Transient and Pseudo-Steady-State Inflow Performance Relationships for Multiphase Flow in Fractured Unconventional Reservoirs

Abstract

The objective of this paper is developing new methodology for constructing the inflow performance relationships (IPRs) of unconventional reservoirs experiencing multiphase flow. The motivation is eliminating the uncertainties of using single-phase flow IPRs and approaching realistic representation and simulation to reservoir pressure–flow rate relationships throughout the entire life of production. Several analytical models for the pressure drop and decline rate as wells productivity index of two wellbore conditions, constant Sandface flow rate and constant wellbore pressure, are presented in this study. Several deterministic models are also proposed in this study for multiphase reservoir total mobility and compressibility using multi-regression analysis of PVT data and relative permeability curves of different reservoir fluids. These deterministic models are coupled with the analytical models of pressure drop, decline rate, and productivity index to construct the pressure–flow rate relationships (IPRs) during transient and pseudo-steady-state production time. Transient IPRs are generated for early-time hydraulic fracture linear flow regime and intermediate-time bilinear and trilinear flow regimes, while steady-state IPRs are generated for pseudo-steady-state flow regime in case of constant Sandface flow rate and boundary-dominated flow regime in case of constant wellbore pressure. The outcomes of this study are as follows: (1) introducing the impact of multiphase flow to the IPRs of unconventional reservoirs; (2) developing deterministic models for reservoir total mobility and compressibility using multi-regression analysis of PVT data and relative permeability curves; (3) developing analytical models for different flow regimes that could be developed during the entire production life of reservoirs; (4) predicting transient and steady-state IPRs of multiphase flow for different wellbore conditions. The study has pointed out: (1) Multiphase flow conditions have significant impact on reservoir IPRs. (2) Multiphase reservoir total mobility and compressibility exhibit significant change with reservoir pressure. (3) Constant Sandface flow rate may demonstrate IPR better than constant wellbore pressure. (4) Late production time is not affected by multiphase flow conditions similar to transient state flow at early and intermediate production time.

Appendix

Consider the formation shown in Fig. 25 where multiple hydraulic fractures propagate in the stimulated reservoir volume (SRV). The formation could have also unstimulated part (USRV) where the stimulation process does not have any impact on the porous media. Formation boundaries are indicated by $\left( {2x_{\text{e}} ,2y_{\text{e}} } \right)$ and fracture half-length is $\left( {x_{\text{f}} } \right)$. Hydraulic fractures considered fully penetrate the formation, i.e., hydraulic fracture height is considered equal to the formation thickness. Stimulated and unstimulated reservoir volumes and hydraulic fracture dimensions are depicted in Fig. 26.

Fig. 25

Schematic drawing for a hydraulically fractured reservoir

Fig. 26

Stimulated and unstimulated reservoir volumes and hydraulic fracture dimensions

Wellbore pressure drop, in dimensionless form, assuming constant Sandface flow rate is given by (Brown et al. 2011; Ozkan et al. 2011):

$$ \overline{{P_{\text{wD}} }} = \frac{\pi }{{sF_{\text{CD}} \sqrt {A_{\text{F}} } \tanh \left( {\sqrt {A_{\text{F}} } } \right)}}, $$

(50)

while the mathematical model for dimensionless Sandface flow rate assuming constant wellbore pressure is given by (van Everdingen and Hurst 1949):

$$ \overline{{q_{\text{D}} }} = \frac{1}{{s^{2} \overline{{P_{\text{wD}} }} }}, $$

(51)

where

$$ F_{\text{CD}} = \frac{{k_{\text{f}} w_{\text{f}} }}{{k_{\text{i}} x_{\text{f}} }}. $$

(52)

The assumptions used in developing the mathematical models given by (50) and (51) are:

• 1. Constant porosity and uniform reservoir thickness.

• 2. Symmetrically distributed hydraulic fractures with symmetrical hydraulic fracture dimensions.

• 3. Fractures fully penetrate the formation in the vertical direction.

Initial reservoir conditions are:

$$ P_{\text{D}} \left( {x_{\text{D}} ,y_{\text{D}} ,t_{\text{D}} = 0.0} \right) = P_{\text{Di}} , $$

(53)

while reservoir inner and outer boundary conditions are:

$$ \left. {\frac{{\partial P_{\text{D}} }}{{\partial x_{\text{D}} }}} \right|_{{x_{\text{D}} = x_{\text{eD}} }} = 0.0, $$

(54)

$$ \left. {\frac{{\partial P_{\text{wD}} }}{{\partial x_{\text{D}} }}} \right|_{{x_{\text{D}} = 0.0}} = \frac{\pi }{{sF_{\text{CD}} }}. $$

(55)

The parameter $\left( {A_{\text{F}} } \right)$ refers to the configurations of the stimulated and unstimulated reservoir volumes in addition to the petrophysical properties of the two volumes. Mathematically, this parameter is written:

$$ A_{\text{F}} = \frac{{2B_{\text{F}} }}{{F_{\text{CD}} }} + \frac{s}{{\eta_{\text{fD}} }}, $$

(56)

where

$$ B_{\text{F}} = \sqrt {A_{\text{r}} } \tanh \left[ {\sqrt {A_{\text{r}} } \left( {y_{\text{eD}} - w_{\text{D}} /2} \right)} \right], $$

(57)

$$ A_{\text{r}} = \frac{{B_{\text{r}} }}{{R_{\text{CD}} y_{\text{eD}} }} + sf\left( s \right) , $$

(58)

$$ B_{\text{r}} = \sqrt {\frac{s}{{\eta_{\text{mD}} }}} \tanh \left[ {\sqrt {\frac{s}{{\eta_{\text{mD}} }}} \left( {x_{\text{eD}} - 1} \right)} \right] , $$

(59)

$$ R_{\text{CD}} = \frac{{k_{i} x_{\text{f}} }}{{k_{\text{m}} y_{\text{e}} }}, $$

(60)

$$ \eta_{\text{mD}} = \frac{{\eta_{\text{m}} }}{{\eta_{i} }}, $$

(61)

$$ \eta_{\text{fD}} = \frac{{\eta_{\text{f}} }}{{\eta_{\text{i}} }}, $$

(62)

$$ x_{\text{eD}} = \frac{{x_{\text{e}} }}{{x_{\text{f}} }}, $$

(63)

$$ y_{\text{eD}} = \frac{{y_{\text{e}} }}{{x_{\text{f}} }}, $$

(64)

$$ w_{\text{D}} = \frac{{w_{\text{f}} }}{{x_{\text{f}} }}, $$

(65)

$$ P_{\text{D}} = \frac{{k_{\text{i}} \left( {\frac{k}{\mu }} \right)_{\text{mp}} h\Delta P}}{{141.2 \left( {q_{\text{t}} B_{\text{t}} } \right)}}, $$

(66)

$$ t_{\text{D}} = \frac{{0.000263k_{\text{i}} \left( {\frac{k}{\mu }} \right)_{\text{mp}} t}}{{\left( {\emptyset c_{\text{t}} } \right)_{\text{f}} x_{\text{f}}^{2} }} . $$

(67)