Reservoir, rock, and fluid properties for simulation and analytical solution.

## Abstract

In this work, we apply the method of matched asymptotic expansions to solve the one-dimensional saturation convection-dispersion equation, a nonlinear pseudo-parabolic partial differential equation. This equation is one of the governing equations for two-phase flow in a porous media when including capillary pressure effects, for the specific initial and boundary conditions arising when injecting water in an infinite radial piecewise homogeneous horizontal medium containing oil and water. The method of matched asymptotic expansions combines inner and outer expansions to construct the global solution. In here, the outer expansion corresponds to the solution of the nonlinear first-order hyperbolic equation obtained when the dispersion effects driven by capillary pressure became negligible. This equation has a monotonic flux function with an inflection point, and its weak solution can be found by applying the method of characteristics. The inner expansion corresponds to the shock layer, which is modeled as a traveling wave obtained by a stretching transformation of the partial differential equation. In the transformed domain, the traveling wave solution is solved using regular perturbation theory. By combining the solution for saturation with the so-called Thompson-Reynolds steady-state theory for obtaining the pressure, one can obtain an approximate analytical solution for the wellbore pressure, which can be used as the forward solution which analyzes pressure data by pressure-transient analysis.

### Keywords

- method of matched asymptotics
- boundary layer approximation
- nonlinear pseudo-parabolic partial differential equation
- convection-dispersion phenomenon
- multiphase flow in porous media

## 1. Introduction

In this chapter, we show how to generate a semi-analytical solution for the wellbore pressure response during a water injection test. In the petroleum industry, well testing is a common practice which consists of wellbore pressure and wellbore flow rate data acquisition in order to estimate parameters that govern flow in the porous media, i.e., the reservoir rock which stores the hydrocarbons. Well tests give an insight into the oil and gas field production potential and profitability and allow the estimation of reservoir parameters. Estimated parameters can be used to calibrate the reservoir numerical simulation model that are used to describe the fluid flow in these reservoirs and forecast their performance as well as to maximize the productivity of the wells. Injections are important tests on reservoirs containing high amount of harmful gases like carbon dioxide and sulfur dissolved in the oil, causing conventional production testing in the exploratory phase of offshore field development inviable. Multiphase flow is the norm in petroleum reservoirs, and an injection test consists of a period of water or gas injection into an oil reservoir (Figure 1), a common technique known as waterflooding or gasflooding that is used to displace oil to a producing well. Data from an injection test can be used to estimate the reservoir rock absolute permeability (

## 2. Mathematical model

The solutions presented assume infinite-acting one-dimensional radial flow from and to a fully penetrating vertical well with no gravity effects. We apply the method of matched asymptotic expansions to solve the one-dimensional saturation convection-dispersion equation, a nonlinear pseudo-parabolic partial differential equation. This equation is one of the governing equations for two-phase flow in a porous media when including capillary pressure effects, for the specific initial and boundary conditions arising when injecting water in an infinite radial piecewise homogeneous horizontal medium containing oil and water. The method of matched asymptotic expansions combines inner and outer expansions to construct the global solution. In here, the outer expansion corresponds to the solution of the nonlinear first-order hyperbolic equation obtained when the dispersion effects driven by capillary pressure became negligible. This equation has a monotonic flux function with an inflection point, and its weak solution can be found by applying the method of characteristics. The inner expansion corresponds to the shock layer, which is modeled as a traveling wave obtained by a stretching transformation of the partial differential equation. By combining the solution for saturation with the so-called Thompson-Reynolds steady-state theory, one can obtain an approximate analytical solution for the wellbore pressure, which can be used as the forward solution which analyzes pressure data by pressure-transient analysis. Let us start by finding the saturation distribution in the reservoir during injection and show how to find pressure.

### 2.1. Saturation

The water mass balance equation, in radial coordinates, leads to the following nonlinear partial differential equation [5]:

where throughout we assume that porosity (*t*, is in hours. Let us use Darcy’s equation in radial coordinates without gravity for the oil (

For field units used throughout,

Rearranging Eq. (3), substituting the capillary pressure p_{c} given by the difference of the oil pressure (

where

where

which usually assumes an S-shape.

and the permeability is a function of radius because we consider a skin-damaged zone:

where

and rewrite Eq. (10) as

For simplicity, we use the Brooks and Corey model [8] given by

to represent capillary pressure. Here,

and the [8] model for the oil phase (nonwetting phase) [13]

Now that we have defined the fractional flow rate and its parameters, let us go back to our governing equation for saturation (Eq. (1)). Inserting Eq. (10) into Eq. (1) and defining

yields

which is the nonlinear “pseudo-parabolic” governing equation for saturation. If we insert same common values for the parameters in Eq. (7) to have an idea of its order of magnitude, we can see that epsilon is a very small number. This suggests that the effect of the third term in Eq. (15) may be treated as a perturbation to the first-order hyperbolic equation [5], given by

where

we divide the domain into two regions, outer and inner regions (Figure 6), where the inner region, the region around the water front, is modeled as a shock layer which propagates with the same speed as the shock would be obtained when

where

#### 2.1.1. Outer solution (S w BL )

The outer solution,

That is the nonlinear hyperbolic convection equation known as the Buckley-Leverett saturation equation given by Eq. (16) which is obtained when capillary and gravity effects are neglected. The well-known unique admissible weak solution of this Riemann problem, with the following initial condition

can be obtained by the application of the method of characteristics and is given by [5].

that is, by a family of rarefaction waves, a semi-shock wave, and a constant saturation zone where water is immobile. The shock jump is caused by the S-shaped form of the fractional flow curve, which leads to a gradient catastrophe and consequently a shock solution. This semi-shock has a constant speed, satisfying the Rankine-Hugoniot condition [25]:

where

The details of this solution can be found in [5]. Figure 7 shows the shock jump slope tangent to the fractional flow curve at *rs*. Ahead of the water front position, there is an immobile water. Figure 8 compares this solution, the outer solution, with the true solution; there is the convection-dispersion saturation profile. Here, we call the true solution the solution obtained from a numerical simulator.

#### 2.1.2. Inner solution (S w SL )

As mentioned, the inner solution intends to represent the saturation profile in the “inner” region around the water front, which is a shock layer (a boundary layer) around the shock traveling with the same speed as the shock itself (Figure 9). In order to find

where *rs* is the shock front position:

w is zero at

The inner solution is obtained by letting

as presented in [26]. Therefore, neglecting the terms of order

Note that here we are treating the permeability *k* as function of the shock position radius, *rs*, only, by assuming that in the limit of the inner solution, *τ* and applying the chain rule gives

where

as the inner solution goes asymptotically to the shock saturations. This necessity of this behavior will be clearer very soon when we compare the inner solution with the matching saturation solution. Using the first boundary condition given by Eq. (32) in Eq. (31) leads to

while using the second boundary condition given by Eq. (32) yields

implying that

Integrating Eq. (35) from

At

#### 2.1.3. Matching solution (S w SH )

The matching saturation

and in the injection case is given by

which is plotted in Figure 10 against the outer and inner solutions. As we were searching for,

#### 2.1.4. Mass balance

Now that we have defined all the three saturations that are composed of the approximate solution for the convection dispersion saturation equation, let us try to find a closed form for the saturation distribution based in the mass balance. Since both the Buckley-Leverett (

must hold. From Eq. (20) and Eq. (39), it follows that

which, upon simplification, gives

Rearranging Eq. (41) using Eq. (38) for

Using Eq. (27) in Eq. (42), it follows that

Transforming Eq. (43) from

From Eq. (35),

Substituting Eq. (45) in Eq. (44) and solving the resulting equation divided by

Setting

As the left sides of Eqs. (46) and (47) are the same, the right sides of these two equations must be equal which gives

Multiplying Eq. (48) by

Once the value *τ*, Eq. (36) is used to determine the saturation profile in the stabilized zone. It is important to note that, as

### 2.2. Wellbore pressure

As mentioned previously, after finding the saturation distribution, we can obtain the wellbore pressure by applying the pressure solutions presented by [2]. During injection at a constant flow rate,

where

where it is assumed that

by assuming that the second term in the right-hand side of Eq. (52) is zero from

where for any practical set of values of physical properties [28] indicate that this assumption is valid. Adding and subtracting the term

Here,

## 3. Validation

We have compared our pressure and saturation solution including capillary pressure effects with the commercial numerical simulator IMEX, using the properties shown in Table 1. Figure 12 compares the saturation distribution obtained from our analytical solution with the one obtained with IMEX, while Figure 13 shows the comparison of the wellbore pressure response from our analytical solution and IMEX during injection test. In order to be able to match saturation and pressure obtained from our solution with IMEX, we have to use a very refined grid (0.01 ft) around the wellbore in the zone invaded by water and then increase it exponentially to a very large external radius (10,000 times the wellbore radius) in order to reproduce an infinite acting reservoir. In addition, we have to start with very short time steps,

Property | Value | Unit | Property | Value | Unit |
---|---|---|---|---|---|

3000 | RB/DAY | 1.003 | RB/STB | ||

60 | ft | 1.002 | RB/STB | ||

0.35 | ft | 1/psi | |||

6800 | ft | 1/psi | |||

300 | ft | 1/psi | |||

0 | mD | 3.0 | cp | ||

0.10 | 0.5 | cp | |||

0.25 | 2 | ||||

2500 | psi | 0.5 | psi | ||

0.22 |

## 4. Conclusions

In this work, an accurate approximate analytical solution was constructed for wellbore pressure during water injection test in a reservoir containing oil and immobile water. Our solution was validated by comparing the bottom hole pressure calculated from the analytical model with the data obtained from a commercial numerical simulator. Our solution presented here for water injection together with the wellbore pressure and flow rate history for subsequent tests as shut-in and flowback can be used as forward model in a nonlinear regression in order to estimate relative permeabilities and capillary pressure curves in addition to the rock absolute permeability, the skin zone permeability, and the water endpoint relative permeability.

## Acknowledgments

This research was conducted under the auspices of TUPREP, the Tulsa University Petroleum Reservoir Exploitation Projects, and it was prepared with financial support from the Coordination for the Improvement of Higher Education Personnel (CAPES) within the Brazilian Ministry of Education.

## Conflict of interest

The authors declare that there is no conflict of interest.

## Nomenclature

β | Unit conversion factor (0.0002637) |

Δpo | Single-phase oil pressure drop (psi) |

λo | Oil mobility (1/cp) |

λt | Total mobility (1/cp) |

λw | Water mobility (1/cp) |

μo | Oil viscosity (cp) |

μw | Water viscosity (cp) |

aw | Water endpoint relative permeability |

Fo | Oil fractional flow |

Fw | Water fractional flow |

fw | Water mobility ratio |

k | Absolute permeability (mD) |

ks | Skin permeability (mD) |

kro | Oil relative permeability (mD) |

krw | Water relative permeability (mD) |

pc | Capillary pressure (psi) |

pi | Reservoir initial pressure (psi) |

po | Oil pressure (psi) |

pt | Pressure threshold (psi) |

pw | Water pressure (psi) |

qo | Oil flow rate (RB/D) |

qt | Total liquid rate (RB/D) |

rs | Shock front position (ft) |

rskin | Skin zone radius (ft) |

rw | Wellbore radius (ft) |

Sw | Water saturation |

Siw | Immobile water saturation |

Sor | Residual oil saturation |

t | Time (h) |

C | Constant given by θqtπhϕ |

D | Shock speed |

h | Reservoir thickness (ft) |

r | Radius (ft) |

w | Traveling wave coordinate |

α | Unit conversion factor (141.2) |

ϵ | Perturbation parameter |

λ | Pore size distribution index |

ϕ | Porosity |

θ | Unit conversion factor (5.6146/24) |

co | Oil compressibility (1/psi) |

cr | Rock compressibility (1/psi) |

cw | Water compressibility (1/psi) |

cto | Single-phase total compressibility (1/psi) |