Equivalent Radius for Well Inflow Calculations at Different Regimes in Reservoir Flow Simulations

A problem of well representation in numerical simulation of fluid flows in porous media is considered. When modelling hydrocarbon reservoirs, underground gas storages and other natural porous media, there’s a large typical difference in scales of wells (about 10 cm) and inter-well distances (from hundreds to thousands meters), so wells aren’t resolved on the grid as explicit boundaries in large-scale reservoir simulations. Instead, wells are represented by sources/sinks. The problem is that the average pressure in a grid block containing the well (well block pressure, WBP) isn’t equal to the well pressure (bottomhole pressure, BHP). D. Peaceman showed that WBP can be interpreted as the pressure at equivalent radius inside the well grid block. He derived the classical formula for the equivalent radius for steady-state (SS) Darcy flow. Though Peaceman showed it could be used for pseudo steady-state (PSS) flow as well, his derivation for PSS was not accurate. In this paper we compute the equivalent radius for PSS and boundary dominated (BD) regimes of Darcy flow and examine the impact of flow regime and distance to external boundary (or inter-well distance) on it. The results are important for accurate well representation in numerical flow simulations of underground reservoirs.


Introduction
Within the problems of numerical simulation of fluid flows in porous media, the correct treatment of wells is of particular interest.The reservoir model is represented by a grid with a certain size.Typical well radius is about 10 -1 m, while the size of a grid block is ~10 1 -10 2 m.Because of this difference in scale, wells are represented by sources or sinks (for injection or production well respectively) located in the penetrated grid blocks.The pressure in such a block (well block pressure, WBP) is significantly different from the real pressure on the well (the bottomhole pressure, BHP).So this pressure difference causes an inflow/outflow to/from the grid block, and a correct model is required to link the well flow rate to this pressure difference.
The problem of transition from WBP to BHP for an oil well was originally considered in the USSR.Those studies were carried out in relation with the solution of two-dimensional flow problems on electrical integrators with RC (resistance-capacitance) grids [1].Until 1977, the most significant study was the one of van Pullen et al. [2].They stated that the WBP should be the areally-averaged pressure in the part of the reservoir represented by the grid block.In 1978, D. Peaceman introduced the concept of the equivalent well radius (Peaceman radius), and also presented the correct interpretation of the WBP and showed how it relates to the current BHP [3].Peaceman showed that for most blocks, the association of pressure in them with the average pressure of the block is appropriate.However, for a block containing a well the pressure in the block is not the average pressure.
In his famous paper [3], D. Peaceman derived the formula for the equivalent radius for steady-state (SS) flow.Though Peaceman showed it could be used for pseudo steady-state (PSS) flow as well, his derivation for this regime was not accurate.In this paper we compute the equivalent radius for PSS and boundary dominated (BD) regimes of the linear Darcy flow and examine the effect of flow regime and distance to external boundary (or inter-well distance).It should be noted that earlier we derived the refined formula for Peaceman radius for the SS nonlinear Forchheimer flow [4].

Peaceman radius for the two flow regimes
Let us revise the formula for equivalent radius for the PSS flow and derive it for the BD flow.Just as in Peaceman's paper [3], the linear Darcy flow of a homogeneous fluid in an isotropic porous medium is considered.
Let's consider inflow to the well with boundary Г w in an isolated domain  of volume  with boundary  = Г e ∪ Г w .The well is located in the center 0 of a numerical finite difference grid with the five-point stencil (figure 1).The pressures in the blocks and the well flow rate can be related by the material balance equation, which is the discrete form of the continuity equation.For unsteady flow of a slightly compressible fluid in the block of size Δ × Δ × ℎ with pressure  0 and volume  0 = Δ 2 • ℎ, containing the well with flow rate , the material balance equation is: where  =   ,  is the reservoir permeability,  is the fluid viscosity,  1 () is the pressure at time  in the laterally adjacent blocks to the well block, ℎ is the reservoir thickness,  is the porosity,  p is the total compressibility,  is the time step.

Pseudo steady-state flow (PSS)
Suppose that the flow corresponds to the PSS regime for the slightly compressible fluid, so the pressure change at each point is determined by the compressibility  p and the volume of the domain .The material balance equation (1) will take the form: where  is assumed constant in time and the time-derivative in the right side of (1) was replaced by its PSS expression (see below).Consider a non-stationary two-dimensional radial inflow of a slightly compressible fluid to the well of radius  w with the given flow rate in an isolated circular reservoir of radius  e : where  ̃=  ℎ .PSS-solution of the problem above in the radial axisymmetric case can be expressed as where () is the solution of the stationary problem: From ( 2) and (6) it follows that to obtain Peaceman radius it is enough to find  0 such that Constituting the axisymmetric solution of the problem (7)-( 9) to (10), we arrive at the following equation for  0 : Equation ( 11) allows to determine the accurate value of Peaceman radius  0 for the PSS regime.
Since the flow domain is the ring with the external radius  e , we find that for  e → ∞ ( e ≫  0 ) the PSS Peaceman radius will equal to Thus, for a sufficiently large flow domain Peaceman radius for PSS is equivalent to Peaceman radius for SS.However, in general this is not the case.Note that the conclusion of Peaceman about the applicability of formula (12) for the PSS case also implicitly assumed a large size of the flow domain.Namely, he relied on the solution for pressure in an unbounded reservoir [3].

Boundary dominated flow (BD)
Now consider the boundary dominated regime of the fluid flow.The material balance equation is the same as in the PSS case, but taking into account the change in the flow rate over time, namely: The initial boundary value problem for the BD flow regime can be defined as = (0,  w ,  e ) = (0,  e )\(0,  w ), Due to linearity, for simplicity, we can assume  w = 0.The solution of the problem ( 14)-( 16) can be presented in the following form: where  0 () is the first eigenfunction and  0 is the first eigenvalue of the following problem in the domain : 0 () Based on the physical meaning of (17), the eigenfunction must be positive.Considering this, we can get  0 () from the problem (18)-(20):  0 () =  0 (√ 0  w ) 0 (√ 0 ) −  0 (√ 0  w ). 0 (√ 0 ), where λ 0 is found from equation Substituting the expression (23) into the material balance equation (13) and using Darcy law to express the flow rate from pressure gradient, we get the following equation for  0 : 0 (√ 0  w ). 0 (√ 0  0 )] • (e It can be shown that in this case  0 ~e −2 → 0 (see [5]).Using asymptotics for the Bessel functions and the arguments, we obtain the following expression: The formula (26) matches the expression (12), which means that Peaceman radius for the BD flow regime is exactly the same as for the SS and PSS flow regimes, but only in the case of sufficiently remote external boundary ( e → ∞).

Calculations
To study the behavior of Peaceman radius for the PSS and BD flow at different values of the external radius  e , we will calculate  0 for these regimes from equations ( 11) and (24).Note that  e can reflect either physical reservoir boundary or virtual no-flow boundary caused by interference of wells.
Let the well radius  w be fixed and equal to 0.1 m.For the PSS regime, finding  0 from (11) at different values of the grid block size Δ and external radius  e is not difficult.We calculated  0 for the values of Δ and  e such that the ratio  e /Δ was in the range from 2 to 50.The results are presented in table 1.Some cells are empty because the value of  e /Δ isn't in the specified range.Based on the results, we see that the ratio  0 /Δ for the PSS flow regime is practically close to the SS value of e −/2 ≈ 0.207879 and tends to it as  e /Δ is increased.This trend is clearly seen from figure 2.
For the BD flow regime, in order to compute the values of Peaceman radius we need to perform some preliminary calculations.Note that there is no exterior radius  e in the equation (24) in explicit form, but there is  0 -the first eigenvalue -which can be found from the equation ( 22) for each pair of values  e and  w .Since the radius of the well was fixed, let's find  0 for some values of the external radius.As an example, graphic solution of the equation ( 22) for  e = 1000 m is presented in figure 3. The first eigenfunction for this value is presented in figure 4.
After finding the values of  0 for each  e , we can construct the graph of  0 versus  e (figure 5).The obtained result confirms that  0 → 0 as  e → ∞.Now we can calculate  0 values for the BD regime similar to the PSS, but using equation (24).The results are presented in table 2.
Based on the results, we can say that the values of  0 /Δ for the BD flow regime are also close to e −/2 and, as for the PSS regime, tend to it.The trend is shown in figure 6.
An interesting picture is seen if we combine the Figures 2 and 6 -figure 7. The both plots, for PSS and BD, tend to the constant SS value of e −/2 .But if the PSS plot tends to this asymptote from above, the BD one -from below.We believe this reflects principal difference of these flows as the effect of different boundary conditions.For the case of sufficiently remote external boundary, we derived theoretically and confirmed numerically that  0 values for the PSS and BD regimes tend to the classical steady-state value.Moreover, our calculations showed that for practical applications the classical formula can be used in the whole range of possible  e values with more than acceptable accuracy.
The results of the study introduce some fundamental background into the engineering practice for treatment of wells at different flow regimes in numerical reservoir flow simulations.

Figure 1 .
Figure 1.Schematic of the five-point stencil on a quadratic grid with the well located in the center grid block 0.  w is the well radius,  0 is the equivalent (well block) radius, ∆ is the grid block size.

Table 1 .
0 /Δ values for a pair of values  e and Δ for the PSS flow regime

Table 2 .
0 /Δ values for a pair of values  e and Δ for the BD flow regime