Analytical, numerical and experimental study of gas coning on horizontal wells

Abstract

Nowadays, new horizontal wells prone to experience gas cone problem usually have multiple production zones. These wells are equipped with inflow control devices (ICDs) or autonomous inflow control devices (AICDs) to mitigate the production of the undesired fluid. In the proposed study, we use analytical calculation, numerical simulations and experimental results to analyze different production strategies when using long horizontal wells in such reservoirs. By the end, we first identified numerical models to represent the dynamic behavior of a gas–oil interface. By validating them through an analytical solution, we used an active control strategy both to obtain and to analyze the associated flow rate for different positions of the interface in steady state from static equilibrium to the well neighborhood. The findings of this research reaffirm there is no strategy that overcomes the production with critical flow rate in steady state. However, the exact critical flow rate is a theoretical value, being hard to assess in the field. In this way, the main contribution of this study is to indicate for the case in consideration the usual production strategy of using multiple zones equipped with AICDs, since it has a cumulative production extremely close to the theoretical maximum and it represents a well-known technology for field applications.

Appendices

Appendix 1: Model equations for the BEM code

1.1 Model equations

Consider a reservoir region close to the well consisting of a homogeneous porous medium with permeability k. This region is filled with two immiscible fluids, one liquid and one gas, each one with different density $\rho _{\mathrm{L,G}}$, dynamic viscosity $\mu _{\mathrm{L,G}}$, and compressibility $c_{\mathrm{L,G}}$. At rest, the fluids are stratified by their density: the bottom layer consists of liquid, while the upper layer consists of gas. Interface between the fluids is considered sharp [16], so under the interface the fluid is composed of $100\%$ liquid and above it, the fluid is $100\%$ gas. Fluid withdrawal occurs permanently on the liquid region at rate Q per unit width. The governing equation of compressible flow in porous media is the Diffusivity Equation:

$$\begin{aligned} \nabla ^2 p = \frac{1}{\eta } \frac{\partial p}{\partial t} + \frac{Q \rho g}{K} \, \delta ({\mathbf {x}} - {\mathbf {x_s}}) \end{aligned}$$

(3)

where

$$\begin{aligned} \eta = \frac{k}{\theta \, \mu \, c_{\mathrm{t}}} \end{aligned}$$

(4)

In Eq. (4), $\theta$ is the average porosity and $c_{\mathrm{t}}$ is the total compressibility. In Eq. (3), $\delta$ is the Dirac delta and $x_{\mathrm{s}}$ is the position of either the source or the sink, i.e., in the coordinates of a sink, the right-hand term is active; in the other points, it is zero.

From Eq. (3), it can be noticed that when the withdrawal dynamics comes to a permanent regimen, the time derivative in the right-hand side comes to zero and the expression becomes

$$\begin{aligned} \nabla ^2 p = \frac{Q \rho g}{K} \, \delta ({\mathbf {x}} - {\mathbf {x_s}}) \end{aligned}$$

(5)

which is known as Poisson equation.

The fluid’s piezometric head is defined as

$$\begin{aligned} \varPhi = \frac{p}{\rho g}+y \end{aligned}$$

(6)

Piezometric head defines a velocity potential function from which the fluid’s velocity can be derived applying Darcy’s Law [18]:

$$\begin{aligned} {\mathbf {u}} = - K \, {\mathbf {\nabla }}\varPhi \end{aligned}$$

(7)

The velocity potential can be rearranged to make the pressure explicit as $p = (\varPhi - y)\rho g$. Substituting this expression in Eq. (5), Poisson equation in terms of the velocity potential is obtained:

$$\begin{aligned} \nabla ^2 \varPhi = \frac{Q}{K} \, \delta ({\mathbf {x}} - {\mathbf {x_s}}) \end{aligned}$$

(8)

In regions on which there are no active sources or sinks, $Q=0$. This leads to the Laplace equation, a particular case of Poisson equation:

$$\begin{aligned} \nabla ^2 \varPhi = 0 \end{aligned}$$

(9)

The use of Poisson equation instead of diffusivity equation is only possible because this study focuses on the particular stage of withdrawal in which the pressure field is totally developed over the considered portion of the reservoir. Since the goal is to determine relations between gravitational and viscous forces in the situation of interface stability, both Poisson and diffusivity equations apply.

In fact, the analytic solution for one-dimensional diffusivity equation (Eq. (3)) in a horizontal slab of porous material with length L along the x-coordinate is given by

$$\begin{aligned} p(x,t) = p_{\mathrm{L}} + \left( p_{\mathrm{R}} - p_{\mathrm{L}}\right) \left[ \frac{x}{L} + \frac{2}{\pi } \sum _{m=1}^{\infty } \frac{1}{m}\hbox {exp}\left( -\frac{n^2 \pi ^2}{L^2} \eta t \right) \sin \left( \frac{m \pi x}{L}\right) \right] \end{aligned}$$

(10)

where $p_{\mathrm{L}}$ represents the pressure on the left side and $p_{\mathrm{R}}$ the pressure on the right side. From this solution, it can be noticed that, when time tends to infinity (in practical terms, when the time is big enough to stabilize the interface), the solution becomes

$$\begin{aligned} p(x,t) = p_{\mathrm{L}} + \left( p_{\mathrm{R}} - p_{\mathrm{L}}\right) \frac{x}{L} \end{aligned}$$

(11)

which is exactly the solution for the Poisson equation in terms of pressure when applied to the same horizontal slab of porous material. This means that, when it comes to stable interface positions, the Poisson equation gives exactly the same result as the diffusivity equation.

It is also important to emphasize that in the considered situations the gas cap expands very slowly, and this expansion plays a particular role on the drive mechanism. This slow gas expansion acts in the way to maintain the pressure over the gas–oil interface and its change of volume is neglected when compared to the simulations and experiments time frame.

1.1.1 Boundary conditions

A section representation of a horizontal well near the reservoir region is shown by Fig. 13.

Fig. 13

Schematic representation of a horizontal well and its neighborhood (two-dimensional view)

In this scheme, 5 boundaries can be seen and they can be classified into 3 categories:

• Boundaries 1 and 5 are impermeable solid walls, in which the normal velocity of the fluid is known to be zero and their potential is unknown.

• Boundaries 2 and 4 are virtual vertical limits (out of the well neighborhood) at a position in which the interface position is no longer affected by the activity of the well on the medium-term range. They establish the limits of the reservoir simulated region. The fluid potential at this wall is known and given in Eq. (6). The fluid normal velocity at these boundaries is unknown.

• Boundary 3 is the moving sharp interface between liquid and gas, in which both potential and velocity are unknown.

The well is positioned at the center of the simulated region (origin of the x-axis) and at an arbitrary height over the y-axis. Mathematically, the above boundary conditions lead to the following set of equations:

$$\begin{aligned} {\texttt {B.C. 1:}}\quad&{\mathbf {u}}_1 \cdot (-\hat{\mathbf {j}}) = 0 \end{aligned}$$

(12)

$$\begin{aligned} {\texttt {B.C. 2:}}\quad&\varPhi _2 = \frac{p_2}{\rho _{\mathrm{L}} g} + y_2 \end{aligned}$$

(13)

$$\begin{aligned} {\texttt {B.C. 4:}}\quad&\varPhi _4 = \frac{p_4}{\rho _{\mathrm{G}} g} + y_4 \end{aligned}$$

(14)

$$\begin{aligned} {\texttt {B.C. 5:}}\quad&{\mathbf {u}}_5 \cdot \hat{\mathbf {j}} = 0 \end{aligned}$$

(15)

where B.C. 3 was left out on purpose due to the fact that it represents the interface boundary condition, which will be now discussed.

Potentials compatibility Since the potential on a fluid depends on its density, it is possible to deduce from Eq. (6) that a potential gap resides in the interface. The pressure field, on the other hand, is continuous throughout the reservoir. Thus, at the interface, $p_{\mathrm{L}} = p_{\mathrm{G}} = p$ and $y_{\mathrm{L}} = y_{\mathrm{G}} = y$. Substituting these values, Eq. (6) leads to:

$$\begin{aligned} (\varPhi _{\mathrm{G}} - y) \rho _{\mathrm{G}} g = (\varPhi _{\mathrm{L}} - y) \rho _{\mathrm{L}} g \end{aligned}$$

(16)

With little effort, Eq. (16) can be rewritten as:

$$\begin{aligned} \varPhi _{\mathrm{G}} - \alpha \varPhi _{\mathrm{L}} = (1 - \alpha ) y \end{aligned}$$

(17)

where

$$\begin{aligned} \alpha = \dfrac{\rho _{\mathrm{L}}}{\rho _{\mathrm{G}}} \end{aligned}$$

(18)

Equation (17) is the potential compatibility equation for an interface between two immiscible fluids. Therefore, for the schematic representation in Fig. 13, it results in:

$$\begin{aligned} {\texttt {B.C. 3 (a):}}\quad \varPhi _{3_{\mathrm{G}}} - \alpha \varPhi _{3_{\mathrm{L}}} = (1 - \alpha ) y_3 \end{aligned}$$

(19)

Velocities balance The interface is an abrupt boundary, where both fluids are always in contact with each other. There is a calculated velocity in the normal direction to the interface. This velocity must have the same magnitude in both sides of the interface. Thus, it is guaranteed that it will not exist either empty spaces or overlaps between the fluids. This leads to the velocities balance equation for an interface between two immiscible fluids:

$$\begin{aligned} {\texttt {B.C. 3 (b):}}\quad {\mathbf {u}}_{3_{\mathrm{G}}} \cdot \hat{\mathbf {n}}_{3_{\mathrm{G}}} = -{\mathbf {u}}_{3_{\mathrm{L}}} \cdot \hat{\mathbf {n}}_{3_{\mathrm{L}}} \end{aligned}$$

(20)

1.2 Boundary element method

The Boundary Element Method presented in [19] was applied in the development of the main simulator used in the numerical analysis of this paper. The implementation is based on the theory presented by Brebbia et al. [20], Brebbia and Dominguez [21], and Banerjee [22]. It follows the guidelines presented by Zhang et al. [8] and Ligget and Liu [18]. This method has the advantage of reducing the analyzed problem in one dimension and decrease the amount of degrees of freedom of the simulated problem.

In the same way that other numerical methods, BEM treats a problem from its differential equations, which in this case is Poisson equation, given in Eq. (8):

$$\begin{aligned} \nabla ^2 \varPhi = \frac{Q}{K} \, \delta ({\mathbf {x}} - {\mathbf {x_s}}) \end{aligned}$$

The peculiarity of this method is that the differential equations are transformed into equivalent integral equations. Once obtained, these equations involving domain integrals are transformed through the Gauss–Green theorem in contour integrals:

$$\begin{aligned} c \, \varPhi (x_d,y_d) = \int _S{\varPhi \, u^* \hbox {d}S} - \int _S{u \, \varPhi ^* \hbox {d}S} \end{aligned}$$

(21)

As a result of this step, the involved variables are evaluated only in the boundaries of the region of integration. Mathematically, the entire domain information is transferred to the boundaries. This is why, in BEM, only the boundary needs to be discretized. Thus, the problem is reduced in one dimension.

In the integral equation, c is a geometric factor with value 1 for points inside the domain and 1/2 for points over the boundary. $x_d$ and $y_d$ are the coordinates of the source point. S is the domain’s boundary. $\varPhi ^*$ and $u^*$ are fundamental solutions to the potential and velocity, respectively, given by:

$$\begin{aligned} \varPhi ^*= & {} - \frac{1}{2 \pi K} \ln (r) \end{aligned}$$

(22)

$$\begin{aligned} u^*= & {} \frac{1}{2\pi r^2}\left[ r_x n_x + r_y n_y \right] \end{aligned}$$

(23)

In the fundamental solutions, $r_x$ and $r_y$ are the components of the distance between source point and field point. $n_x$ and $n_y$ are components of the normal vector, which points outward the S boundary.

The continuous boundary S is discretized into boundary elements. Equation (21) is, then, applied to each of these elements. This generates a system of equations, which is solved, obtaining the potential and the velocity of every element of the boundary and any desired point inside the domain.

Because of the characteristics of boundary-only discretization, BEM is especially advantageous when the problem involves moving boundaries, as when a boundary moves, there is no need to recreate the mesh for the entire domain. Another characteristic that led to the adoption of this method was its capability to easily treat the condition of sharp interface that is assumed by the mathematical model. This easiness resides on the fact that in BEM formulations, the sharp interface is represented by the boundary elements themselves.

1.2.1 Interface shape and evolution

From its horizontal position at rest, the interface shape evolves with time. This evolution is obtained from the velocity of each point that forms the interface between gas and oil. The interface position is calculated in two steps. In the first one, the numerical method calculates the potential gradient and the associated velocity of each material point over the gas–oil interface. The second step is to obtain the new position of these material points by integrating its velocities in time. This is calculated in terms of finite differences, regarding the past position of the interface, the velocity of each point and the time steps:

$$\begin{aligned} y_{m+1} = y_m + \frac{1}{\theta \, \cos \beta } (u_y)_{m} \, \Delta t \end{aligned}$$

(24)

In this equation, y is the y-coordinate of an interface’s point, $\theta$ is the average porosity, $\beta$ is the angle between the boundary’s normal vector and the y-axis, $u_y$ is the vertical component of the interface’s point velocity, $\Delta t$ is the size of the time step, and the subindexes m and $m\!+\!1$ indicates the present time step and the immediately next one.

1.2.2 Transient validation

Aiming at the transient regime validation, another comparison was made observing the returning path of the interface from deformed to undeformed position once the fluid withdrawal stopped.

Results obtained with the numerical simulators were compared with experimental ones, as shown in Fig. 14. It can be seen that the results obtained by the numerical simulators agree with experimental results.

Fig. 14

Interface’s central point position over time when returning to initial condition

As shown in Eq. (11), if the Diffusivity equation is applied until the time at which the diffusion stabilizes, the result at this instant of time is the same as that obtained by the Poisson equation. Therefore, in moving boundary problems, this means that by using small enough time steps, the simulation treats transient regime as a succession of quasi-equilibrium states in which the Poisson equation sufficiently describes the transient regime. This can be done because in moving boundary problems, the successive changes in the boundary position cause successive changes in the boundary conditions which Poisson equation is dealing with. In other words, each small time step of simulation deals with a completely new problem stated by the new position of the interface and the correspondent boundary conditions. This is why, in this particular case, it is possible to simulate the transient regime of a moving boundary problem using the Poisson equation.

Appendix 2: Analytical approach

1.1 Interface shape for a critical extraction rate

From the schematic representation shown in Fig. 13, the well is positioned at the origin of the Cartesian coordinate system, above the solid surface. It produces a total flux Q per unit time.

1.1.1 Governing equation

Equation (9) is the governing equation. Velocities inside the considered domain are given by Darcy’s Law (Eq. 7). Once it is a linear relation and the hydraulic conductivity K is constant, it can be rewritten that

$$\begin{aligned} u = \nabla \varPhi \end{aligned}$$

(25)

if the velocities potential $\varPhi$ is rewritten as

$$\begin{aligned} \varPhi = - \frac{k}{\mu } \left( p + \rho g y \right) \end{aligned}$$

(26)

1.1.2 Boundary conditions

Taking into account that the density of the gas is much smaller than that of the liquid, the pressure at the interface is considered to have a constant value, and this value is assumed to be zero (phreatic interface). Substituting $p=0$ at Eq. (26) leads to

$$\begin{aligned} \varPhi&= -\dfrac{k}{\mu }\Delta \rho g y \end{aligned}$$

(27)

$$\begin{aligned} \varPhi&= - K_0 y \end{aligned}$$

(28)

at the interface, where

$$\begin{aligned} K_0 = \dfrac{k \Delta \rho g}{\mu } \end{aligned}$$

(29)

with the term $\Delta \rho$ representing the difference between the densities of the two immiscible fluids.

Since the flow is incompressible and two-dimensional, one can use the stream function $\psi$, which in terms of the velocity components u and v, in the x and y directions, respectively, is represented by:

$$\begin{aligned} u = \frac{\partial \psi }{\partial y}; \quad v = -\frac{\partial \psi }{\partial x} \end{aligned}$$

(30)

As defined in this way, it is verified that the stream function also satisfies the Laplace equation:

$$\begin{aligned} \nabla ^2 \psi = 0 \end{aligned}$$

(31)

At permanent flow, the interface is stationary and there is no velocity component in the direction normal to it. The velocity vector is always tangent to it. Then, another boundary condition is that in the interface the stream function is constant, i.e., the interface is a streamline. As the expression for $\psi$ is unknown, the shape of the interface is also not known a priori.

The interest here is to find an equation for the shape of the interface. For this purpose, it was used the Hodograph Method. With this mathematical artifice, in simple words, the unknown interface in the physical plane is transformed into known shapes in auxiliary planes. This method is best detailed below.

1.1.3 Hodograph method

In the following, consider that s is the distance, q is the module of the velocity vector along the interface with its respective components u and v along the x- and y-axes, respectively. Since this analysis regards only the steady-state flow, it is known that the speed is always tangential to the interface:

$$\begin{aligned} q = \frac{\partial \varPhi }{\partial s} \end{aligned}$$

(32)

or

$$\begin{aligned} q = -K_0 \frac{\partial y}{\partial s} \end{aligned}$$

(33)

Knowing that $q^2=u^2+v^2$ and that $v=q \partial y/ \partial s$, it can be demonstrated that:

$$\begin{aligned} u^2 + \left( v + \frac{K_0}{2}\right) ^2 = \left( \frac{K_0}{2}\right) ^2 \end{aligned}$$

(34)

Therefore on the hodograph plane, w-plane $(w=u+iv)$, the interface which had an unknown shape in the z-plane, is mapped by a circle which passes through the origin and center point $(0,-K_0/2)$. Due to the symmetry of the problem, it will be considered the only right half of the z-plane, shown in Fig. 15, and the w-plane is shown in Fig. 16.

Fig. 15

The z-plane

Fig. 16

The w-plane

There is a singularity at the point where the sink is located (point A), the flow velocity at this point becomes infinite.

The velocity potential and the stream function of a two-dimensional potential flow $\varPhi$ and $\psi ,$ respectively, are related as follows [23]:

$$\begin{aligned} u&= \frac{\partial \varPhi }{\partial x} = \frac{\partial \psi }{\partial y} \end{aligned}$$

(35)

$$\begin{aligned} u&= \frac{\partial \varPhi }{\partial y} = -\frac{\partial \psi }{\partial x} \end{aligned}$$

(36)

Showing that $\varPhi$ and $\psi$ of a two-dimensional potential flow satisfies the Cauchy–Riemann equations. Moreover, both $\varPhi$ and $\psi$ satisfies their Laplaces equations.

Also with regards to these potentials, it is known that the interface represents a streamline for the steady flow. Thus, one can form a complex variable $f = \varPhi + i\psi$, called complex potential, where the interface has a constant value for the stream function. Therefore, it is possible to represent a different plane, called f-plane, shown in Fig. 17. The distance between two streamlines represents a flow rate. In this case, the distance between the line that represents the interface and the line that represents the impermeable boundary is Q / 2, because the flow rate is the same on both sides of the considered domain.

Fig. 17

The f-plane

Differentiating the complex potential f in relation to z:

$$\begin{aligned} \dfrac{df}{dz}&= \dfrac{\partial \varPhi }{\partial x} + i \dfrac{\partial \psi }{\partial x} = \dfrac{\partial \psi }{\partial y} - i \dfrac{\partial \varPhi }{\partial y} \end{aligned}$$

(37)

$$\begin{aligned} w&= \frac{df}{dz} = u - iv \end{aligned}$$

(38)

where w is the complex velocity [23].

Note that Eq. (38) represents the conjugate of the fluid velocity in the z-plane and is useful to determine velocity of the flow at any point. The hodograph plan is also useful to show that the maximum local velocity (magnitude) at the interface is $K_0$ and that this vector has its direction only along the y-axis, as it is clearly shown in Fig. 17. So, according to Eq. (34):

$$\begin{aligned} \left| V_{\mathrm{max}}\right| = \frac{k \left( \rho _{\mathrm{L}} - \rho _{\mathrm{G}}\right) g}{\mu _{\mathrm{L}}} = \frac{k \Delta \rho g}{\mu _{\mathrm{L}}} \end{aligned}$$

(39)

At first, the problem can be solved, however, is still difficult to work with the circle in the w-plane. Hence it is necessary to find a conformal transformation to map the interface, represented by the circle in the w-plane, into a straight line in another plane, which will be called $\zeta$-plane $(\zeta = \eta + i\xi )$. For this particular case, the inverse transformation [24] is used:

$$\begin{aligned} \zeta = \frac{1}{w} \end{aligned}$$

(40)

It may be noted that the transformation specifies the relationships:

$$\begin{aligned} \eta&= \frac{u}{u^2 + v^2} \quad \xi = - \frac{v}{u^2 + v^2} \end{aligned}$$

(41)

$$\begin{aligned} u&= \frac{\eta }{\eta ^2 + \xi ^2} \quad v = - \frac{\xi }{\eta ^2 + \xi ^2} \end{aligned}$$

(42)

Then, through this conformal transformation, the interface that is circle in the w-plane is mapped in a known straight line in the $\zeta$-plane, as shown in Fig. 18.

Fig. 18

The $\zeta$-plane

Fig. 19

The t-plane

As the flow region of both $\zeta$-plane and f-plane are bounded by a polygon, the Schwarz–Christoffel transformation can be used [24]. Considering the polygon located in the $\zeta$-plane, semi-infinite strip FDAC of width $1/K_0$, the Schwarz–Christoffel transformation has the general formula [24]:

$$\begin{aligned} \zeta = M \int \frac{\hbox {d}t}{\left( t-t_F\right) ^{1-\frac{\zeta _F}{\pi }} \left( t-t_D\right) ^{1-\frac{\zeta _D}{\pi }} \left( t-t_A\right) ^{1-\frac{\zeta _A}{\pi }} \left( t-t_{\mathrm{C}}\right) ^{1-\frac{\zeta _{\mathrm{C}}}{\pi }}} + N \end{aligned}$$

(43)

where M and N are complex constants, $\zeta _F$, $\zeta _D$, $\zeta _A$ and $\zeta _{\mathrm{C}}$ are the internal angles in radians, of the respective vertices F, D, A and C of the polygon in the $\zeta$-plane, and $t_F$, $t_D$, $t_A$ and $t_{\mathrm{C}}$ are the points of the r axis in the t-plane (Fig. 19) corresponding to the respective vertices.

Making the appropriate substitutions in Eq. (43) and determining the constants by the coordinates of the points in the $\zeta$-plane and in the t-plane, the required transformation is:

$$\begin{aligned} \zeta = \frac{1}{K_0 \pi }\cosh ^{-1}(t) \end{aligned}$$

(44)

Now considering the polygon located in the f-plane, the Schwarz–Christoffel transformation can also be used for conformal mapping on the upper half t-plane. Making the appropriate substitutions in Eq. (43) and determining the constants by the coordinates of the points in the f-plane and in the t-plane, the required transformation is:

$$\begin{aligned} f = \frac{Q}{2\pi }ln \left( \frac{t-1}{2}\right) \end{aligned}$$

(45)

Once those transformations have been determined, z can be expressed in terms of t:

$$\begin{aligned} \hbox {d}z = \frac{Q}{2 K_0 \pi ^2}\frac{\cosh ^{-1}(t)}{(t-1)}\hbox {d}t \end{aligned}$$

(46)

Once the conformal mapping transformations of the f-plane and of the $\zeta$-plane into the z-plane were found, Eq. (46) can be integrated to give the shape of the interface on physical plane. This is the essence of the hodograph method.

The shape of the interface ${\mathop {DF}\limits ^{-}}$ is obtained by integrating Eq. (46) for $-\infty \le t\le -1$, since this is the range that represents the interface in the t-plane. However, as the integral of Eq. (46) does not converge in this interval, the indefinite integral of Eq. (46) was calculated and the result plotted in the same range. Therefore, the following equation is obtained:

$$\begin{aligned} \frac{z K_0}{Q} = \frac{\cosh ^{-1}(t)\left[ \cosh ^{-1}(t) + 4 ln \left( 1 - e^{-\cosh ^{-1}(t)}\right) \right] -4Li_2\left( e^{-\cosh ^{-1}(t)}\right) }{4\pi ^2} + \frac{1}{6} \end{aligned}$$

(47)

where $Li_2$ is the second order poly-logarithm function [25].

Note that the result of the integral has real and imaginary parts. A parametrization to obtain the shape of the interface was necessary. Figure 20 shows the calculated interface on the z-plane, with the axes normalized by the factor $Q/K_0$.

Fig. 20

Calculated interface on the z-plane