# Dynamic Drainage Volume (DDV)

It is essential to develop efficient, reliable, consistent and scalable techniques for predicting the well performance of unconventional oil and gas reservoirs. These methods should be able to accurately model on individual well level and leverage the benefits of both data-driven approaches, such as speed, emphasis on data, and adaptability, with those of physics-based models, including robust prediction, generalization, and interpretability.

We develop a comprehensive, hybrid reservoir modeling methodology for transient well performance (TWP). The method is based on a novel formulation that combines diffusive time of flight, succession of pseudo-steady state material balance and transient productivity index concepts for performing production diagnostics and forecasting (Molinari et al. 2019a, Molinari et al. 2019b, Molinari and Sankaran 2021). This method has been successfully applied to thousands of wells across multiple shale plays and tight reservoirs in North America.

The key concept behind TWP is that the drainage volume increases over time, but geometry is unknown. The diffusive time of flight approach to calculate drainage volume ensures no-flow outer boundary and the reservoir withdrawal is known through cumulative volume produced. Under the succession of pseudo-steady state (SPSS) assumption, material balance can be performed at any point in time and at high resolution (e.g. daily). This allows for the calculation of pressure depletion (i.e. average reservoir pressure) in the contacted drainage volume at any timestep.

Routinely available production rates along with PVT and flowing bottomhole pressure are utilized in TWP workflow. The method takes the dynamics of reservoir system into consideration, with handling the variable flow rates and bottomhole pressure, various pressure drawdown, compaction impacts, pressure-dependent PVT properties, and various production methods (e.g. natural flow, gas lift, ESP, sucker rod pumps etc.).

The diffusivity equation in heterogeneous porous media may be written as:

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The diffusive time of flight (DTOF)

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is physically associated with the peak propagation of a pressure pulse for an impulse source. The 3D diffusivity equation can be reduced to a 1D diffusivity equation (King et al. 2016, Zhang et al. 2016), if we assume that the pressure gradients are aligned with the time of flight

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(i.e.).

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Drainage volume is obtained by solving for the boundary condition of the asymptotic form of the 1D diffusivity equation. In the absence of a well and reservoir model, the drainage volume is calculated using the pressure and rate data as follows (Xue et al. 2018, Molinari et al. 2019b):

(2)

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The calculated drainage volume in equation (2) represents the contacted reservoir pore volume due to the propagation of the pressure front at any given time step in the reservoir. Analogous to the concept of investigation radius in homogeneous fields, it tracks the DTOF contour of an irregular geometry due to the draining of a lumped fracture system, stimulated and unstimulated matrix. This RNP formulation represents the production behavior that would be observed if the well were produced at a constant reference rate.In equation (2), material balance time and rate-normalized pressure are calculated as follows:

(3)

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For liquids, (4)

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(4)Though the original DTOF approach was applicable to single phase flow problem, it is important to consider the presence of multiple phases. The multiphase flow is approximated by total equivalent rate (Perrine 1956, Martin 1959), estimated by recombining all phases into an equivalent single composite phase at reservoir conditions. The total equivalent rate defined in equation (5) is used to estimate material balance time in equation (3) and pressure normalized rate in equation (4), which are applied in equation (2) to determine drainage volume.

(5)

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(6)

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In unconventional wells, it is rare to reach steady-state or pseudo-steady-state conditions (when a fixed volume has been drained and all reservoir boundaries have been contacted by a producing well) where material balance becomes strictly valid. This is because even when the well contacts all fractures, known as the stimulated rock volume, or pseudo-boundary dominated flow, the low permeability matrix is still being incrementally contacted.

We solve the problem by using an expanding control volume approach. This is analogous to an expanding balloon, where the outer boundary expands at the rate of our pressure contours. Basically, assume that transient flow (where the drainage volume is dynamic and continuously increasing) can be approximated as “snapshots” or succession of instantaneous pseudo-steady states (i.e. transient states represented as a series of consecutive pseudo-steady states for small time intervals), as shown in Figure 1. The drainage volume, already calculated in equation (2), defines the size of the “container” in each time step. By our definition of drainage volume based on DTOF, there’s no flow into the container at each timestep. However, we know how much volume is taken out of the system through production at the well. By applying material balance, we can estimate the loss of energy in the contacted drainage volume, manifested as reservoir depletion. Cumulative production is tracked daily in the field and these small timesteps also help in improving the accuracy of our material balance.

Figure 1: Material Balance applied to succession of pseudo-steady-state conditions with expanding drainage volume

For liquid systems, the general material balance expression represents the following fundamental effects:(Liquid expansion) + (Liberated gas expansion) + (Change in pore volume due to connate water/residual oil expansion and pore volume reduction) = (Underground withdrawal) (7)

Using first principles, each term can be derived to obtain a representative material balance expression for an oil-producing unconventional well in the absence of free gas cap and aquifer in the reservoir.

The liquid expansion term is defined as:

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where:

N – total oil volume originally in place

W – total water volume originally in placeoil formation volume factor at pressure of interest

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– oil formation volume factor at pressure of interest

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– oil formation volume factor at initial reservoir pressure

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– water formation volume factor at pressure of interest

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– water formation volume factor at initial reservoir pressure

The liberated gas expansion term is defined as:

(9)

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where:

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– total oil volume originally in place

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– initial solution gas-oil ratio

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– solution gas-oil ratio at pressure of interest

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– gas formation volume factor at pressure of interest

The change in pore volume due to the expansion of connate water and residual oil, and pore volume reduction, is derived as follows: (10)

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where oil, water and pore volumes are defined as:

(11)

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(12)

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(13)

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By combining equations (12), (13) and (14) into equation (11), we finally obtain:

(14)

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where:

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– total change in pore volume

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– expansion in connate water volume

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– expansion in residual oil volume

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– reduction in pore volume due to formation compressibility

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– connate water saturation

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– residual oil saturation

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– reduction in reservoir pressure due to depletion

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– total oil volume originally in place

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– total water volume originally in place

The last term in the material balance equation, underground withdrawal, is obtained from the cumulative produced volumes, calculated as follows.

(15)

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where:

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– cumulative oil production

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– cumulative water production

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– cumulative gas-oil ratio (calculated as cumulative gas divided by cumulative oil)

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– solution gas-oil ratio at pressure of interest

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– oil formation volume factor at pressure of interest

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– gas formation volume factor at pressure of interest

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– water formation volume factor at pressure of interest

We need a closure relationship for material balance relating oil (N) and water (W) volumes originally in place. However, we only estimate a total drainage volume (V_d) combining all phases, as defined in equation (2). It is necessary to split the drainage volume into oil and water components, which can be done by defining ω, the ratio of water to oil in place volumes, as shown in equation (16). The ratio ω can be directly provided as an input, from volumetrics, or it can also be approximated from PVT and petrophysical properties, as defined in equation (17).

(16)

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(17)

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Putting it all together, equation (14) describes the pressure change representing reservoir depletion for liquid systems. Note here,

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which is the total pore volume contacted which continuously expands as a function of time.

(18)

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The delta-pressure in equation (14) represents the reservoir pressure drop from initial pressure:

(19)

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The average reservoir pressure represents the volumetric-averaged pressure in the contacted drainage volume at any instantaneous time, and it approximates the depletion in the reservoir due to production.

Equation (18) can be re-arranged to solve for drainage volume (𝑉𝑑):

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(20)

Equations (2) through (20) represent the key steps involved in the workflow to estimate drainage volume and reservoir pressure depletion, the two key metrics to capture transient well performance.

The following steps outline the overall algorithm (Figure 2) that incorporates pressure dependent PVT properties (e.g., formation volume factor, compressibility, and solution GOR) and is posed as an optimization problem.

- Assume average reservoir pressure (𝑃𝑎𝑣𝑔) for all evaluated time steps, as a function of cumulative liquid production, as an initial guess. The reservoir pressure trend must start from initial reservoir pressure and be monotonically decreasing.
- Calculate all required PVT properties (𝐵𝑜, 𝐵𝑔, 𝐵𝑤, 𝑅𝑠, 𝑐𝑜, 𝑐𝑤) at each input pressure value.
- Estimate total compressibility (𝑐𝑡) for all given pressures with equation (8), with initial watersaturation (it is assumed that the average water saturation does not change in time).
- Calculate the total equivalent rate (𝑞𝑡𝑒) at all evaluated pressures, using equation (7).
- Estimate material balance time (𝑡𝑒) and rate-normalized pressure (RNP) with equations (3) and (4).
- Apply filters to eliminate outliers and fit RNP vs 𝑡𝑒 with a hyperbolic equation.
- Calculate drainage volume (𝑉𝑑) using equation (2) with the smoothened RNP vs 𝑡𝑒 trend.
- Finally, estimate average reservoir pressure using equations (17) and (18). The required ωparameter can either be provided as a direct input or estimated with equation (16).
- The above steps are repeated until the calculated average reservoir pressure converges withinrequired error tolerance.

Similarly, for gas systems:

(Gas expansion) + (Water expansion) + (Change in pore volume due to connatewater/residual gas expansion and pore volume reduction) = (Undergroundwithdrawal) (21)

The drainage volume equation can be written as follows:

(22)

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Equation (22) is applicable only for dry gas cases. In the presence of liquid condensate, the cumulative gas term (𝐺𝑝) is modified to account for the condensate production.

(23)

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We can substitute the gas formation volume factor (𝐵𝑔) by a two-phase formation volume factor which explicitly includes the impact of retrograde condensate on the gas properties, using an approach such as proposed by Rayes et al. (1992).

Instantaneous recovery ratio (IRR) can be defined as the recovery factor at a given moment in time, given the producing rates and contacted drainage volume, which is a useful diagnostic metric that estimates completion effectiveness. Asymptotically, IRR will reach the final recovery factor when the entire well drainage volume is contacted. However, the shapes of IRR trends can be quite insightful in evaluating completions effectiveness.

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- Molinari, D., Sankaran, S., Symmons, D., and M. Perrotte. "A Hybrid Data and Physics ModelingApproach Towards Unconventional Well Performance Analysis." SPE-196122-MS. Paperpresented at the SPE Annual Technical Conference and Exhibition, Calgary, Alberta, Canada,September 2019a. doi: https://doi.org/10.2118/196122-MS
- Molinari, D., Sankaran, S., Symmons, D., Perrotte, M., Wolfram, E., Krane, I., Han, J., and N.Bansal. "Implementing an Integrated Production Surveillance and Optimization System in anUnconventional Field." URTEC-2019-41-MS. Paper presented at the SPE/AAPG/SEGUnconventional Resources Technology Conference, Denver, Colorado, USA, July 2019b. doi:https://doi.org/10.15530/urtec-2019-41
- Molinari, D., and S. Sankaran. "A Reduced Physics Modeling Approach to Understand MultiphaseWell Production Performance for Unconventional Reservoirs." URTEC-2021-5023-MS. Paperpresented at the SPE/AAPG/SEG Unconventional Resources Technology Conference, Houston,Texas, USA, July 2021. doi: https://doi.org/10.15530/urtec-2021-5023
- King, M. J., Wang, Z., and A. Datta-Gupta. "Asymptotic Solutions of the Diffusivity Equation andTheir Applications." Paper presented at the SPE Europec featured at 78th EAGE Conference andExhibition, Vienna, Austria, May 2016. doi: https://doi.org/10.2118/180149-MS
- Zhang, Y., Yang, C., King, M. J., and A. Datta-Gupta. "Fast-Marching Methods for Complex Gridsand Anisotropic Permeabilities: Application to Unconventional Reservoirs." SPE-163637-MS.Paper presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA,February 2013. doi: https://doi.org/10.2118/163637-MS
- Xue X., Yang C., Park J., Sharma V. K., Datta-Gupta, A., and M. J. King. "Reservoir and FractureFlow Characterization Using Novel Diagnostic Plots." SPE J. 24 (2019): 1248–1269. doi:https://doi.org/10.2118/194017-PA
- Perrine, R.L. "Analysis of Pressure-buildup Curves." Paper presented at the Drilling and Production Practice, New York, New York, January 1956.
- Martin, J. C., "Simplified Equations of Flow in Gas Drive Reservoirs and the TheoreticalFoundation of Multiphase Pressure Buildup Analyses." Trans. 216 (1959): 321–323. doi:https://doi.org/10.2118/1235-G
- Rayes, D.G., Piper, L.D., McCain, W.D., and S.W. Poston. "Two-Phase Compressibility Factorsfor Retrograde Gases." SPE Form Eval 7 (1992): 87–92. doi: https://doi.org/10.2118/20055-PA