Recovery Factor Prediction

1. Theory

One can determine forecast fluid rates in a daily basis by extrapolating the straight line from multiphase FMB analysis and assuming all the future data will honor the line.

The idea is to solve a quadratic equation to get predicted gas rates. Then, multiplying these gas rates with instantaneous oil-gas ratio \((ogr)\) and water-gas ratio \((wgr)\) to get oil and water rates, respectively. Since original hydrocarbons in place is known from previous analysis, recovery factor (RF) can be easily determined.

1.1 Requirements

This module only works if and only if a multiphase FMB analysis has been performed before. Additionally, this module requires:

1. Contacted pore volume (\(V_p\)) and multiphase productivity index (\(b_M\)) from multiphase FMB analysis module.
2. Extrapolated bottomhole pressure \((p_{wf})\) vs. time.
3. Extrapolated flowing instantaneous oil-gas ratio \((ogr)\) vs. \(p_{wf}\).
4. Extrapolated flowing instantaneous water-gas ratio \((wgr)\) vs. \(p_{wf}\).

Flowing ogr and wgr are obtained from multiphase FMB analysis.

1.2 Quadratic Equation Derivation

First, start with cumulative producing oil-gas ratio \((OGR)\) and water-gas ratio \((WGR)\) at day \(t+1\).

$$\begin{equation} \label{Cum-OGR} % (STB/Mscf) OGR^{t+1}=\frac{N_p^{t+1}}{G_p^{t+1}}=\frac{N_p^{t}+ogr^{t+1}q_g^{t+1}}{G_p^{t}+q_g^{t+1}} \end{equation}$$

$$\begin{equation} \label{Cum-WGR} % (STB/Mscf) WGR^{t+1}=\frac{W_p^{t+1}}{G_p^{t+1}}=\frac{W_p^{t}+wgr^{t+1}q_g^{t+1}}{G_p^{t}+q_g^{t+1}} \end{equation}$$

Please note that uppercase OGR and WGR have different definitions than lowercase ogr and wgr.

As explained in multiphase flowing material balance section, cumulative producing OGR and WGR can also be written in terms of component concentrations as

$$\begin{equation} \label{Cum-OGR-MFMB} % (STB/Mscf) OGR^{t+1}=\frac{\zeta_{oi}-\left(\frac{S_o^{t+1}}{B_o^{t+1}}+\frac{S_g^{t+1}R_v^{t+1}}{B_g^{t+1}}\right)}{\zeta_{gi}-\left(\frac{S_o^{t+1}R_s^{t+1}}{B_o^{t+1}}+\frac{S_g^{t+1}}{B_g^{t+1}}\right)} \end{equation}$$

$$\begin{equation} \label{Cum-WGR-MFMB} % (STB/Mscf) WGR^{t+1}=\frac{\zeta_{wi}-\left(\frac{S_w^{t+1}}{B_w^{t+1}}\right)}{\zeta_{gi}-\left(\frac{S_o^{t+1}R_s^{t+1}}{B_o^{t+1}}+\frac{S_g^{t+1}}{B_g^{t+1}}\right)} \end{equation}$$

Where the component concentrations of oil, gas and water (\(\zeta_o\), \(\zeta_g\) and \(\zeta_w\)) are defined

$$\begin{equation} \label{Component-Concentration-Oil} % (STB/bbl) \zeta_{o}=\frac{S_o}{B_o}+\frac{S_gR_v}{B_g} \end{equation}$$

$$\begin{equation} \label{Component-Concentration-Gas} % (Mscf/bbl) \zeta_{g}=\frac{S_oR_s}{B_o}+\frac{S_g}{B_g} \end{equation}$$

$$\begin{equation} \label{Component-Concentration-Water} % (STB/bbl) \zeta_{w}=\frac{S_w}{B_w} \end{equation}$$

Subscripts \(o\), \(g\) and \(w\) represent oil, gas and water properties; while subscript \(i\) is for initial condition. Definitions and units for all variables can be seen in the nomenclature.

Combining these two definitions results in $$\begin{equation} \label{Cum-OGR-Combined} % (STB/Mscf) \frac{N_p^{t}+ogr^{t+1}q_g^{t+1}}{G_p^{t}+q_g^{t+1}}=\frac{\zeta_{oi}-\left(\frac{S_o^{t+1}}{B_o^{t+1}}+\frac{S_g^{t+1}R_v^{t+1}}{B_g^{t+1}}\right)}{\zeta_{gi}-\left(\frac{S_o^{t+1}R_s^{t+1}}{B_o^{t+1}}+\frac{S_g^{t+1}}{B_g^{t+1}}\right)} \end{equation}$$

$$\begin{equation} \label{Cum-WGR-Combined} % (STB/Mscf) \frac{W_p^{t}+wgr^{t+1}q_g^{t+1}}{G_p^{t}+q_g^{t+1}}=\frac{\zeta_{wi}-\left(\frac{S_w^{t+1}}{B_w^{t+1}}\right)}{\zeta_{gi}-\left(\frac{S_o^{t+1}R_s^{t+1}}{B_o^{t+1}}+\frac{S_g^{t+1}}{B_g^{t+1}}\right)} \end{equation}$$

Total saturation is denoted by

$$\begin{equation} \label{Saturation} S_o^{t+1}+S_g^{t+1}+S_w^{t+1}=1 \end{equation}$$

Thus, oil and water saturation (\(S_o\) and \(S_w\)) in terms of gas rate can be reckoned by rearranging Eqs. \eqref{Cum-OGR-Combined} and \eqref{Cum-WGR-Combined}.

$$\begin{equation} \label{Oil-Saturation} % (-) S_o^{t+1}=\frac{A_o^{t+1}q_g^{t+1}+C_o^{t+1}}{A_d^{t+1}q_g^{t+1}+C_d^{t+1}} \end{equation}$$

$$\begin{equation} \label{Water-Saturation} % (-) S_w^{t+1}=\frac{A_w^{t+1}q_g^{t+1}+C_w^{t+1}}{A_d^{t+1}q_g^{t+1}+C_d^{t+1}} \end{equation}$$

in which the constants are

$$\begin{equation} \label{Ao} A_o^{t+1}=B_o^{t+1}\left[\zeta_{gi}\left(B_g^{t+1}ogr^{t+1}+B_w^{t+1}R_v^{t+1}wgr^{t+1}\right)-\zeta_{oi}\left(B_g^{t+1}+B_w^{t+1}wgr^{t+1}\right)+\left(ogr^{t+1}-R_v^{t+1}\right)\left(\zeta_{wi}B_w^{t+1}-1\right) \right] \end{equation}$$

$$\begin{equation} \label{Aw} A_w^{t+1}=B_w^{t+1}\left\{wgr^{t+1}\left[\zeta_{gi}\left(B_g^{t+1}-B_o^{t+1}R_v^{t+1}\right)+\zeta_{oi}\left(B_o^{t+1}-B_g^{t+1}R_s^{t+1}\right)-\left(1-R_s^{t+1}R_v^{t+1}\right) \right]\\+\zeta_{wi}\left[ogr^{t+1}\left(B_g^{t+1}R_s^{t+1}-B_o^{t+1}\right)+B_o^{t+1}R_v^{t+1}-B_g^{t+1} \right]\right\} \end{equation}$$

$$\begin{equation} \label{Ad} A_d^{t+1}=B_g^{t+1}\left(ogr^{t+1}R_s^{t+1}-1\right)+B_o^{t+1}\left(R_v^{t+1}-ogr^{t+1}\right)+B_w^{t+1}wgr^{t+1}\left(R_s^{t+1}R_v^{t+1}-1\right) \end{equation}$$

$$\begin{equation} \label{Co} C_o^{t+1}=B_o^{t+1}\left[\zeta_{gi}\left(B_g^{t+1}N_p^{t}+B_w^{t+1}R_v^{t+1}W_p^t\right)-\zeta_{oi}\left(B_g^{t+1}G_p^t+B_w^{t+1}W_p^t\right)+\left(N_p^t-G_p^tR_v^{t+1}\right)\left(\zeta_{wi}B_w^{t+1}-1\right) \right] \end{equation}$$

$$\begin{equation} \label{Cw} C_w^{t+1}=B_w^{t+1}\left\{W_p^t\left[\left(R_s^{t+1}R_v^{t+1}-1\right)-\zeta_{gi}\left(B_o^{t+1}R_v^{t+1}-B_g^{t+1}\right)-\zeta_{oi}\left(B_g^{t+1}R_s^{t+1}-B_o^{t+1}\right) \right]\\+\zeta_{wi}\left[B_g^{t+1}\left(N_p^tR_s^{t+1}-G_p^t\right)-B_o^{t+1}\left(N_p^t-G_p^tR_v^{t+1}\right) \right]\right\} \end{equation}$$

$$\begin{equation} \label{Cd} C_d^{t+1}=N_p^t\left(B_g^{t+1}R_s^{t+1}-B_o^{t+1}\right)+G_p^t\left(B_o^{t+1}R_v^{t+1}-B_g^{t+1}\right)+W_p^tB_w^{t+1}\left(R_s^{t+1}R_v^{t+1}-1\right) \end{equation}$$

Second, mixture density is defined as $$\begin{equation} \label{Mixture-Density} % lbm/bbl \beta=5.615\rho_{osc}\left(\frac{S_o}{B_o}+\frac{S_gR_v}{B_g}\right)+1000\rho_{gsc}\left(\frac{S_oR_s}{B_o}+\frac{S_g}{B_g}\right)+5.615\rho_{wsc}\left(\frac{S_w}{B_w}\right) \end{equation}$$

Subscript \(sc\) indicates standard condition. Substituting \(S_o\) and \(S_w\) in Eq. \eqref{Mixture-Density} with Eqs. \eqref{Oil-Saturation} and \eqref{Water-Saturation} in terms of \(q_g^{t+1}\) yields $$\begin{equation} \label{Beta} \beta^{t+1}=\frac{D^{t+1}q_g^{t+1}+E^{t+1}}{B_g^{t+1}B_o^{t+1}B_w^{t+1}\left(A_d^{t+1}q_g^{t+1}+C_d^{t+1}\right)} \end{equation}$$

where $$\begin{equation} \label{alpha-g} \alpha_g^{t+1}=5.615\rho_{osc}R_v^{t+1}+1000\rho_{gsc} \end{equation}$$

$$\begin{equation} \label{alpha-o} \alpha_o^{t+1}=5.615\rho_{osc}+1000\rho_{gsc}R_s^{t+1} \end{equation}$$

$$\begin{equation} \label{alpha-w} \alpha_w=5.615\rho_{wsc} \end{equation}$$

$$\begin{equation} \label{D} D^{t+1}=\alpha_g^{t+1}A_d^{t+1}B_o^{t+1}B_w^{t+1}+A_w^{t+1}B_o^{t+1}\left(\alpha_wB_g^{t+1}-\alpha_g^{t+1}B_w^{t+1}\right)\\+A_o^{t+1}B_w^{t+1}\left(\alpha_o^{t+1}B_g^{t+1}-\alpha_g^{t+1}B_o^{t+1}\right) \end{equation}$$

$$\begin{equation} \label{E} E^{t+1}=B_g^{t+1}\left(\alpha_wB_o^{t+1}C_w^{t+1}+\alpha_o^{t+1}B_w^{t+1}C_o^{t+1}\right)+\alpha_g^{t+1}B_o^{t+1}B_w^{t+1}\left(C_d^{t+1}-C_o^{t+1}-C_w^{t+1}\right) \end{equation}$$

Recalling multiphase FMB equation at day \(t+1\), $$\begin{equation} \label{Multiphase-FMB} \frac{\beta_i-\beta^{t+1}}{\dot{m}^{t+1}}=\frac{1}{V_p}{MBT}_M^{t+1}+\frac{1}{b_M} \end{equation}$$

\(MBT_M\) indicates mass material balance time, $$\begin{equation} \label{Mass-Material-Balance} % day {MBT}_M^{t+1}=\frac{M_T^{t+1}}{\dot{m}^{t+1}} \end{equation}$$

The mass flow rate is identified as $$\begin{equation} \label{Total-Mass-Production} % lbm/d \dot{m}^{t+1}=5.615\rho_{osc}q_o^{t+1}+1000\rho_{gsc}q_g^{t+1}+5.615\rho_{wsc}q_w^{t+1}\\=q_g^{t+1}\left(5.615\rho_{osc}ogr^{t+1}+1000\rho_{gsc}+5.615\rho_{wsc}wgr^{t+1}\right) \end{equation}$$

and \(M_T\) depicts the total cumulative mass of oil, gas and water, $$\begin{equation} \label{Cumulative-Mass-Produced-Total} % lbm M_T^{t+1}=M_o^{t+1}+M_g^{t+1}+M_w^{t+1}=5.615\rho_{osc}N_p^{t+1}+1000\rho_{gsc}G_p^{t+1}+5.615\rho_{wsc}W_p^{t+1} \\M_T^{t+1}=M_T^{t}+\dot{m}^{t+1} \end{equation}$$

Multiplying both sides in Eq. \eqref{Multiphase-FMB} by mass flow rate (\(\dot{m}\)) gives $$\begin{equation} \label{Multiphase-FMB-2} \beta_i-\beta^{t+1}=\frac{1}{V_p}M_T^{t+1}+\frac{\dot{m}^{t+1}}{b_M} \end{equation}$$

Assume $$\begin{equation} \label{alpha-c} \alpha_c^{t+1}=5.615\rho_{osc}ogr^{t+1}+1000\rho_{gsc}+5.615\rho_{wsc}wgr^{t+1} \end{equation}$$

Replacing \(\beta^{t+1}\) with Eq. \eqref{Beta}, \(M_T^{t+1}\) with Eq. \eqref{Cumulative-Mass-Produced-Total} and \(\dot{m}^{t+1}\) with Eq. \eqref{Total-Mass-Production}, then Eq. \eqref{Multiphase-FMB-2} can be rewritten as $$\begin{equation} \label{Multiphase-FMB-3} \beta_i-\frac{D^{t+1}q_g^{t+1}+E^{t+1}}{B_g^{t+1}B_o^{t+1}B_w^{t+1}\left(A_d^{t+1}q_g^{t+1}+C_d^{t+1}\right)}=\frac{M_T^t}{V_p}+\alpha_c^{t+1}\left(\frac{1}{V_p}+\frac{1}{b_M}\right)q_g^{t+1} \end{equation}$$

By rearranging Eq. \eqref{Multiphase-FMB-3} in terms of \(q_g^{t+1}\), a quadratic equation is formed. $$\begin{equation} \label{Quadratic-Equation} A^{t+1}\left(q_g^{t+1}\right)^2+B^{t+1}q_g^{t+1}+C^{t+1}=0 \end{equation}$$

Where the constants are expressed as

$$\begin{equation} \label{A} A^{t+1}=\alpha_d^{t+1}B_g^{t+1}B_o^{t+1}B_w^{t+1}\left[B_g^{t+1}\left(1-ogr^{t+1}R_s^{t+1}\right)+B_o^{t+1}\left(ogr^{t+1}-R_v^{t+1}\right)+B_w^{t+1}wgr^{t+1}\left(1-R_s^{t+1}R_v^{t+1}\right) \right] \end{equation}$$

$$\begin{equation} \label{B} B^{t+1}=-D^{t+1}+B_g^{t+1}B_o^{t+1}B_w^{t+1}\left\{B_g^{t+1}\left[\alpha_d^{t+1}\left(G_p^t-N_p^tR_s^{t+1}\right)+\left(\frac{M_T^t}{V_p}-\beta_i\right)\left(1-ogr^{t+1}R_s^{t+1}\right) \right]\\+B_o^{t+1}\left[\alpha_d^{t+1}\left(N_p^t-G_p^tR_v^{t+1}\right)+\left(\frac{M_T^t}{V_p}-\beta_i\right)\left(ogr^{t+1}-R_v^{t+1}\right) \right]\\+B_w^{t+1}\left[1-R_s^{t+1}R_v^{t+1} \right]\left[wgr^{t+1}\left(\frac{M_T^t}{V_p}-\beta_i\right)+\alpha_d^{t+1}W_p^t \right] \right\} \end{equation}$$

$$\begin{equation} \label{C} C^{t+1}=-E^{t+1}+B_g^{t+1}B_o^{t+1}B_w^{t+1}\left[\frac{M_T^t}{V_p}-\beta_i \right]\left[B_g^{t+1}\left(G_p^t-N_p^tR_s^{t+1}\right) \\+B_o^{t+1}\left(N_p^t-G_p^tR_v^{t+1}\right)+B_w^{t+1}W_p^t\left(1-R_s^{t+1}R_v^{t+1}\right) \right] \end{equation}$$

$$\begin{equation} \label{alpha-d} \alpha_d^{t+1}=\alpha_c^{t+1}\left(\frac{1}{V_p}+\frac{1}{b_M}\right) \end{equation}$$

1.3 Recovery Factor Calculation

After solving the quadratic equation, oil and water rate can be calculated as follows

$$\begin{equation} \label{Oil-Rate} % (STB/d) q_o^{t+1}=q_g^{t+1}ogr^{t+1} \end{equation}$$

$$\begin{equation} \label{Water-Rate} % (STB/d) q_w^{t+1}=q_g^{t+1}wgr^{t+1} \end{equation}$$

Having calculated fluid rates, the cumulative productions as well as the recovery factor can be easily resolved.

$$\begin{equation} \label{Oil-Cumulative} % (STB) N_p^{t+1}=N_p^t+q_o^{t+1} \end{equation}$$

$$\begin{equation} \label{Gas-Cumulative} % (Mscf) G_p^{t+1}=G_p^t+q_g^{t+1} \end{equation}$$

$$\begin{equation} \label{Water-Cumulative} % (STB) W_p^{t+1}=W_p^t+q_w^{t+1} \end{equation}$$

$$\begin{equation} \label{Oil-RF} % (-) RF_o^{t+1}=\frac{N_p^{t+1}}{OIP} \end{equation}$$

$$\begin{equation} \label{Gas-RF} % (-) RF_g^{t+1}=\frac{G_p^{t+1}}{GIP} \end{equation}$$

2. Workflow

EUR Prediction

1. Extrapolate \(pwf\) vs. time starts from the last day of history data to the desired time.
2. Extrapolate flowing \(ogr\) vs. \(pwf\) and flowing \(wgr\) vs. \(pwf\), ensure the extrapolations cover the range of \(pwf\) from step 1. These will be the "predicted" data.
3. At the start of the forecast, \(N_p^t\), \(G_p^t\) and \(W_p^t\) are the cumulatives at the last time of the historical data.
4. Calculate the following properties for every time step:
   • \(OGR\) and \(WGR\) by interpolating predicted data at \(pwf\)
   • Fluid properties (\(B_{o}\), \(B_{g}\), \(B_{w}\), \(R_{s}\) and \(R_{v}\)) by interpolating BOT data at \(pwf\)
5. Find fluid rates for every time step:
   • Calculate saturation constants with Eqs. \eqref{Ao} to \eqref{Cd}
   • Calculate mixture density constants with Eqs. \eqref{alpha-g} to \eqref{E}
   • Calculate all constants for quadratic equation with Eqs. \eqref{A} to \eqref{alpha-d}
   • Solve Eq. \eqref{Quadratic-Equation} by choosing the positive and largest root
   • Calculate \(q_o\) and \(q_w\) using Eqs. \eqref{Oil-Rate} and \eqref{Water-Rate}, respectively
6. Calculate cumulative production (\(N_p\), \(G_p\) and \(W_p\)) with Eqs. \eqref{Oil-Cumulative}, \eqref{Gas-Cumulative} and \eqref{Water-Cumulative}.
7. Calculate recovery factor by using Eqs. \eqref{Oil-RF} and \eqref{Gas-RF}.

How to forecast CGR and WGR vs Pressure?

Regarding extrapolating OGR and WGR

R_v and 1/R_s curves are generated from the EoS based on the PVT initialization.

R_v or the condensate gas ratio represents the minimum oil that will be recovered at the surface as condensate dropout from gas production.
This represents the minimum oil evolved from gas that would still be recoverable when the oil phase stops being mobile and only gas is flowing. Hence, this volatilized Oil-Gas ratio, forms the minimum bound of the OGR expectation.

1/R_s or the oil gas ratio represents the maximum oil that has to be recovered to produce a unit volume of gas from solution. This essentially makes the upper bound of the ratio forecast.

Your forecasts will always be well within these bounds.

OGR data above the saturation pressure will be very close to the 1/Rs line and the forecast needs to follow the data and reasonably extrapolate to lower pressures (upto the abandonment pressure). Water Gas Ratio is typically declining with decline in pressure.

Reducing Uncertainty in Ratio Forecasting - See analogs!

In case the yield/GOR is a strong function of flowing bottomhole pressure - The case may not vary much with the type of analog chosen (parent/edge/fully bound), instead the analog must have similar producing history of sufficient length.

In case the yield/GOR is a strong function of average reservoir pressure - Split the data from analog wells by the type and select the trend that closely matches the type of the well, since parent wells will have a lowest, flat GOR while an edge and fully bound cases have different, but both sharply increasing trends in comparison. The forecast built should be bounded by the analog wells with the fully bounded tight infill yields the sharpest increase in GOR while the parent well offers a minimum bound for the magnitude and the rate of change of GOR.

Finally, always use the Cumulative GOR vs Cumulative Oil to sanity check the forecast - realistically, you expect a linear increase with a concave up trend.

Impact of these forecasts on recovery factors

The oil EUR and recovery factors are far less sensitive and gas EUR may be slightly more sensitive to the constructed forecast in WGR.
The most practical uncertainty you might face is the OGR ratio forecasting, which also has the highest impact on calculated recovery rates.

MFMB vs Numerical Model recovery factors

In MFMB, we forecast the ratios and use simple material balance equations to calculate Recovery Factors and EUR. We proceed in the opposite direction compared to the Numerical Simulations, where we run a numerical simulation to get the same ratios over time to ultimately calculate the Recovery factors and EUR.

Under the assumption that most unconventional wells produce from reservoirs with an expansion-drive, if the forecasted ratios and abandonment pressures are exactly the same in Numerical Model and MFMB, we get the exact same recovery factors and EUR as well.

Nomenclature

Variable	Definition	Units
\(B_g\)	Gas formation volume factor	RB/Mscf
\(b_M\)	Multiphase productivity index	RB/day
\(B_w\)	Water formation volume factor	RB/STB
\(B_o\)	Oil formation volume factor	RB/SRB
\(GIP\)	Gas in place	Mscf
\(G_p\)	Cumulative produced gas	Mscf
\(\dot{m}\)	Mass flow rate	lbm/d
\(M_g\)	Cumulative mass of surface produced gas	lbm
\(M_o\)	Cumulative mass of surface produced oil	lbm
\(M_T\)	Cumulative total mass produced	lbm
\(M_w\)	Cumulative mass of surface produced water	lbm
\(MBT_M\)	Mass material balance time	day
\(N_p\)	Cumulative produced oil	STB
\(OGR\)	Cumulative producing oil-gas ratio	STB/Mscf
\(ogr\)	Instantaneous producing oil-gas ratio	STB/Mscf
\(OIP\)	Oil in place	STB
\(p\)	Pressure	psia
\(p_{wf}\)	Flowing bottomhole pressure	psia
\(q_g\)	Gas production rate	Mscf/d
\(q_o\)	Oil production rate	STB/d
\(q_w\)	Water production rate	STB/d
\(R_s\)	Solution gas-oil ratio	Mscf/STB
\(R_v\)	Solution oil-gas ratio	STB/Mscf
\(RF_g\)	Gas recovery factor	-
\(RF_o\)	Oil recovery factor	-
\(S_g\)	Gas saturation	-
\(S_o\)	Oil saturation	-
\(S_w\)	Water saturation	-
\(t\)	Time	day
\(V_p\)	Contacted reservoir pore volume	RB
\(WGR\)	Cumulative producing water-gas ratio	STB/Mscf
\(wgr\)	Instantaneous producing oil-gas ratio	STB/Mscf
\(W_p\)	Cumulative produced water	STB
\(\beta\)	Mixture density	lbm/RB
\(\rho\)	Density	lbm/cuft
\(\zeta_g\)	Component comcentration of gas	Mscf/RB
\(\zeta_o\)	Component comcentration of oil	STB/RB
\(\zeta_w\)	Component comcentration of water	STB/RB