production prediction of hydraulically fractured reservoirs
TRANSCRIPT
doi.org/10.26434/chemrxiv.8206214.v1
Production Prediction of Hydraulically Fractured Reservoirs Based onMaterial BalancesRaul Velasco, Palash Panja, Milind Deo
Submitted date: 30/05/2019 β’ Posted date: 31/05/2019Licence: CC BY-NC-ND 4.0Citation information: Velasco, Raul; Panja, Palash; Deo, Milind (2019): Production Prediction of HydraulicallyFractured Reservoirs Based on Material Balances. ChemRxiv. Preprint.
Transient flow is dominant during most of the productive life of unconventional wells in ultra-low permeabilitiesresources such as shales. As a result, traditional reservoir performance analysis such as the conventionalmaterial balance have been rendered inapplicable. A new novel semi-analytical production predictive toolbased on application of material balance on a transient linear flow system is developed.
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Production Prediction of Hydraulically Fractured Reservoirs Based on Material Balances Raul Velascoa, Palash Panja a,b, *, Milind Deo b a Energy & Geoscience Institute, b Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, the United States * Corresponding author [email protected]
Abstract Transient flow is dominant during most of the productive life of unconventional wells in ultra-low permeabilities resources such as shales. As a result, traditional reservoir performance analysis such as the conventional material balance have been rendered inapplicable. A new novel semi-analytical production predictive tool based on application of material balance on a transient linear flow system is developed. This method considers two important regions during transient production of oil reservoirs: the saturated region where gas evolves and flows with oil, and the undersaturated region where only oil flows. These regions evolve in extent as pressures diffuse from the fractures into the reservoir while production takes place. First, the proposed semi-analytical method and concept are introduced and validated using numerical simulations. After verification is done, the method is applied to three field cases including black and volatile oil examples where production prediction is highlighted. We observed that application of this method on tight reservoirs around the U.S. shows very good forecasting capabilities. Input requirements for this multiphase method are PVT data, relative permeability ratios, and average pressure estimations. Unlike conventional predictive tools, this method is based on material balance application to several evolving flow zones during production and can accommodate for variable bottom-hole pressures. We present a method simple enough for quick application on a spreadsheet, but comprehensive enough to capture important multiphase reservoir physics not present in most traditional production analysis. There are several applications of this tool, ranging from production performance evaluation to prediction of hydrocarbon production.
Introduction One of the main objectives regarding reservoir engineering is to analyze production performance such as fluid rates, cumulative production, pressure declines, and to determine hydrocarbon estimated ultimate recovery (EUR). Numerous methods have been developed and modified since the early studies of reservoir behavior (Millikan,1926, Coleman et al.,1930). These studies can be divided into few broad categories such as decline curve and type curve analysis, semi-analytical methods, surrogate models, material balance methods and numerical simulation. The decline curve analysis method has been used widely due to its quick predictive capabilities based on available production data. DCA has a long history after Arps (Arps,1945) developed the first mathematical model where a curve is fitted on a plot of flow rate vs time. The fitted curve then is used to forecast fluid rates as well as cumulative production and eventually used to determine EUR. Many new and modified versions of Arpβs method (Mead,1956, Mannon,1965, Fetkovich,1980) were developed consequently. Most decline curve analyses are suitable for single phase flow in conventional reservoir with high permeabilities. However, Arpβs original model has
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been unsuccessful for analysis of production from unconventional and tight permeability reservoirs (Duong,2010). Arp's empirical formula is constructed on the assumption that the reservoir has a boundary dominated flow. Unconventional reservoirs have predominantly linear flow with long transient flow regimes due to the very low matrix permeability and presence of induced fractures. These resulted in value of decline exponent (b) greater than one which overestimated production. As alternatives, a series of techniques that embody the advantages of the DCA have been developed in recent years to address unconventional reservoirs. A few examples include the Power Law Exponential (PLE) (Ilk et al.,2008), Stretched-Exponential model(SE) (Valko,2009), Logistic Growth Analysis (LGA) (Clark et al.,2011)and Duong Method (Duong,2010). Rate transient analysis (Palacio and Blasingame,1993) using type curves was successfully applied in analysis of gas production data from tight formations. However, multiphase flow of oil and gas cannot be captured using this type curves. Concept of transient productivity index (PI) (Medeiros et al.,2010) was first introduced to analysis production data from fractured and unfractured horizontal wells. The better estimations are obtained with longer data which includes multiple flow regime. One of the disadvantages of these techniques, however, is the sensitivity to variation in production data due to changes in flowing bottom-hole pressures. Calculations of EUR also vary with the selection of fitted curves, a problem that only be alleviated by fitting more production data. Various methods such as Arp's, PLE, SE, RTA and transient PI method were compared in estimating oil-in-place in fractured tight formation (Kabir et al.,2011). Dynamic material balance was developed in estimating original-gas-in-place when production rate is variable (Mattar and Anderson,2005). This method is only applicable during boundary-dominated flow in gas reservoir (Ismadi et al.,2011). It fails to predict original-hydrocarbon-in-place for multi-phase (oil and gas) flow at transient state conditions. Therefore, the dynamic material balance method is not an efficient tool for unconventional reservoirs.
Analytical methods based on continuum fluid mechanics have been developed for several geometries and production conditions. Researchers (Levine and Prats,1961, Miller,1962, Fetkovich,1980, El-Banbi and Wattenbarger,1998, Wattenbarger et al.,1998, Clarkson and Beierle,2010) have developed expressions derived from the conservation of mass and Darcyβs law to tie the physics of fluid flow and surface production. Analysis using these expressions is made universal by use of dimensionless groups that combine production, completion, operational, petrophysical, and fluid parameters. The equations provided by governing partial differential equations give us great insight into the physics of fluid. In spite of the single-phase assumption in these equations, conventional and unconventional production analysis has been successful as long as the well has a stable production life. This approach, however, is not applicable for complex and heterogeneous reservoirs.
State-of-the-art numerical simulations is the to-go approach for highly complex multi-phase systems. In many cases, simulation has become the ultimate reservoir performance tool. The drawback, however, is the extensive data required for a simulation model to be meaningful. Required input data for unconventional wells include seismic data interpretation, petrophysical properties, fluid data, relative permeabilities from cores, etc. Any uncertainty in the input data can lead to misinterpretation of the simulation results. As a consequence, reservoir simulations tend to be expensive and time consuming for extensive projects.
Surrogate based models (Amorim and Schiozer,2012, Dahaghi et al.,2012) such as polynomial response surface (Parikh,2003, Li and Friedmann,2005, Carreras et al.,2006, Panja and Deo,2016) and artificial intelligence method such as least square support vector machine (E. El-Sebakhy,2007, Fatai Adesina Anifowose,2011, Panja et al.,2016) and artificial neuron network are
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gaining interest due to high computational efficiency. In surrogate models, important parameters (Panja et al.,2016) are selected based on their hierarchy to affect the production. The main disadvantage of this approach is that the models are developed based on simplified reservoir geometries.
The material balance method is a very useful tool to estimate initially hydrocarbon in place, forecast reservoir performance and to calculate individual contributions from different drive mechanisms. The general material balance equations which are based on hydrocarbon volume balance are initially presented by Schilthuis (Schilthuis,1936). Material balance comprises three variables namely average reservoir pressure, recovery factor and cumulative GOR. Other variables such as compressibilities and fluid PVT properties are function of average pressure.
For prediction of production performance, many smart approaches (Tarner,1944, Muskat,1945, Tracy,1955) were tried in order to solve the material balance equation progressively by small pressure steps in solution-gas-drive reservoir. These methods are applicable to well depleted saturated reservoirs.
Another use of material balance equation is to calculate initial hydrocarbon in place by using field production data. For this approach, the equation is rearranged in the form of straight line. The straight line form was first proposed by Havlena and Odeh (Havlena and Odeh,1963) by re-arranging the material balance equations. The linear form is convenient for various calculations such as initial oil in place, gas cap size, water influx and other drive mechanisms.
The material balance method is advantageous for requiring only production data and fluid PVT properties. The existence of kerogen in shale plays that might alter the PVT properties(Pathak et al.,2015) are not considered in this study. Fluid properties are determined at average reservoir pressures i.e., a uniform pressure throughout the entire reservoir. Application of material balance using average reservoir pressure is suitable for high permeability conventional reservoirs. The assumption of average reservoir pressure leads to erroneous predictions if the variation of pressure is significant across reservoir. This is the case for transient flow in ultra-low permeability (10 nD to 1000 nD) reservoirs such as shales.
Unconventional reservoirs such as shales, tight formation etc. behave differently from conventional reservoirs (Orangi et al. , Panja et al.,2016, Panja and Deo,2016). Long transient state prevails in ultra-low permeable reservoirs where the total average pressure does not go below bubble point for long time. Conventional material balance fails to capture unconventional production behavior such as produced GOR, liquid flowrates etc. Modification of conventional material balance is essential to address this problem. In this paper, we approach transient flow by dividing the reservoir into meaningful regions; a similar concept was introduced before for condensates (Fevang and Whitson,1996). The difference between conventional material balance and the proposed method is depicted in Figure 1.
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Figure 1 β Pressure distribution for a transient flow system
The detailed description of different regions is provided in the next section. A simple spreadsheet for rapid and simple calculations was developed for ultra-low
permeability, hydraulically fractured reservoirs during transient and boundary dominated linear flow. The results from the model are compared with fine grid reservoir simulations using IMEX (Computer Modeling Group, Calgary, Canada).
In the following sections, we introduce the methodβs core concept, provide simulation verification, and present application to three field cases. Since this method is based on material balances, the same assumptions apply.
Methodology We begin by defining the system and determining the important components of multi-phase flow behavior in porous media. The system may be of any regular geometry, but for the purpose of application to shale plays we define a horizontal well with a number of vertical and equally spaced fractures draining from a constant thickness pay zone as shown by Figure 2.
For practical purposes, most analysis techniques assume that the fractures along the horizontal well behave identically. By taking advantage of symmetry, the entire length of the horizontal well is condensed into one representative fracture draining from one of its sides as shown by the red volume in Figure 2.
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Figure 2 β Idealized multi-stage horizontal well model
Reservoir Segmentation Depending on the fluids originally present in the reservoir, initial reservoir pressure and flowing bottom-hole pressures, a number of different multi-phase configurations evolve as fluids drain into the fracture. As a working example, we consider an initially undersaturated reservoir producing at a bottom-hole pressure lower than the bubble point pressure. As a consequence, transient flow develops two important regions: the undersaturated region and the saturated region.
Both regions are a consequence of the pressure behavior inside the system as described in Figure 3 (a). It should be noted, however, that even though gas evolves in the saturated portion of the system, not all of the gas flows. Gas critical saturations determine the point at which gas starts flowing with the oil as described in Figure 3 b).
(a)
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(b)
Figure 3 β System segmentation for an initially undersaturated reservoir during transient flow: (a) Typical pressure profile describing evolution of gas near the fracture (b) Typical gas saturation profile describing immobile and mobile gas portions inside the saturated
region Figures 3(a) and (b) show a key parameter: the bubble point position,π₯π₯ππ, defined as the distance
from the fracture at which the pressure is at the bubble point. During transient flow, the bubble point position (xb) travels away from the fracture, effectively increasing the size of the saturated region, and decreasing the undersaturated portion of the reservoir. Undersaturated Region (π·π·ππ β€ π·π·οΏ½πΌπΌ β€ π·π·ππ) This region is characterized by the flow of oil and water without gas saturation. The inner boundary of this zone is located at bubble point position, π₯π₯ππ, and outer boundary is equal to half fracture spacing, πΏπΏππ. By defining a pressure gradient for a slightly compressible fluid, the average undersaturated pressure, πππ’π’οΏ½ , is calculated as shown below.
πππποΏ½οΏ½οΏ½ =1
πΏπΏππ β π₯π₯πποΏ½ ππ πππ₯π₯
πΏπΏππ
ππππ
(1)
where 0 β€ π₯π₯ππ β€ πΏπΏππ and π₯π₯ππis a function of time. As the bubble point position progresses, the undersaturated region shrinks while providing the
saturated region a larger hydrocarbon production potential. We define the hydrocarbon contribution from the undersaturated region to the saturated portion as the bubble point position travels away from the fracture. To calculate this contribution, we must first determine the βexcessβ volume of fluid in the undersaturated region as oil expands and the regionβs volume shrinks. The excess volume in the undersaturated region due to oil expansion and region shrinkage is evaluated as:
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πΈπΈπ’π’ =2π₯π₯ππβπ΅π΅ππ(πποΏ½ππ)
5.615οΏ½ππππ(πποΏ½ππβ² )ππ(πποΏ½ππβ² )
π΅π΅ππ(πποΏ½ππβ²)οΏ½πΏπΏππ β π₯π₯ππβ² οΏ½ β
ππππ(πποΏ½ππ)ππ(πποΏ½ππ)π΅π΅ππ(πποΏ½ππ)
οΏ½πΏπΏππ β π₯π₯πποΏ½οΏ½ (2)
where the apostrophe denotes values evaluated at the previous step and porosity (if not
constant) can be determined with: ππ(ππ) = ππππ οΏ½1 + πππποΏ½ππ β πππππππποΏ½οΏ½ for a linear elasticity compaction model or a pore volume lookup tables for a non-linear elasticity model. The detailed derivation of equation 2 is provided in Appendix B.
Due to material balance, the undersaturated excess volume, Eu, can no longer be contained in the undersaturated region and accounts for the contribution from the undersaturated region to the saturated region. The oil and gas contribution from the undersaturated to the saturated region are determined as:
πΆπΆπ’π’βπ π ,ππ = πΈπΈπ’π’ (3.a)
πΆπΆπ’π’βπ π ,ππ = πΈπΈπ’π’π΅π΅ππ(πποΏ½ππ)π΅π΅ππ(πποΏ½ππ)
[π π π π ππ β π π π π (πποΏ½ππ)] (3.b)
Once π₯π₯ππ = πΏπΏππ, the undersaturated region disappears terminating the oil contribution to the
saturated region as shown by equation (2).
Saturated Region (π·π·ππππ β€ π·π·οΏ½πΊπΊ β€ π·π·ππ) In this region, gas evolution occurs as pressures develop below the bubble-point. The pressure at the fracture boundary is the flowing bottom-hole pressure and the pressure at the undersaturated-saturated interface (π₯π₯ππ) is the bubble point pressure. The average pressure in this region is calculated as:
πππποΏ½ =1π₯π₯πποΏ½ ππ πππ₯π₯
π₯π₯ππ
0
(4)
In this case, as the bubble point position progresses, the saturated region grows by tapping into
the undersaturated oil potential. The oil contribution from the undersaturated region has been determined earlier. As the average pressure in this region changes, so do the oil and gas volumes at reservoir conditions. The oil and gas volumes at reservoir conditions are calculated as follows:
πππ π ,ππ = πππ π ,ππβ² π΅π΅ππ(πποΏ½ππ)π΅π΅ππ(πποΏ½ππβ²)
(5.a)
πππ π ,ππ = πππ π ,ππβ² π΅π΅ππ(πποΏ½ππ)π΅π΅ππ(πποΏ½ππβ²)
+ πππ π ,ππβ² π΅π΅ππ(πποΏ½ππ)π΅π΅ππ(πποΏ½ππβ²)
[π π π π (πποΏ½ππβ²) β π π π π (πποΏ½ππ)] (5.b)
Once new saturated volumes are determined, the saturated region total volumes are calculated at the new step as:
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πππ π ,ππ = πππ π ,ππ + πΆπΆπ’π’βπ π ,ππ (6.a) πππ π ,ππ = πππ π ,ππ + πΆπΆπ’π’βπ π ,ππ (6.b)
Similarly to the undersaturated excess volume, the saturated excess volume is the difference
between the saturated volume expansion and the size of the region.
πΈπΈππ = πππ π ,ππ + πππ π ,ππ β2π₯π₯ππβπ₯π₯ππππ(πποΏ½ππ)
5.615 (7)
Similar to the undersaturated region procedure, the excess fluid in the saturated region is
contribution from the region to the surface, i.e. production. Because the excess fluid is calculated as a combination of oil and gas barrels at reservoir conditions, we need a function that determines the portion of the excess fluid that flows as oil and the portion that flows as gas. Fractional flow has been used widely for CO2 and waterflooding analyses (Willhite,1986). We introduce the fractional flow concept for this case as:
ππππ =
1
1 +ππππππ(π π ππ)ππππ(πποΏ½ππ)ππππππ(π π ππ)ππππ(πποΏ½ππ)
(8.a)
ππππ = 1 β ππππ (8.b)
where gas saturation is simply calculated as: π π ππ = πππ π ,ππ
πππ π ,ππ+πππ π ,ππ
Note that besides oil and gas, water flow can be easily implemented by addition of πππ€π€. Surface production is readily calculated as shown below:
ππππ = πΆπΆπ π βπ π π’π’πππππ π π π ππ,ππ = πππππΈπΈππ
π΅π΅ππ(πποΏ½ππ) (9.a)
ππππ = πΆπΆπ π βπ π π’π’πππππ π π π ππ,ππ = ππππ
πΈπΈπππ΅π΅ππ(πποΏ½ππ)
+ πππππ π π π (πποΏ½ππ) (9.b)
Equations (1) to (9) close the system of equations and solve for cumulative oil and gas
production by advancing π₯π₯ππ β πΏπΏππ. When the reservoir becomes fully saturated, production continues by declining the average pressure in the saturated region. The total reservoir average pressure can be expressed by combination of equations (1) and (4):
πππ π οΏ½οΏ½οΏ½ =1πΏπΏπποΏ½οΏ½ ππ πππ₯π₯
ππππ
0
+ οΏ½ ππ πππ₯π₯
Lf
Xb
οΏ½ =π₯π₯ππ πππποΏ½ + (πΏπΏππ β π₯π₯ππ) πππποΏ½οΏ½οΏ½
πΏπΏππ (10)
Saturated and undersaturated average pressures depend on the particular case study. Estimates
can be found in the Appendix.
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Region Integration Once all regions and PVT properties have been defined within a system, all contributive volumes from all regions have to be accounted for up until the producing region. In cases where we encounter high gas oil ratios at high bubble point pressures, gas buildup is substantial, especially once the reservoir is fully saturated. In such cases, it is encouraged that the saturated region be split into a number of sequential saturated portions. The volume excess and volume contribution equations presented earlier still apply simply by introducing a new travelling variable similar to π₯π₯ππ. In an ideal situation, we wish to introduce variable π₯π₯π π πππ π defined as the distance where gas saturation reaches its critical point. However, π₯π₯π π πππ π introduces more unknowns to the system and has been found to be of little effect in our experience. Similarly, Whitson and Fevang (Fevang and Whitson,1996) concluded that critical condensate saturations had no effect on condensate gas deliverability.
For practical purposes, we choose to divide the saturation region into parts that grow equally as π₯π₯ππ grows. An example of a saturated region partition is shown Figure 4.
Figure 4 β Initially undersaturated reservoir segmentation with two saturated portions
evaluated at a π«π«ππππ
If the saturated region is divided into equal portions, each portionβs volume changes and contributions to other portions are calculated as:
ππππ,ππ = ππππ,ππβ²π΅π΅ππ(πποΏ½ππ)π΅π΅ππ(πποΏ½ππβ²)
(11.a)
ππππ,ππ = ππππ,ππβ²π΅π΅ππ(πποΏ½ππ)π΅π΅ππ(πποΏ½ππβ²)
+ πππ π ,ππβ² π΅π΅ππ(πποΏ½ππ)π΅π΅ππ(πποΏ½ππβ²)
[π π π π (πποΏ½ππβ²) β π π π π (πποΏ½ππ)] (11.b)
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πΈπΈππ = ππππ,ππ + ππππ,ππ β
2π₯π₯ππβ οΏ½π₯π₯ππππ οΏ½ππ(πποΏ½ππ)
5.615
(12)
πΆπΆππβ(ππ+1),ππ = ππππ,πππΈπΈπππ΅π΅ππ(πποΏ½ππ+1)π΅π΅ππ(πποΏ½ππ)
(13.a)
πΆπΆππβ(ππ+1),ππ = ππππ,πππΈπΈπππ΅π΅ππ(πποΏ½ππ+1)π΅π΅ππ(πποΏ½ππ)
+ πΆπΆππβ(ππ+1),πππ΅π΅ππ(πποΏ½ππ+1)π΅π΅ππ(πποΏ½ππ+1)
[π π π π (πποΏ½ππ) β π π π π (πποΏ½ππ+1)] (13.b)
where ππ β (ππ + 1) denotes fluid contribution from portion ππ to portion (ππ + 1) Equations (11) through (13) are adaptations of the equations presented earlier. They are used
for each saturated portion up until the last portion where the excess fluid contribution becomes the production as shown in equation (9). Because this is a direct method, a simple spreadsheet can be created by implementing all equations into the system and allowing π₯π₯ππ to travel horizontally. In summary, spreadsheet steps for an initially undersaturated, linear flow case are shown in Figure 5.
Note that water flow can easily be implemented to the methodology by a similar fashion to oil expansion and contribution to neighboring regions. Relative permeabilities would then need to be integrated using Stone II model.
Figure 5 β Method flowchart where βnβ is the number of partitions in the saturated region
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Validation: Comparison with numerical simulation The proposed segmentation approach is compared to numerical simulation as a mean of
validation for a black oil and a volatile oil model. We consider a simple, vertical and infinite conductivity fracture model that penetrates the entire thickness of the formation as shown in Figure 2. A numerical model was built and ran using a finite difference simulator with a logarithmic cell distribution (Panja et al.,2013) around the fracture to capture steep pressure declines as shown in Figure 6.
Figure 6 βPressure distribution of a hydraulic fracture draining from the matrix at an
arbitrary time
Along the horizontal well, a significant difference in reservoir pressure is observed. Pressure along the fracture decline to the bottom-hole pressure of 1000 psi while pressure just a few feet away is at the initial conditions as shown in Figure 6. The main consideration in conventional material balance is the calculation of fluid properties at average reservoir pressure which is determined at the moment when the production data is collected. It is also assumed that the flow is boundary dominated i.e., reservoir has been depleted enough to produce from entire reservoir. To measure the reservoir pressure, well is shut in for hours or days for equilibration of pressure throughout the reservoir. On the other hand, in case of low permeability reservoir, initial pressure front reaches only few feet far from the fracture even after few years. In some cases, initial pressure front even never reaches boundary in the entire life of the well. Therefore, average pressure which represents the entire reservoir is not applicable for partially depleted reservoir such as tight formations. Under these circumstances, conventional material balance calculation cannot be applied.
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Model key parameters for both cases are shown in Table 1 and Figure 7.
Table 1 β Reservoir simulation parameters Reservoir Parameter Black
Oil Volatile Oil
Thickness, ft 200 200 Hydraulic fracture half-length, ft 500 500 Fracture spacing, ft 200 200 Initial pressure, psi 5500 6500 API gravity 36.5 45 Bubble point pressure, psi 1615 5015 Initial solution gas-oil ratio, scf/stb 745 2000 Initial oil formation volume factor, bbl/stb 1.08 1.80 Oil formation volume factor at the bubble point, bbl/stb 1.20 1.87 Initial oil viscosity, cp 0.47 0.19 Bottom-hole pressure, psi 1000 1000 Matrix porosity 0.08 0.08 Matrix permeability, md 0.0005 0.0005 Critical gas saturation, fraction 0.01 0.005 krg at connate liquid, fraction 0.87 0.74 kro at connate gas, fraction 0.95 0.74 ng 1.8 1.86 nog 2 4
Figure 7 β Gas formation volume factor and viscosity for the black oil and volatile oil cases
The entire study is focused on the two phase flow of oil and gas. Therefore, the initial water saturation is kept equal to one and water saturation is zero.
Similar to numerical simulation gridding, we must make a choice of Ξxb values. Fortunately, we can afford small Ξxb steps since the calculations are direct and no iterative solvers are necessary. Choosing a bubble point distance step size of 0.5 feet and a saturated reservoir pressure step size of 5 psi, we built a multiphase cumulative production spreadsheet with PVT functions
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and relative permeability ratios based on the steps shown in Figure 5. Note that bottom-hole pressures decline from initial pressures to their constant values in table 1.
Black Oil Case
Figure 8 shows a good agreement in terms of oil and gas cumulative production for the black oil case considering that the method does not require of permeability as an input.
(a) (b)
Figure 8 β (a) Cumulative oil and gas production and (b) cumulative gas oil ratio simulation results compared to the proposed method for a black oil case
If the number of hydraulic fractures is uncertain and their extent unknown, these values may
be assumed to be one. Essentially, simplifying reservoir dimensions to unity leads to calculation of recovery factors instead of cumulative production. The number and extent of hydraulic fractures can be estimated by iterating initial guesses until the calculated production matches historical production. This procedure can be used to estimate the original oil in place in the stimulated reservoir volume.
The proposed method predicts a maximum recovery factor of 29.5% if the reservoir average pressure declines close to the bottom-hole pressure (assuming no other production enhancement techniques are applied). This value matches the simulation recovery factor under similar conditions. The average pressures are also compared in Figure 9.
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Figure 9 β Comparison of average reservoir pressure obtained from flow simulation and proposed model in black oil case
The average pressure from the proposed model matches very well with the average pressure from commercial simulator. The match is almost perfect at higher pressure side due to bigger undersaturated region with less gas evolution.
We found that the reservoir becomes fully saturated at an average pressure of almost 1500 psi. Note that even though the saturation pressure is 1615 psi, the entire reservoir becomes fully saturated only when xb reaches the boundary of the reservoir. When this happens, there is no point in the reservoir with a pressure value above the bubble point, hence the average reservoir pressure must be less than the bubble point pressure. Volumetric integration of the pressure profile results in an average pressure of 1468 psi which is just below the value predicted by the proposed method.
Figure 8 (a) shows a very good agreement in terms of cumulative oil and gas production
proving that accurate results can be achieved with only two saturated regions. The reservoir becomes fully saturated at an average pressure of around 4070 psi compared to 4130 psi obtained from simulation. The excess volumes of fluid of 2 segments for saturated section (Es1 and Es2) and 1 segment for undersaturated section (Eu) are calculated using equation 12 and 2 respectively as shown in Figure 10
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Figure 10 β The changes in excess volume in two saturated segments and the single undersaturated segment in black oil case Boundary (xb) of saturated and undersaturated sections moves away from fracture with time. Consequently, the average pressure of each segment decreases and this causes the reduction of excess volumes for both section. Excess volume in undersaturated section i.e., Eu is always greater than excess volume in saturated sections (Es) because of higher average pressure in undersaturated section. At higher pressure the formation volume factor as well as the solution gas oil ratio is higher.
Accuracy can be enhanced by partitioning the saturated region many times to capture steep
saturation changes, especially for cases with high gas oil ratios. This practice, however, turns this methodology into a framework akin to a finite-difference simulator. In an effort to keep the workflow as simple as possible, we partitioned the saturated region into two equal portions.
The pressure and gas saturation profiles for the simulation model are shown in Figure 11 to draw comparison with the conceptual Figure 3 (b).
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Figure 11 β Gas saturation and pressure profiles from simulation
Volatile Oil Case
Two distinct production trends are shown in Figure 8(a) and Figure 12 (a).
(a) (b) Figure 12 β (a) Cumulative oil and gas production and (b) cumulative gas oil ratio simulation results compared to the proposed method for a volatile oil case
Figure 8 (b) and Figure 12 (b) shows a cumulative GOR plot that features an initial plateau higher than the initial solution gas-oil ratio. This behavior is seen in many wells in the Bakken and other tight oil linear flow reservoirs and discussed in the literature (Jones, 2016).
In the black oil case, when the reservoir becomes fully saturated (xb=Lf) both oil and gas
production see a steep production slope versus pressure where cumulative oil production increases by about 70% in a span of 500 psi. In the volatile oil case, production after the reservoir becomes fully saturated increases by over 100% in the span of 3000 psi. This difference in behavior is a direct result of different fluid expansion characteristics (black oil versus volatile oil) from the PVT data. The absolute maximum recovery with no production enhancement is calculated to be 22%
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which matches the output from simulation. A comparison of average pressure from proposed model and commercial simulator is given in Figure 13.
Figure 13 β Comparison of average reservoir pressure obtained from flow simulation and
proposed model in volatile oil case Similar to black oil case, the average pressure calculated using the proposed model matches very well with results from commercial simulator in case of volatile oil.
The excess volumes of 2 saturated segments (Es1 and Es2) and 1 undersaturated segment (Eu) for volatile oil case are shown in Figure 14
Figure 14 β The changes in excess volume in two saturated segments and the single
undersaturated segment in volatile oil case
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The decrease in excess volume is less in undersaturated segment in volatile oil case because of
the fact that the higher amount of dissolved gas is evolved which sustains the pressure of the section.
Field Applications We begin with the analysis of an Eagle Ford shale horizontal well that produced for over a year (Well A). PVT data as well as other parameters are shown in table 2. Bottomhole pressure data as well as production as shown in Figure 15.
(a) (b)
Figure 15-Data for Well A (a) bottom-hole pressure and (b) fluid production.
Table 2 β Well A parameters Parameters Values Initial Pressure, psi 5500 Bubble Point Pressure, psi 2800 Initial solution GOR, scf/stb 500 Initial Boi, rb/stb 1.4 Number of hydraulic fractures 60 Estimated fracture half-length, ft
~400
Formation thickness, ft 180-200 Average porosity, fraction 0.05
Similarly, to the simulation examples, a spreadsheet was set up with a bubble point distance
step size of 0.5 feet and a saturated reservoir pressure step size of 5 psi. Relative permeability ratios were obtained from in table 1 where critical gas saturation is 0.05 for this case. Production data as a function of average reservoir pressure is not available for this well. To overcome this obstacle, we plotted cumulative oil versus cumulative gas and compared that to the proposed methodβs results as shown in Figure 16 (a). Cumulative oil and cumulative gas are calculated from proposed model using the PVT data, initial conditions, operational parameters, and completion
19
parameters for the field. Predicted cumulative fluid production versus average pressure are shown in Figure 16 (b).
(a) (b)
Figure 16-Well A (a) predicted cumulative production using different fracture half-lengths and (b) fluid cumulative production versus average reservoir pressure.
As shown in Figure 16 (a), we plotted cumulative fluid production for different hydraulic fracture sizes (half-lengths of 200, 400, and 500 ft). Assuming that the reported fracture half-lengths of 400 ft are representative of the system, we can calculate OOIP volumetrically for the stimulated portion of the reservoir. As shown in Figure 16(a), the predicted production for a fracture half-length of 200 feet does not match the historical production. Based on this the minimum average half-length fracture is 200 ft, and the SRV OOIP is at least 2.4 MMSTB. In this fashion, uncertain hydraulic fractures can be determined by plotting cumulative data versus the results from this method. Only future production will decidedly corroborate the reported 400 ft half-lengths and the actual OOIP. Using the methodology, we are also able to predict a 10.3% maximum oil recovery regardless of fracture dimensions. This is due to high bottom-hole pressures and can be improved by the addition of gas lift.
We now apply the methodology to two prolific shale/tight formation plays, namely Bakken in North Dakota /Montana and Eagle Ford in Texas where information is scarce. Field parameters are given in Table 3.
Table 3 β Key field parameters Parameters Bakken
(Well B) Eagle Ford
(Well C) Initial Pressure, psi 4600 7000 Flowing Bottom-Hole Pressure, psi ~970 ~1000 Bubble Point Pressure, psi 2800 3644 Initial solution GOR, SCF/STB 700 1577 Initial Boi, rb/STB 1.44 1.70
As shown in Figure 17, production data from both wells match the predicted data using proposed the model. This shows good applicability of the method to multi-phase flow in field cases. Calculation of original oil in place (OOIP) is the direct implication of this results if the geometry of the reservoir is unknown. If field data is available along with initial reservoir conditions, a unit-size system can be created using the proposed method. The unit-size results
20
would then be multiplied by a guess of OOIP until a match like the ones shown in Figure 16 (a) and 17 is achieved.
(a) (b)
Figure 17-Comparison of field data with the proposed method for a (a) Bakken well and an (b) Eagle Ford well.
Conclusions We have introduced a quick and direct method based on material balances with the aim to analyze multiphase reservoir performance during transient and boundary dominated flow. The method makes use of a traveling variable that divides the reservoir into regions depending on the developing phases. This technique was verified by comparing oil and gas production using several numerical simulation models, two of which were presented here for black and volatile oils.
Field applications have been explored and performance analysis suggested even when data is scarce. One clear outcome from the use of this method is the estimation of OOIP which entails knowledge of fracture surface area for horizontal wells. OOIP is based upon the stimulated reservoir volume (SRV) defined by the dimensions of hydraulic fractures. The method relies on the availability of PVT data, relative permeability ratios, and assumptions regarding average pressures. Application of this method is not limited to geometry and technically not limited to type of fluids. Condensates, for instance, may be coupled into the method by introduction of Rv data. The proposed methodology does not require absolute permeabilities as input. The price we pay is that the current framework does not account for time.
Nomenclature Symbol Description Units
β Apostrophe denoting parameter is evaluated at the previous step - Ξ± Diffusivity coefficient ft2/sec Ο Porosity - Ξ³ Constant in equation A-1 1/ft πποΏ½ Average pressure psi πποΏ½ππ1 Average pressure of first saturation part psi
21
πποΏ½ππ1 Average pressure of second saturation part psi πποΏ½ππ2 Average pressure of second saturation part psi πππ π οΏ½οΏ½οΏ½ Average reservoir pressure psi πππποΏ½ Average pressure in saturation region psi πποΏ½ππβ² Average pressure in saturation region at previous step psi πππποΏ½οΏ½οΏ½ Average pressure of the undersaturated region psi πποΏ½ππβ² Average pressure of the undersaturated region at previous step psi
πΆπΆππβ(ππ+1),ππ Volume of gas contributed by ith portion to (i+1)th portion in saturated region rb
πΆπΆππβ(ππ+1),ππ Volume of oil contributed by ith portion to (i+1)th portion in saturated region rb
πΆπΆππβπ π π’π’ππππ,ππ Volume of gas from nth portion or last portion of saturated region to surface scf
πΆπΆππβπ π π’π’ππππ,ππ Volume of oil from nth portion or last portion of saturated region to surface stb
πΆπΆπ π βπ π π’π’ππππ,ππ Volume of oil from saturated region to surface scf πΆπΆπ π βπ π π’π’ππππ,ππ Volume of oil from saturated region to surface stb
πΆπΆπ’π’βπ π ,ππ Volume of gas contributed by undersaturated region to saturated region rb
πΆπΆπ’π’βπ π ,ππ Volume of oil contributed by undersaturated region to saturated region rb
Β΅ Viscosity cp Β΅g Gas viscosity cp Β΅o Oil viscosity cp Bg Gas formation volume factor rb/scf Bo Oil formation volume factor rb/stb c1 Constant in equation A-4 psi/ft cf Formation compressibility 1/psi ct Total compressibility 1/psi ES Excess volume in the saturated region rb Eu Excess volume in the undersaturated region rb
f(xb) Pressure gradient at the bubble point pressure psi/ft fg Fractional gas flow - fo Fractional oil flow - fw Fractional water flow - h Formation thickness ft k Absolute permeability mD
krg Gas relative permeability - kro Oil relative permeability -
22
l1, l2 Limits of integration if the saturated region is divided into smaller portions ft
Lf Half fracture spacing ft P Pressure psi Pb Bubble point pressure psi Pi Initial reservoir pressure psi
Pref Reference Pressure psi Pwf Flowing Bottom Hole pressure psi Qg Volume of gas produced at surface scf Qo Volume of oil produced at surface stb Rs Solution Gas to Oil Ratio scf/stb Rsi Initial solution gas to oil ratio scf/stb Sg Gas saturation - Sgc Critical gas saturation - So Oil saturation - Vi,g Volume of gas in ith portion of saturation region rb ππππ,ππβ² Volume of gas in ith portion of saturation region at previous step rb Vi,o Volume of oil in ith portion of saturation region rb ππππ,ππβ² Volume of oil in ith portion of saturation region at previous step rb Vs,g Volume of gas in saturation region rb πππ π ,ππβ² Volume of gas in saturation region at previous step rb
Vs,o Volume of oil in saturation region rb πππ π ,ππβ² Volume of oil in saturation region at previous step rb
xb Bubble point distance from fracture ft π₯π₯ππβ² Bubble point distance from fracture at previous step ft xf Fracture half length ft xs2 Distance of boundary of first saturation portion ft xsgc Distance where gas saturation reaches its critical point ft
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Clark, Aaron James, Larry Wayne Lake and Tadeusz Wiktor Patzek (2011). Production Forecasting with Logistic Growth Models. SPE Annual Technical Conference and Exhibition. Denver, Colorado, USA, Society of Petroleum Engineers. SPE-144790-MS. Clarkson, Christopher R. and Joshua Beierle (2010). Integration of Microseismic and Other Post-Fracture Surveillance with Production Analysis: A Tight Gas Study. SPE Unconventional Gas Conference. Pittsburgh, Pennsylvania, USA, Society of Petroleum Engineers. SPE-131786-MS. Coleman, Stewart, H. D. Wilde, Jr. and Thomas W. Moore (1930). "Quantitative Effect of Gas-oil Ratios on Decline of Average Rock Pressure." Dahaghi, Amirmasoud Kalantari, Soodabeh Esmaili and Shahab D. Mohaghegh (2012). Fast Track Analysis of Shale Numerical Models. SPE Canadian Unconventional Resources Conference. Calgary, Alberta, Canada, Society of Petroleum Engineers. SPE 162699. Duong, Anh N. (2010). An Unconventional Rate Decline Approach for Tight and Fracture-Dominated Gas Wells. Canadian Unconventional Resources and International Petroleum Conference. Calgary, Alberta, Canada, Society of Petroleum Engineers. SPE-137748-MS. E. El-Sebakhy, T. Sheltami, S. Al-Bokhitan, Y. Shaaban, I. Raharja, Y. Khaeruzzaman (2007). Support Vector Machines Framework for Predicting the PVT Properties of Crude-Oil Systems, Kingdom of Baharin, 15th SPE Middle East Oil & Gas Show and Conference. El-Banbi, Ahmed H. and Robert A. Wattenbarger (1998). Analysis of Linear Flow in Gas Well Production. SPE Gas Technology Symposium. Calgary, Alberta, Canada, Society of Petroleum Engineers. SPE-39972-MS. Fatai Adesina Anifowose, AbdlAzeem Oyafemi Ewenla, Safiriyu Ijiyemi (2011). Prediction of Oil and Gas Reservoir Properties using Support Vector Machines, Bangkok, Thailand, International Petroleum Technology Conference,. Fetkovich, M. J. (1980). "Decline Curve Analysis Using Type Curves." Journal of Petroleum Technology 32(06). Fevang, Γivind and C.H. Whitson (1996). "Modeling Gas-Condensate Well Deliverability." SPE Reservoir Engineering 11(4): 221-230. Havlena, D. and A.S. Odeh (1963). The Material Balance as an Equation of a Straight Line. Ilk, Dilhan, Jay Alan Rushing, Albert Duane Perego and Thomas Alwin Blasingame (2008). Exponential vs. Hyperbolic Decline in Tight Gas Sands: Understanding the Origin and Implications for Reserve Estimates Using Arps' Decline Curves. SPE Annual Technical Conference and Exhibition. Denver, Colorado, USA, Society of Petroleum Engineers. Ismadi, Danu, C. Shah Kabir and A. Rashid Hasan (2011). The Use of Combined Static and Dynamic Material-Balance Methods in Gas Reservoirs. SPE Asia Pacific Oil and Gas Conference and Exhibition. Jakarta, Indonesia, Society of Petroleum Engineers: 16. Kabir, Shah, Faisal Rasdi and B. Igboalisi (2011). "Analyzing Production Data From Tight Oil Wells." Journal of Canadian Petroleum Technology 50(05): 48-58. Levine, J. S. and M. Prats (1961). "The Calculated Performance of Solution-Gas-Drive Reservoirs." Society of Petroleum Engineers Journal 1(3). Li, B. and F. Friedmann (2005). Novel Multiple Resolutions Design of Experiment/Response Surface Methodology for Uncertainty Analysis of Reservoir Simulation Forecasts. SPE Reservoir Simulation Symposium. The Woodlands, Texas, 2005,. Society of Petroleum Engineers Inc. SPE 92853. Mannon, Robert W. (1965). Oil Production Forecasting By Decline Curve Analysis, Society of Petroleum Engineers.
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Mattar, L. and D. Anderson (2005). Dynamic Material Balance(Oil or Gas-In-Place Without Shut-Ins). Canadian International Petroleum Conference. Calgary, Alberta, Petroleum Society of Canada: 10. Mead, Homer N. (1956). Modifications to Decline Curve Analysis, Society of Petroleum Engineers. Medeiros, Flavio, Basak Kurtoglu, Erdal Ozkan and Hossein Kazemi (2010). "Analysis of Production Data From Hydraulically Fractured Horizontal Wells in Shale Reservoirs." SPE Reservoir Evaluation & Engineering 13(03): 559-568. Miller, G. Frank (1962). "Theory of Unsteady state Influx of Water in Linear Reservoirs." Journal of the Institute of Petroleum 48. Millikan, C. V. (1926). "Gas-oil Ratio as Related to the Decline of Oil Production, with Notes on the Effect of Controlled Pressure." Transactions of the AIME SPE-926147-G. Muskat, Morris (1945). "The Production Histories of Oil Producing GasβDrive Reservoirs." Journal of Applied Physics 16(3): 147-159. Orangi, Abdollah, Narayana Rao Nagarajan, Mehdi Matt Honarpour and Jacob J. Rosenzweig Unconventional Shale Oil and Gas-Condensate Reservoir Production, Impact of Rock, Fluid, and Hydraulic Fractures, Society of Petroleum Engineers. Palacio, J. C. and T. A. Blasingame (1993). Decline-Curve Analysis With Type Curves - Analysis of Gas Well Production Data. Low Permeability Reservoirs Symposium. Denver, Colorado, Society of Petroleum Engineers: 1. Panja, Palash, Tyler Conner and Milind Deo (2013). "Grid sensitivity studies in hydraulically fractured low permeability reservoirs." Journal of Petroleum Science and Engineering 112: 78-87. Panja, Palash, Tyler Conner and Milind Deo (2016). "Factors Controlling Production in Hydraulically Fractured Low Permeability Oil Reservoirs." International Journal of Oil, Gas and Coal Technology (In Press). Panja, Palash and Milind Deo (2016). "Factors That Control Condensate Production From Shales: Surrogate Reservoir Models and Uncertainty Analysis." SPE Reservoir Engineering and Evaluation SPE-179720-PA. Panja, Palash and Milind Deo (2016). "Unusual Behavior of Produced Gas Oil Ratio in Low Permeability Fractured Reservoirs." Journal ofPetroleumScienceandEngineering 144: 76-83. Panja, Palash, Manas Pathak, Raul Velasco and Milind Deo (2016). Least Square Support Vector Machine: An Emerging Tool for Data Analysis. SPE Low Perm Symposium. Denver, Colorado, USA, Society of Petroleum Engineers. SPE-180202-MS. Parikh, Harshal (2003). Reservoir Characterization Using Experimental Design And Response Surface Methodology Master of Science, Texas A&M University. Pathak, Manas, Milind Deo, Palash Panja and Levey Raymond (2015). The Effect of Kerogen-Hydrocarbons Interaction on the PVT Properties in Liquid Rich Shale Plays. SPE/CSUR Unconventional Resources Conference. Calgary, Alberta, Canada. SPE-175905-MS. Schilthuis, Ralph J. (1936). "Active Oil and Reservoir Energy." Tarner, J (1944). "How different size gas caps and pressure maintenance programs affect amount of recoverable oil." Oil Weekly 144. Tracy, G. W. (1955). "Simplified Form of the Material Balance Equation." Transaction of AIME 204. Valko, Peter P. (2009). Assigning value to stimulation in the Barnett Shale: a simultaneous analysis of 7000 plus production hystories and well completion records. SPE Hydraulic Fracturing Technology Conference. The Woodlands, Texas, Society of Petroleum Engineers: 19.
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Wattenbarger, Robert A., Ahmed H. El-Banbi, Mauricio E. Villegas and J. Bryan Maggard (1998). Production Analysis of Linear Flow Into Fractured Tight Gas Wells. SPE Rocky Mountain Regional/Low-Permeability Reservoirs Symposium, Society of Petroleum Engineers. Willhite, G.P. (1986). Waterflooding. Dallas, texas, USA, Society of Petroleum Engineers.
Appendix: A Average Pressure Estimations
Average pressure approximations rely heavily upon the particular case study and should be derived individually. In the present work, we used modified versions of known pressure equations for single phase linear flow as presented by Miller (Miller,1962) .These equations were initially derived from the study of heat transfer with different boundary conditions and accommodated for linear flow in infinite and bounded systems. The exact solution to the single phase, incompressible flow is calculated as:
ππ(π₯π₯, π‘π‘) β πππ€π€ππππππ β πππ€π€ππ
=4πποΏ½οΏ½
πππ₯π₯ππ(βπΎπΎ2 πΌπΌ π‘π‘) π π ππππ(πΎπΎπ₯π₯)2ππ β 1
οΏ½ β
ππ=1
(A-1)
where, πΎπΎ = (2ππβ1)
2πππΏπΏππ
, and πΌπΌ = πππππππ π π‘π‘
The exact pressure profile for the undersaturated portion of a transient system can be determined with by modifying π₯π₯ and πΏπΏππand iteratively solving for the bubble point pressure distance. The average pressure can be determined by analytical integration as shown in equation (1). Undersaturated average pressure calculations using this equation were done with a simple Newtonian solver, however, the process takes away from the proposed direct method and requires substantial additional information such as time. We found that an approximation to this equation must comply with the following conditions:
πποΏ½ππ(π₯π₯ππ = 0) = ππππ
πποΏ½πποΏ½π₯π₯ππ = πΏπΏπποΏ½ = ππππ
An approximation based on these conditions can be used for a number of cases as:
πποΏ½ππ(π₯π₯ππ) = πππ€π€ππ + (ππππ β πππ€π€ππ)οΏ½1 βπ₯π₯ππππ
πΏπΏπππποΏ½ (A-2)
where ππ = 1 for linear pressure decline cases. The linear form of equation (A-2) is a decent approximation to the analytical solution for
several cases. This a preferred way to estimate undersaturated average pressures since it requires to iterative methods.
The saturated portion of the reservoir can be estimated using a variety of ways. In our analysis, we start at the diffusivity equation for a linear system:
26
πππππππ‘π‘
= πΌπΌππ2πππππ₯π₯2
(A-3)
For a saturated portion of a linear system, we estimate the pressure profile by assuming that ππππ
ππππ
is constant for a particular value of π₯π₯ππ. This condition has been used before for pseudo-steady state flow. Letting 1
πΌπΌππππππππ
= ππ, rearranging terms and integration gives:
πππππππ₯π₯
= πππ₯π₯ + ππ1 (A-4)
At this point, pseudo-steady state dictates that ππππ
πππ₯π₯οΏ½π₯π₯=π₯π₯ππ
= 0, however, we deviate from this
condition because the undersaturated region contributes to the flow. Hence, the pressure gradient at π₯π₯ππis not zero, but a function of π₯π₯ππ. Letting ππππ
πππ₯π₯οΏ½π₯π₯=π₯π₯ππ
= ππ(π₯π₯ππ) and solving for ππ1:
πππππππ₯π₯
= πππ₯π₯ + ππ(π₯π₯ππ) β πππ₯π₯ππ (A-5)
Integration and application of bottom-hole boundary condition yields:
ππ(π₯π₯) = πππ€π€ππ + π₯π₯ππ(π₯π₯ππ) + ππ οΏ½π₯π₯π₯π₯ππ βπ₯π₯2
2οΏ½ (A-6)
where π₯π₯ππ is a constant at each step. There is one more boundary condition that must be met: ππ(π₯π₯ππ) = ππππ, this can be achieved by
solving for ππ or by scaling the pressure function. Both methods yield the same solution. Keeping in mind that the saturated region may be divided in n parts, application of scaling and equation (1) gives us an average pressure expression for a single phase incompressible fluid. The saturated, region, however contains important amounts of gas that grow during transient flow and affect production substantially once the reservoir is fully saturated. By application of square-pressures analogy:
πποΏ½ππ = οΏ½πππ€π€ππ2 +
οΏ½ππππ2 β πππ€π€ππ2οΏ½(ππ1 β ππ2)π₯π₯ππ
(ππ12 β ππ22)ππ(π₯π₯ππ) + οΏ½(ππ12 β ππ22)π₯π₯ππ β(ππ13 β ππ23)
3 οΏ½
2ππ(π₯π₯ππ) + π₯π₯ππ
(A-7)
where, ππ1 and ππ2are the limits of integration if the saturated region is divided into smaller
portions. If the saturated region consists of only one cell, then ππ1 = π₯π₯ππand ππ2 = 0. Recalling that ππ(π₯π₯ππ) is the pressure gradient at the bubble point pressure as the saturated region
grows, we invoke two conditions: πππππππ₯π₯οΏ½π₯π₯ππβ0
=ππππ β πππ€π€ππ
π₯π₯ππ
27
πππππππ₯π₯οΏ½π₯π₯ππ=πΏπΏππ
= 0
According to these conditions, ππ(π₯π₯ππ) can be approximated as:
ππ(π₯π₯ππ) =ππππ β πππ€π€ππ
π₯π₯ππβ π₯π₯ππ
οΏ½ππππ β πππ€π€πποΏ½πΏπΏππ2
(A-8)
Equation (A-2) was used for the majority of cases, whereas a few required the iterative method
for the estimation of the undersaturated average pressure. However, Equation (A-7) worked very well for all cases ranging from black oils to volatile oils.
Appendix: B Calculation of Excess Volume
Oil occupying the reservoir volume extended 'x' distance from fracture face is calculated as
ππππππ π£π£π£π£πππ£π£ππππ πππ‘π‘ πππππ π πππππ£π£π£π£ππππ πππ£π£πππππππ‘π‘πππ£π£πππ π =2π₯π₯ππ β π₯π₯ ππππ(πποΏ½)ππ(πποΏ½)
5.615 (ππππ) (B-1)
πποΏ½ denotes the average reservoir pressure in the section where the calculation is performed.
Conversion of the volume at reservoir conditions to stock tank conditions using formation volume factor is shown in equation B-2
ππππππ π£π£π£π£πππ£π£ππππ πππ‘π‘ π π π‘π‘π£π£ππππ π‘π‘ππππππ πππ£π£πππππππ‘π‘πππ£π£πππ π =2π₯π₯ππ β π₯π₯ ππππ(πποΏ½)ππ(πποΏ½)
5.615 π΅π΅ππ(πποΏ½) (π π π‘π‘ππ) (B-2)
A simplified version of Figure 4 is shown in Figure B.1 to understand the volume change in
saturated portion due to pressure depletion below bubble point in the undersaturated portion.
28
Figure B.1- Calculation of excess volume in the saturated portion
The length of saturated segment is increased from x'b to xb after a small time period. The average pressures of the saturated and undersaturated segments at current time step are PοΏ½S and PοΏ½U respectively. The average pressure of undersaturated portion at previous time step was πποΏ½ππβ² . From the figure B.1, it is evident that the volume of oil in undersaturated zone at previous time step was
Oil volume in undersaturated segment at previous time step =2π₯π₯ππβ οΏ½πΏπΏππ β π₯π₯ππβ² οΏ½ππππ(πποΏ½ππβ² )ππ(πποΏ½ππβ² )
5.615π΅π΅ππ(πποΏ½ππβ² ) (B.3)
Similarly the volume of oil in undersaturated zone at current time step is calculated as shown
in Equation B.4
Oil volume in undersaturated segment at current time step =2π₯π₯ππβ οΏ½πΏπΏππ β π₯π₯πποΏ½πππ£π£οΏ½πποΏ½πποΏ½πποΏ½πποΏ½πποΏ½
5.615 π΅π΅π£π£οΏ½πποΏ½πποΏ½ (B.4)
The change in oil volume in these two consecutive time steps is calculated as
29
Change in Oil volume in undersaturated zone
=2π₯π₯ππβ οΏ½πΏπΏππ β π₯π₯πποΏ½πππ£π£οΏ½πποΏ½πποΏ½πποΏ½πποΏ½πποΏ½
5.615 π΅π΅π£π£οΏ½πποΏ½πποΏ½β
2π₯π₯ππβ οΏ½πΏπΏππ β π₯π₯πποΏ½πππ£π£οΏ½πποΏ½πποΏ½πποΏ½πποΏ½πποΏ½5.615 π΅π΅π£π£οΏ½πποΏ½πποΏ½
=2π₯π₯ππβ5.615
β£β’β’β‘πππ£π£ οΏ½πποΏ½ππ
β² οΏ½ππ οΏ½πποΏ½ππβ² οΏ½
π΅π΅π£π£ οΏ½πποΏ½ππβ² οΏ½
οΏ½πΏπΏππ β π₯π₯ππβ² οΏ½ βπππ£π£οΏ½πποΏ½πποΏ½πποΏ½πποΏ½πποΏ½
π΅π΅π£π£οΏ½πποΏ½πποΏ½οΏ½πΏπΏππ β π₯π₯πποΏ½
β¦β₯β₯β€
(B.5) The section (as marked by light green in Figure B.1) where this change occurred has now
become the part of saturated region and the reservoir condition has also been changed. The change in oil volume calculated in equation B.5 is at stock tank conditions. This volume which is referred as excess volume (Eu) is converted back to reservoir conditions which is basically the conditions of saturated regions now. Therefore, the formation volume factor of oil at average pressure at saturated region at current time step is used in the conversion as shown in equation B.6
πΈπΈπ’π’ =2π₯π₯ππβπ΅π΅ππ(πποΏ½ππ)
5.615οΏ½ππππ(πποΏ½ππβ² )ππ(πποΏ½ππβ² )
π΅π΅ππ(πποΏ½ππβ²)οΏ½πΏπΏππ β π₯π₯ππβ² οΏ½ β
ππππ(πποΏ½ππ)ππ(πποΏ½ππ)π΅π΅ππ(πποΏ½ππ)
οΏ½πΏπΏππ β π₯π₯πποΏ½οΏ½ (B.6)
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