1. Introduction
Natural resources such as water, hydrocarbons and geothermal heat occurring in subsurface reservoirs are massively exploited by humans for a multitude of industrial and agricultural purposes [
1,
2,
3]. Similarly, depleted reservoirs are widely used for seasonal storage of natural gas, hydrogen and liquids, as well as for permanent sequestration of carbon dioxide, CO
2 [
4,
5]. The effective engineering of fluid withdrawal from, and plume movements of fluid injected into, porous rock formations requires practical expressions that describe the pressure changes in the reservoir [
5,
6]. This is needed, a.o., to prevent formation damage and potential leakage from occurring when the CO
2 pressure exceeds the seal failure pressure [
7].
The prior scientific literature records a progressive development of insight, with solutions for fluid withdrawal rates via wells from subsurface reservoirs. The Fourier solution of the diffusivity equation for heat [
8] was modified and scaled by Fick [
9] for molecular diffusion, and then (in an indirect way) to pressure diffusion problems by Darcy [
10]. From this general insight, specific solutions of well behavior for fluid withdrawal from porous reservoirs in the subsurface were formulated by Dupuit [
11] and Thiem Sr. [
12], and were tested with field data by Thiem Jr. [
13]. Drawing analogy from heat diffusion solutions of Carlslaw [
14], Theis [
15] gave analytical solutions for the pressure drawdown profile around a well in an aquifer.
The prior work cited so far on the theoretical well performance was commonly focusing on wells sourcing water from a subsurface aquifer. Next, a modified Dupuit–Thiem equation was developed for application in hydrocarbon well test modeling by Raghavan [
16] and Zimmerman [
17,
18], largely modeled after a textbook solution of heat diffusion equations by Jaeger and Carslaw [
19]. The analytical solution of the diffusivity equation based on a constant well rate boundary condition is widely used in practical well tests assessing flow rates in newly drilled exploration wells [
20,
21]. The aim of well testing is to establish how fast the discovered hydrocarbons can be produced, and if the fluid production rate of the exploration well can meet the required economic threshold for additional wells to be drilled in order to validate the full-scale development of costly surface installations.
There have been recent advancements in both numerical and analytical frameworks aimed at elucidating the transport dynamics of hydrocarbon fluids [
22,
23,
24] and delineating the movement of CO
2 plumes during both injection and post-injection phases in subsurface aquifer storage [
5,
25]. In the physical entrapment of CO
2, the dominant transport mechanisms include horizontal plume migration at rates controlled by viscous forces resulting from pressure gradients. Subsequently, buoyancy and capillary forces play crucial roles in post-injection trapping when the upward migration of CO
2 may occur [
5]. Sbai and Azaroual [
25] utilized numerical methods to predict potential formation damage caused by CO
2 injection. Weijermars [
22,
23] introduced a novel Gaussian pressure-transient-based solution to quantify the spatial and temporal progression of pressure and the associated fluid depletion, both in unfractured and fractured wells in unconventional reservoirs. Okoroafor et al. [
24] utilized numerical methods to simulate hydrogen injection, storage and recovery from both depleted methane gas reservoirs and saline aquifers.
The present study derives new analytical solutions for fluid injection, storage and withdrawal via a vertical wellbore in a conventional reservoir from a horizontal, permeable reservoir, assuming a constant bottomhole pressure. Such a solution did not exist until now. The diffusivity equation was revisited, and the well rate equations were derived for a constant bottomhole pressure boundary condition; such a situation applies to many oil and gas wells produced with artificial lift systems (using a pump which imposes a certain constant bottomhole pressure). The new method does not require an elaborate time-stepped pressure-matching procedure as in nodal analysis [
26,
27,
28,
29], the only other physics-based analytical method currently available to compute the well rate decline when a constant bottomhole pressure system is used.
This research paper is structured as follows.
Section 2 provides a historical review and details the workflow used in this study; after first tracing the evolution of analytical methods from modeling basic groundwater behavior to describing complex hydrocarbon flow behavior in oil and gas reservoirs.
Section 3 revisits the governing diffusion equation [
30] and presents pressure transient solutions applicable to vertical wells with artificial lift in reservoirs with no lateral boundaries. Wells produced with artificial lift have a constant bottomhole pressure and are comprehensively modeled in the present study.
Section 4 presents pressure gradient solutions for such wells.
Section 5 explores well rate solutions, including a complementary solution for a vertical well in a bounded reservoir, illustrated with case studies using real production data from two wells (UK offshore gas fields), which are history-matched with the new method using specific reservoir and well conditions.
2. Review of Prior Solutions
This section lays the groundwork for the present study. We begin by introducing foundational solutions for modeling how water table levels decline in aquifers (drawdown) in
Section 2.1. We then transition to the pivotal pressure transient test equation used in oil and gas well analysis (
Section 2.2). In
Section 2.3, we present the crucial assumption of steady-state well rates that underpins the traditional well rate solution. Finally, considering the limitations of prior research, we restate the objective and workflow of this study in
Section 2.4.
2.1. Hydraulic Head Drawdown
All prior solutions governing the injectivity and productivity of water and petroleum wells have assumed a constant well rate. Landmark solutions for practical use include the Thiem [
12,
13] solution, the Theis [
15] solution and the pressure transient solution of the diffusion equation by Carslaw and Jaeger [
19]. Thiem [
13] provided a practical equation based on earlier work by Dupuit [
11], which relates the constant well rate,
, to the respective water table drawdown to heights (
,
) in nearby observation wells at radial distance (
,
) from a well in the origin with original water table height,
, in the aquifer of study as follows:
The parameter to be determined from the measurements in the observation well(s) is the hydraulic conductivity,
, of the aquifer. The solution of Equation (1) is based on comparing the mass balance of fluid flow using Darcy’s Law [
30,
31,
32].
Separately, Theis [
15] gave a continuous solution for the drawdown profile of the fluid surface in aquifers, based on the 2D diffusion solution for heat dissipation as given in Carslaw [
14]. The change in the hydraulic head,
, in radial direction around the well in the origin is given by the exponential integral
[
33] as follows:
The well rate in Equation (2) is again assumed constant, and the rate of drawdown of the hydraulic head is controlled by the aquifer transmissivity,
T, and storativity,
S. The practical purpose of Theis’s [
15] solution is that it solves for the transient change in hydraulic head due to a well pumping at a constant rate from a confined aquifer [
34].
2.2. Well Test Equation
In petroleum engineering, new oil and gas reservoirs are tested in exploration wells over relatively short flow periods to measure what well rate can be achieved in response to a pressure differential between the reservoir and the wellbore. This solution is again based on Darcy’s Law, and the assumed boundary condition is a steady flow in a porous media, at rate
:
Equation (3) features the fluid viscosity, , and reservoir permeability, ; both are initially assumed to be constant and uniform. Note that [L T−1] in Equation (3) is used here for Darcy flux from the porous formation into the open space of the wellbore across a unit area of wellbore exposed; should not be confused with the dimensionless well rate and is distinct from the volumetric flux into the well, [L3 T−1] in Equation (2). The actual velocity, , in the porous reservoir is always greater than the Darcy flux and will be given by = , with representing the connected porosity.
The associated steady-state pressure profile
for radial flow near the well producing at a constant rate,
, with original reservoir pressure,
, is as follows [
18]:
where
is payzone thickness, and
is wellbore radius. The formation volume factor,
, accounts for the volume change in the fluid mostly due to drawdown pressure when the fluid withdrawn from the reservoir is transferred to the surface. The solution of Equation (4) itself does not account for friction losses in the wellbore’s vertical section, the exclusion of which is a routine practice in well test equations. Such friction may be considered in a more comprehensive analysis of the later production system, which can account for the wellbore friction if critically needed. Provide the well flows, the solution of Equation (4) is correct, even if the wellbore fiction I left out of the equation, because the dynamics is computed at the reservoir level as controlled by Darcy’s law, with
specified for the reservoir level.
The radial flow solution of Equation (4) is termed a ‘modified’ Dupuit–Thiem equation for the steady-state pressure profile, because the original equations by Dupuit [
11], Thiem Sr. [
12] and (his son) Thiem Jr. [
13] do not include the formation volume factor; for hydrological problems in shallow aquifers,
indeed can be safely assumed close to unity.
2.3. Steady-State Pressure Profile
The assumption of well production at steady state underlying Equation (4) is useful, because this way a steady-state pressure profile may exist, where the reservoir pressure increases logarithmically with distance to the well to converge to the original reservoir pressure
at a certain distance from the well. The Dupuit–Thiem equation describes the steady-state pressure profile and still features centrally in today’s well-testing literature [
20], with small modifications such as the skin factor,
s, a fidgeting factor which accounts for near-wellbore damage or stimulation of the reservoir:
The steady well rate and constant pressure profile assumptions are only physically reasonable for reservoirs produced with pressure replenishment by balanced water injection [
35]. When no fluid replenishment occurs, the pressure profile around the wells will be transient and a function of the hydraulic diffusivity [
19]:
The purpose of most well tests is primarily to measure the rate of the well for an imposed pressure drop at the wellbore, where
. Assuming the well rate
stays constant, reservoir parameters can be estimated by coupling the Zimmerman [
18] hydraulic diffusivity,
, expression (Equation (7)) into Equation (6).
is simultaneously estimated by constraining one or more of the primary values that jointly control its value, such as the intrinsic properties of the reservoir (,,) and of the fluid () in its pores of total compressibility, , made up of rock compressibility, , and fluid compressibility, . Representative values of , ,, can be obtained from lab tests on sampled reservoir fluids and core test data, as routinely conducted by oil companies during field appraisal programs. Alternatively, the value of can be abstracted from history matching of well rates as demonstrated later in this study.
2.4. Study Objective and Research Workflow
The prior mathematical solutions cited above all assumed a vertical well pumping at a constant rate from a horizontal aquifer confined between an upper and a lower impervious boundary, with infinite lateral extent, uniform thickness, homogeneous conductivity, storativity and transmissivity. In the analyses by Thiem [
13] and Theis [
15], the diameter of the well was assumed to be negligibly small so that fluid storage in the well may be neglected.
In order to compute the well rate decline when a constant bottomhole pressure production system is used, one must currently use an elaborate time-stepped pressure-matching procedure called nodal analysis [
26,
27,
28,
29], which follows a history-matching procedure based on analytical solutions for transient and boundary dominated flow. The procedure is reasonably accurate but elaborate and technically complex due to the need for time-stepped history matching [
36]. The benefits and limitations of the nodal analysis procedure are acknowledged in this present study, but the procedure is not further detailed.
Figure 1 visually summarizes the key steps of our research workflow. It begins with the governing partial differential equation (PDE) that incorporates an hydraulic diffusivity term. Next, we show how reservoir and well boundary conditions are applied to this equation. The present study focuses on vertical wells in the transient-flow regime of reservoirs without lateral boundaries (unbounded) and with boundary-dominated flow effect [
37].
Figure 1 illustrates how subsequent steps lead to the development of the proposed analytical solutions for well performance, starting from the transient pressure field solutions of the diffusivity equation. The pressure transient solutions in laterally unbounded reservoirs are provided in
Section 3 of the present study.
3. Diffusion of Pressure Transients in Unbounded Reservoirs
This section focuses on pressure field solutions, which are the basis for the subsequent pressure gradient solutions’ development in
Section 4, and is structured as follows. The classical diffusivity equation is revisited in
Section 3.1. Many oil and gas wells are produced with artificial lift systems (using a pump) which impose a certain constant bottomhole pressure. Thus, the pressure transient diffusion solution is first derived for a constant bottomhole pressure change in vertical well systems in laterally unbounded reservoirs (
Section 3.2). one- and two-dimensional pressure diffusion solutions are provided. We further distinguish between producer wells and injection wells.
3.1. Governing Diffusion Equation
Consider the classical diffusion equation [
38], adapted for pressure diffusion in three-dimensional space:
The first Del operator
in Equation (8) takes the dot product or divergence of
[MT
−3], and
denotes the pressure gradient. If normalized pressure is initiated at initial time, and gravity gradients are negligible, the spatial changes in
are governed by Equation (8) as would apply to a perfectly homogenous, isotropic porous medium with slow, laminar flow due to the pressure gradient (
Figure 2a).
For a free flow of liquid media, inertia and gravity force terms will quickly dominate the spatial changes in the pressure field, which is why the pressure gradients of free flow with turbulence may quickly appear extremely complex (
Figure 2b). The pressure diffusion Equation (8) does not account for such additional terms. Also, only diffusion mass transfer due to the pressure gradient is considered. If molecular diffusion is included as well, the relative importance of the convective mass transfer due to the pressure gradient and that due to molecular diffusion can be described by a Fokker–Planck (convection–diffusion) equation [
23,
39]. A Gaussian Péclet number has been proposed to compare the relative importance of the two processes for application in gas reservoir/well dynamics [
40].
Next, assume the pressure diffusion process of interest occurs radially outward from a cylindrical well source placed vertically in a tabular homogeneous reservoir with an impervious upper and lower boundary (
Figure 3) and at a certain initial pressure,
. The pressure changes
occur exclusively in the (
x,
y) plane, due to the no-flow condition at the upper and lower boundary. Specifically, 2D diffusion starting at the well is governed by:
where
is modeled according to Zimmerman’s [
18] hydraulic diffusivity equation (Equation (7)). The simplified PDE expression in Equation (9) is for an unbounded reservoir, meaning no lateral boundaries occur in the radial direction away from the well.
3.2. Steady Pressure Transient Diffusion (1D) for Production and Injection Wells in an Unbounded Reservoir
The key differences between injection and production wells are first established prior to presenting the solutions for such possible applications.
Figure 4 illustrates a schematic of both types alongside their bottomhole pressures relative to the initial reservoir pressure. This helps visualize the distinct well designs used for injection and production purposes.
Injection wells: These wells have a higher bottomhole pressure than the original reservoir pressure. This pushes fluids outward away from the well landing point, following the pressure gradient, when injected into the formation.
Production wells: In contrast, production wells have a lower bottomhole pressure compared to the initial reservoir pressure. This pressure difference draws fluids inward towards the wellbore, enabling their extraction, and the drainage patterns enlarges as time goes.
For injection wells, we strive for a boundary condition where the injection pressure is initiated at
with a value
(
Figure 4), and for
t > 0, the source maintains a constant pressure differential
, given by:
For production wells, the injection pressure is initiated at
with a value
(
Figure 4), and for
t > 0, the source maintains a constant pressure differential
, given by:
We also distinguish between well systems based on types of technical completion: wells drilled vertically and completed with a single longitudinal fracture (bi-wing fracture,
Figure 5a) and those completed with simply a vertical, open-hole wellbore (
Figure 5b). The bi-wing fractured wells are commonly used in low-permeability reservoirs; their pressure transient is governed by 1D diffusion from the fracture plane. The open wellbore without fracturing is used in high-permeability reservoirs; such wells act as 2D, cylindrical diffusion sources for the pressure transient. Studying the 1D and 2D cases jointly is instructive, in particular because we show in the present study how the 1D diffusion solution can be transformed to a 2D diffusion solution.
3.3. Fractured Well Solution (1D Diffusion)
The pressure change component
in the
x-direction due to a planar, 1D diffusion source along the
y-axis (
Figure 5a) is as follows:
A valid 1D solution of Equation (12) with the boundary condition of Equation (10) was given by ([
38], Equation 2.45):
Outside the well system, the pressure transient will change the reservoir pressure according to:
The time-dependent part of Equation (14) is given by Equation (13), which after normalizing gives the non-dimensional pressure change function [using for ] at any time and given position in the reservoir as follows.
For the production well case, the initial reservoir pressure at the well location is changed by the well according to a step function, which lowers the pressure of the reservoir at
with a value
; for 1D diffusion from a production well with a single fracture. The equation for the normalized pressure transient that travels perpendicular to the vertical fracture plane acting as the pressure change source is (
Figure 6a):
For a 1D diffusion from an injection source, the step function of the initial pressure increase in the well advances in the lateral reservoir space (
x-direction) as follows (
Figure 6b):
Figure 6a shows curves of the pressure transient progressively advancing across the reservoir space for time steps separated by 10 time units for the 1D diffusion of a hydraulically fractured
production well, using Equation (15). The top border of the graph (
y = 0) is where the original reservoir pressure,
, remains unchanged. However, the non-dimensional pressure change initiated from the production well advances into the reservoir space and results in progressive pressure depletion associated with the fluid withdrawal process. Ultimately, after infinite time, the pressure everywhere in the reservoir will attain the bottomhole pressure,
, which for the production case is given by the non-dimensional pressure change maximum at the bottom border of the graph (at
y = −1) of
Figure 6a.
Figure 6b shows the advance of the pressure change profiles at different times for the 1D diffusion of a fractured
injection well, according to Equation (16). In this case, the bottom border of the graph (with
y = 0) is where the original reservoir pressure,
, remains unchanged. However, at a particular time, say
t = 0.1 (depicted with yellow line in
Figure 6b), the reservoir pressure has changed close to the wellbore, but asymptotically, no pressure change has occurred (yet) further away from the well. For the injector case, the pressures near the well location are increased, and the imposed pressure changes dissipate into the reservoir space as graphed in
Figure 6b. Ultimately, after infinite time, the pressure everywhere in the reservoir will reach the bottomhole pressure,
, which is given by the normalized pressure change top border of the graph (at
y = 1) of
Figure 6b.
3.4. Cylindrical Well Solution (2D Diffusion)
The advancing pressure change and normalized pressure depletion profiles for uni-directional 1D diffusion were given in
Figure 6a,b. If we were to obtain the 2D solution (
Figure 7b) by an orthogonal superposition of 1D pressure change sources (
Figure 7a), it would give radial pressure gradients useful for flow path mapping. However, the rate of advance of the 2D diffusion process would be inaccurately scaled by the 1D diffusion process (as in
Figure 6a,b), due to which the divergence of the flow paths and the required conservation of mass during the mass transport process will not be satisfied.
In order to correct for the divergence deficit, we apply a coordinate transformation of the 1D diffusion space to a 2D cylindrical diffusion space. The required transformation of the Cartesian reference system to a polar reference system can be achieved by conformal mapping (
Figure 8), such that the normalized spatial coordinate
x in domain
ε(
x) is mapped to ln
r in the domain
ε(
x,
y):
For the production case, the solution of Equation (15) becomes (
Figure 9a):
For the injector case, the solution of Equation (16) becomes (
Figure 9b):
It is important to note that the inclusion of “+1” in the numerator of Equations (18) and (19) serves to accurately represent the wellbore as a point source at
r = 0. However, in practical wellbore applications, the spatial distance between the wellbore and the boundary typically precludes the necessity of this correction. To plot
Figure 6a,b, no inputs are required other than normalized pressure change at the wellbore location and assuming a unit diffusivity. In
Section 5, we will show how the well rate declines as a consequence of the lowered bottomhole pressure for a production well case with a specific dimensional hydraulic diffusivity. Unlike classical well test analysis, the well rate is not a boundary condition but an output of the applied bottomhole pressure, original reservoir pressure and hydraulic diffusivity (and well radius, which is assumed to be negligible in
Figure 6a,b). The dimensional example of well rate computation and production forecasting using real field data from the early well life is included in
Section 5.4.
Comparison of
Figure 6a,b and
Figure 9a,b reveals the slower advance of the pressure gradient due to the cylindrical source assumed as a consequence of the applied transformation.
Equations (18) and (19) are valid solutions of the general diffusivity equation in polar coordinates (Equation (20)):
6. Discussion
6.1. Importance of Well Rate Forecasting Methods
Accurate production forecasts and reserve estimations are particularly important for the global petroleum industry in order for individual companies to estimate their remaining hydrocarbon reserves which underpin the book value of their fixed assets in the balance sheet and serves as collateral for credit lines. Reporting of the proved reserves (P90 certainty) is mandatory as per the guidelines of the US Security and Exchange Commission (SEC) and applies to all petroleum companies listed on the New York Stock Exchange (NYSE).
Two types of practical tools for well rate forecasting are widely used.
One type of tool is the commercial reservoir simulators (GMC, Kappa, Eclipse, etc.), of which many different varieties are offered by service companies, which provide slick interfaces with ready-to-use input boxes. The actual computations are invariably based on numerical, finite difference computational solution methods. While the results are reasonably accurate, the user licenses for the commercial simulators are costly, the methods and assumptions involved are non-transparent and filling out the detailed well data in the user interface requires quite a bit of practical experience. Also, the runtime to obtain output may take several hours, especially when probabilistic determination of reserve categories is required.
The other principal type of tools popular for determining hydrocarbon well performance and related reserves reporting is based on empirical decline curve analysis. Numerous decline curve analysis (DCA) formulas have been proposed in the past, as reviewed in considerable detail elsewhere [
48,
49]. These tools are not physics-based and largely phenomenological and use real well data to match decline curve formulas and then project the future well performance based on the historic trend. Analogy and type-curve well assumptions are extensively used to book reserves based on still undrilled but vacant well locations in the acreage leased by the companies.
The solutions for well rate forecasting for fractured wells (Equation (28)) and cylindrical wells (Equation (29)) in unbounded reservoirs and vertical wells in bounded reservoirs (Equation (31)) provide physics-based analytical solutions to predict well performance. The key unknown parameter for new reservoirs to be drilled is the hydraulic diffusivity. The diffusivity can be computed either using Equation (7) with primary parameter inputs, or by history matching early pilot well data, as demonstrated in [
50].
Table 2 lists the strengths and weaknesses of the existing types of tools for well analysis, production forecasting and reserve estimation. Because nodal analysis is rather technical and requires specialist knowledge, companies apply the tool only sparsely. Strengths and weaknesses of the new Gaussian pressure transient solution are also included. A major strength of the Gaussian method is that historic well data are not needed to predict the performance of a new well, provided the hydraulic diffusivity is already known from applying Zimmerman’s Equation (7).
6.2. Relation to Prior Work
In previous research, various analytical solutions have been devised for interpreting well testing data and calculating well rates resulting from pressure changes induced by production in aquifers and oil reservoirs [
17,
18,
20,
21]. However, these earlier solutions typically assumed a constant well rate as the boundary condition. This assumption does not align with real-world scenarios where fluid withdrawal or injection is managed by pump devices that maintain a constant bottomhole pressure, a common practice in oil and gas wells equipped with artificial lift systems. Consequently, the ability to rapidly compute the decline and associated changes in the well rate over time based on the pressure differential applied at the well by the lift system (pump) relative to initial reservoir conditions has been constrained by the absence of concise expressions accommodating the constant bottomhole pressure boundary conditions.
The present study refined and expanded upon the analytical solutions established in recent works [
22,
42]. The earlier contributions provided valuable insights into fluid dynamics within subsurface reservoirs and introduced initial techniques for analytical modeling to compute well rates based on hydraulic diffusivity derived from real field data and production-induced pressure changes, with a particular emphasis on unconventional reservoirs.
By introducing a new analytical solution that accommodates constant bottomhole pressure boundary conditions and incorporates pressure interference from neighboring wells or external boundaries through the superposition of a cylindrical image source, the present study represents a significant leap forward from prior work. This advancement bridges the gap between theoretical models and practical applications by offering a practical and efficient approach for analyzing well testing data and computing well rate changes under constant bottomhole pressure conditions. Moreover, the new solution eliminates the need for time-intensive pressure-matching procedures, such as nodal analysis, for determining well rate declines in systems operating under constant bottomhole pressures. Through a reassessment of the diffusivity equation and the derivation of tailored well rate equations, this study provides a practical and efficient means to compute well rate changes based on fundamental physical parameters, while also accounting for pressure interference effects on diffusion and offering insights into the sensitivity of engineering parameters to outcomes.
The following practical recommendations can be formulated based on the insights from this study:
Analyze real-world historical reservoir pressure and production measurements with instantaneous and steady pressure analytical solutions. Refer to well test paper success in providing valuable estimates of parameters relevant for modeling real-world reservoir behavior.
Expanding analytical models to incorporate multi-phase flow phenomena, such as gas–liquid interactions or phase behavior effects, would advance our understanding of complex reservoir systems and enable more accurate predictions of production rates.
Explore methods for incorporating uncertainty analysis into analytical models, allowing for probabilistic assessments of reservoir performance and providing decision-makers with more robust risk assessments.
Explore the application of analytical models to emerging reservoir types, such as geothermal reservoirs, to expand the scope of their applicability and address pressing challenges in energy production.
Conduct comparative studies to validate the evolving analytical models and benchmark their performance against empirical data to enhance confidence in their predictive capabilities and facilitate their adoption in practical engineering applications.
These recommendations represent potential avenues for further research aimed at advancing our understanding of reservoir behavior and improving the effectiveness of reservoir management practices. By addressing these research areas, we can meaningfully contribute to the sustainable development of energy resources and the optimization of reservoir performance.
7. Conclusions
This study introduces novel analytical solutions for computing well rates in aquifers and oil reservoirs under the condition of constant bottomhole pressure, a scenario often encountered in practical applications involving pump devices, addressing a significant gap in the prior literature. Previous solutions assumed a constant well rate. The new solutions presented in this study enhance the understanding of system dynamics and ensure the precision of production metric estimations. This represents a significant step forward in the field of analytical modeling for well testing and reservoir management. This new approach enables the rapid computation of well rate changes based on pressure differences, reservoir permeability, diffusivity and fluid properties. This study derives a solution for the pressure diffusion equation with a constant bottomhole pressure boundary condition, allowing for the computation of well rate changes without the need for elaborate time-stepped procedures.
Our research builds upon the foundation laid by previous works, as evidenced by the contributions of the Weijermars research group [
22,
23,
42]. The emphasis in the present study is on applying the diffusion equation to a vertical wellbore in a horizontal, permeable reservoir, assuming a constant bottomhole pressure. Furthermore, our approach provides a comprehensive framework for understanding reservoir dynamics and optimizing production strategies.
The discussion section highlights the advantages of the proposed solution over nodal analysis, a traditional method for computing declining well rates. The new solution eliminates the need for time-stepped pressure matching and provides a more straightforward and rapid approach for forecasting well rates. The paper also compares the new solution to other forecasting tools, such as reservoir simulators, decline curve analysis and nodal analysis, outlining the strengths and weaknesses of each approach. The derived Gaussian pressure transient solutions offer practical benefits, particularly for wells produced with artificial lift systems, and provide a valuable alternative and complement to existing methods used in the petroleum industry. In conclusion, our study represents a significant advancement in the field of reservoir engineering, offering a robust analytical toolkit for estimating well rates and paving the way for more accurate and efficient reservoir management in the oil and gas industry.