for a commingled well produced at various constant wellbore pressures (pwD = (pwf/zwf)/ 0 3 (pi/zi) = 0.1), with k1/k2 varying from 1x10 to 1x10 . For Eqs. 1-3, and Eqs. 7-14, the index and layer counters are given by: j = layer index n = number of layers
Definitions of Dimensionless Variables: In addition to the dimensionless rate and pressure solutions, the following dimen- sionless definitions are used:
0.00634 k j t 1 t Ddj μ c r 2 2 j ij ti wa 1 re re 1 ...... (15) - 1 ln - 2 r r 2 wa wa 15
0.00634 k t 1 t Dd c r 2 2 i ti wa 1 re re 1 ...... (16) - 1 ln - 2 r r 2 wa wa n k j h j j 1 k ...... (17) n h j j 1 n j h j j 1 ...... (18) n h j j 1
( pwf /zwf ) pwD ...... (19) ( pi /zi ) ( p/z) pD ...... (20) ( pi /zi )
Pseudosteady-State Flow Equation:
2 2 qgj C j ( p/z) - ( pwf /zwf ) ...... (21)
where:
k j h j z C j pc re 3 t p ...... (22) 1.4232 T ln - ref rwa 4
Rate and Cumulative Production Relations:
— Total gas production rate: (qg,t) n qg,t qgj ...... (23) j 1
— Cumulative gas production for each layer: (Gp,n)
G p,n G p,n-1 (qgj Δ t) ...... (24)
— Cumulative gas production for the total (multilayer) system (Gp,t) 16
n G p,t G pj ...... (25) j 1
2 2 (p/z) – (pwf/zwf) Versus Gas Production Rate : (pwD = 0.1)
2 2 Fig. 6 - (p/z) - (pwf/zwf) versus gas production rate for a commingled well produced at various constant wellbore pressures (pwD = (pwf/zwf)/(pi/zi) = 0.1), with k1/k2 varying from 1x100 to 1x103.
2. Construction of generalized plots/type curves for layered reservoirs using Eqs. 1–14. The following plots are included for reference: (variations and extensions of these plots will be provided in the dissertation)
Fig. 1: p/z versus Total Cumulative Gas Production (pwD = 0.1 case) Fig. 2: Total Cumulative Gas Production versus Production Time (pwD = 0.1 case) Fig. 3: Dimensionless p/z difference function versus Dimensionless Total
Cumulative Gas Production function (pwD = 0.1 to 0.5)
Fig. 4: Type Curve — Layered (No Crossflow) Gas Reservoir (pwD = 0.1 to 0.5) Fig. 5: Dimensionless Cumulative Production — Dimensionless Time (pwD = 0.1) 17
2 2 Fig. 6: (p/z) – (pwf/zwf) Versus Gas Production Rate (pwD = 0.1) 3. Application of the analytical liquid flow solutions and the quasi-analytical gas flow solutions as mechanisms for characterizing the performance of layered gas reser-voirs —in particular, cases of commingled production (no crossflow in the reservoir).
The properties to be estimated include: Decline Type Curve Analysis: The total original gas-in-place (G). The permeability ratio (2-layer case). The total flow capacity (kh product).
Specialized Analysis: (p/z versus Gp multilayer analysis plots) The total original gas-in-place (G). The permeability ratio (2-layer case). The moveable (or recoverable) reserves in each layer.
Estimated Ultimate Recovery (EUR) Analysis: (qg versus Gp plots)
The moveable (or recoverable) reserves in each layer (EURj).
4. To investigate the sensitivity of individual layer properties on the depletion perfor- mance of layered reservoirs. The reservoir properties to be investigated include:
The permeability ratio (2-layer case), Skin factors for individual layers, Reservoir layer volumes, and The effect of drawdown (i.e., the magnitude of the wellbore flowing pressure). This investigation will include the use of synthetic data (derived using numerical simulation), as well as a variety of field data.
5. Analysis, interpretation, and correlation of gas well performance data taken from the Hugoton Field, Kansas, USA. The following data are required. Production rates and wellhead pressures (on a monthly basis). Shut-in bottomhole pressures. Fluid properties. Reservoir layer properties (porosity, thickness, etc.). A complete database of "parent" and "infill" wells. 18
We will use these data to create a systematic analysis and interpretation of the reservoir, and in particular, we will attempt to characterize the "dominant" and "subordinate" layer systems. Specifically, we will perform the following subtasks:
a. p/z versus Gp analysis for each well where a sufficient data are available. — Standard p/z versus Gp analysis (straight-line approach). — Multilayer p/z versus Gp analysis, use of the quasi-analytical solution to analysis the total p/z versus Gp performance. Regression and (possibly) type curve methods will be used.
b. New rate decline type curve analysis approach for the analysis of rate-time data from layered (no crossflow) gas reservoir systems. This will require new type curves specifically constructed for multilayered reservoir systems. c. Direct extrapolation technique for the estimated ultimate recovery (EUR). d. Correlation of results using areal mapping of the results.
6. Apply these results of this study for the purpose of reservoir managementsuch as infill drilling, well stimulation, optimizing production practices, future reservoir performance etc. 19
Organization of the Research The outline of the proposed research dissertation is as follows: 1. Chapter I Introduction 1.1 Background of Analysis Layered Gas Reservoir 1.2 Research objectives 1.3 Summary of results — layered gas reservoirs 1.3.1 p/z analysis 1.3.2 Decline type curve analysis 1.3.3 EUR analysis 1.4 Organization of the dissertation 2. Chapter II Literature Review 2.1 Analysis of gas reservoir performance using material balance 2.2 Decline type curve analysis (single-layer reservoir) 2.3 Depletion performance of a layered gas reservoir 3. Chapter III Development of Semi-Analytical Solutions for the Analysis of Reser- voir Performance Data from a Layered Gas Reservoir 3.1 Summary 3.2 Description of the reservoir model 3.3 Development of solutions for a multilayer gas reservoir 3.3.1 Gas productivity/stabilized flow equation (rigorous model) 3.3.2 (p/z)2 gas flow equation (proposed model) 3.3.3 p/z analysis for layered gas reservoir 3.3.4 Dimensionless variables/parameters for a new rate decline type curve for layered gas reservoir systems 3.4 Discussion 4. Chapter IV Development of Decline Type Curve Analysis Techniques for Layered Gas Reservoir Systems 4.1 Decline Type Curve Analysis 4.1.1 Plotting functions for decline type curve analysis (general approach) 4.1.2 Decline type curve model for a layered gas reservoir 4.1.3 Orientation/methodology for decline type analysis techniques applied to layered gas reservoir systems 4.2 Illustrative applications of the new rate decline type curve for layered gas reser- voir systems 4.3 Discussion 20
5. Chapter V Validation and Application of the New Production Performance Type Curve for a Layered Gas Reservoir 5.1 Summary 5.2 Numerical Simulation Case 5.2.1 p/z Performance Analysis 5.2.2 Decline type curve analysis 5.2.3 EUR analysis 5.3 Application — Field Case (Hugoton Gas Field, Kansas, USA) 5.3.1 p/z Performance Analysis 5.3.2 Decline type curve analysis 5.3.3 EUR analysis 5.3 Discussion 6. Chapter VI Summary and Conclusions 6.1 Summary 6.2 Conclusions 6.3 Recommendations for Future Work 7. Nomenclature 8. References 9. Appendices: Appendix A Derivation of the Gas Well Productivity Index and Derivation of the Gas Material Balance Equation for Layered Reservoirs. Appendix B Derivation of an Approximate Relation for Gas Flow Behavior at a Constant Bottomhole Pressure (Pseudosteady-State Flow Conditions). Appendix C Derivation of the Dimensionless Solutions for Production Rate and Average Reservoir Pressure Behavior in a Multilayer Gas Reservoir. Appendix D Summary of Field Case Analyses — Hugoton Gas Field, Kan- sas, USA 21
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