Performance of a horizontal well in a bounded anisotropic reservoir: Part I: Mathematical analysis

: To enhance the productivity of horizontal wells, it is of necessity to ensure that they perform optimally. This requires an understanding of how the reservoir ’ s geometry, anisotropy and well design a ﬀ ect the pressure response. Mathematical formulations can be used to simu - late pressure response in the wellbore and the data obtained can be analysed to obtain well and reservoir parameters that can aid performance and evaluation. In this study, a mathematical model that can be used to approximate pres - sure response in a horizontal well is formulated, and a detailed mathematical analysis that can be used to obtain well and reservoir parameters are provided. A horizontal well inside a rectangular drainage volume with sealed boundaries is considered and the e ﬀ ect of each boundary on pressure throughout its productive life is studied. In the analysis, investigations on how the reservoir parameters can be approximated over a given period of production are conducted. This is achieved by identi ﬁ cation of the appropriate source and Green ’ s functions. These source functions allow us to formulate a mathematical model for dimensionless pressure. Considering the diagnostic plots for both dimensionless pressure and dimensionless pressure derivative, mathematical analysis studies the possible beha - viour of the plots. Analysis indicates that the reservoir ani - sotropy can be approximated during the in ﬁ nite - acting ﬂ ow at early times when other parameters are known. Further, when the ﬁ rst boundary is felt, in this case the vertical boundary, the horizontal permeability can be approximated during the transition ﬂ ow periods at middle times. Finally, at late times when all the boundaries have been felt and a pseudosteady state ﬂ ow is evident, reservoir dimensions can be approximated. These results can signi ﬁ cantly improve well test analysis and enhance the performance evaluation of a horizontal well.


Introduction
Horizontal wells are replacing vertical wells in modern exploration of oil due to their improved production over a given drainage volume. Their ability to reach larger areas of a drainage volume places them at an advantage compared to vertical wells. With their drilling complexity, this comes with more cost and thus it is important that proper evaluation and a good understanding of their performance is improved so that they can be more productive. The length of the well to be drilled in a given drainage volume so as to maximise the pressure response is of importance compared to the dimensions of the reservoir. Mathematical models are used to simulate pressure response and thus can be used to analyse the data obtained from well tests.
In a monograph [1], the advances in well test analysis with discussions on the use of diffusivity equation in solving fluid flows are discussed. The pressure behaviour during infinite-acting flow is discussed in detail in ref. [2], in which, the exponential integral is applied in the approximation of pressure. In a very detailed paper [3], instantaneous source and Green's functions are studied and it is demonstrated that a solution of the diffusivity equation could be obtained using the product of the appropriate source and Green's functions when the Newman's product method was used. This provided a way of solving unsteady flows. In refs [4][5][6][7][8], studies using these source and Green's functions were conducted and mathematical models which could simulate pressure response in reservoirs were developed. In these studies, infiniteacting models are considered, with some considering certain boundaries as sealed. Isotropic cases were employed in most studies to simplify calculations, thus not considering each directional permeability. These approximations and assumptions continue to limit the models and affect the accuracy of the results obtained. In an attempt to delineate flow periods, authors [9,10] developed strategies of delineating flow periods and developed equations that could be used to approximate the time when flow periods started or ended in a rectangular drainage system. With more horizontal wells being drilled, it became even more necessary to account for these flow periods and how the pressure response during a certain flow period was affected by the well design and the reservoir geometry. This has seen development of more models to simulate pressure response and investigate how well and reservoir parameters influence the performance of a horizontal well. Source and Green's functions have been considered further in the development of these models. Authors [11][12][13][14][15][16][17][18][19][20] have developed models and used them to study horizontal well performance. The consideration of only the infinite-acting flow, sealing of certain boundaries and isotropic cases to ease the complexity of anisotropy considerations in three dimensional drainage volumes continue to limit these models. In this study, the performance of a horizontal well in an anisotropic reservoir when all the boundaries are sealed is investigated.

Reservoir physical model description
A completely sealed rectangular drainage volume of dimensionless length, x eD , dimensionless width, y eD and dimensionless thickness, h D is considered. A horizontal well of dimensionless length, L D drilled in the x-direction is considered. For mathematical analysis, the well is considered to be parallel to the x-boundary and perpendicular to the y-boundary. The well is centrally located in the reservoir such that from the centre of the reservoir located at (x wD , y wD , z wD ), the well stretches to a length L D /2 in both the directions along the x-axis as shown in Figure 1.

Mathematical description
The heterogeneous three dimensional diffusivity equation accounting for directional permeability as given by ref. [9] is shown in equation (1), Where k x , k y and k z are the axial permeabilities in x, y and z directions, Φ is porosity, μ is reservoir fluid viscosity, c t is total compressibility and P is pressure. Equation (1) finds a lot of applications in solving unsteady flows and its solution for dimensionless pressure P D as given by ref. [11] is shown in equation (2), and the dimensionless pressure derivative P D ′ is given by equation (3): where t D is dimensionless time and τ D is a dummy variable for time.
In equation (2), s(i D , τ D ) is the appropriate instantaneous source and Green's function in the respective axial direction given by: • An infinite-slab source of thickness L located at x = x w in an infinite-slab reservoir given by ref. [3] and simplified by inserting dimensionless variables is approximated as shown in equation (4) at early time and as shown in equation (5) for late time.
• An infinite plane-source located at y = y w in an infiniteslab reservoir of thickness y e given by ref. [3] and simplified by inserting dimensionless variables is approximated as shown in equation (6) at early time and as shown in equation (7) for late time.
• An infinite plane-source at z = z w in an infinite-slab reservoir of thickness h given by ref. [3] and simplified by inserting dimensionless variables is approximated as shown in equation (8) at early time and as shown in equation (9) for late time. Substituting these instantaneous source and Green's functions in equation (2), solving for equation (3) and simplifying, we obtain the following mathematical models to simulate dimensionless pressure response at early time as reported in ref. [21]; (i) During infinite-acting flow when no boundary has been felt, the dimensionless pressure is given by equation (10) and the dimensionless pressure derivative by equation (11). where r wD is defined by equation (12).
The exponential integral Ei(−x) in equation (10) is defined as (ii) When the vertical boundary is felt, the dimensionless pressure is given by equation (14) and the dimensionless pressure derivative by equation (15).
(iii) When the horizontal boundary parallel to the well is felt before the horizontal perpendicular boundary, the dimensionless pressure is given by equation (16) and the dimensionless pressure derivative by equation (17).
(iv) In case the horizontal boundary perpendicular to the well is felt before the horizontal parallel boundary, dimensionless pressure is given by equation (18) and the dimensionless pressure derivative by equation (19).
When all the boundaries have been felt and a pseudosteady state behaviour is evident, the dimensionless pressure is given by equation (20) and the dimensionless pressure derivative by equation (21) as reported in ref. [22].
P π x y In this study, the performance of the well from inception to date is considered. To determine the dimensionless pressure response from inception to date, the pressure response models from the infinite-acting flow when no boundary has been felt, through the transition flows as specific boundaries are felt up to the point when all the boundaries are felt and a pseudosteady state flow is evident are superposed. Since the formation thickness is considered far much smaller compared to the length and width of the reservoir, it is expected that the vertical boundary will be felt earlier than the horizontal boundaries. Depending on which of the two horizontal boundaries is felt earlier after the vertical boundary has been felt, the mathematical model for computing dimensionless pressure and its dimensionless pressure derivative starting from inception to date will have two cases. First, for a case where the horizontal boundary parallel to the well (yboundary) is felt first, dimensionless pressure is given by equation (22) Second, the horizontal boundary perpendicular to the well (x-boundary) is felt first. In this case, the dimensionless pressure is given by equation (24) and dimensionless pressure derivative given by equation (25) In these models, the integral limits are calculated using the strategies developed in ref. [9], such that t De is the dimensionless time when the first boundary is felt indicating an end to the infinite-acting flow given by (28) The next integral limit, t D2 is the dimensionless time when all the boundaries of the reservoir have been felt given by the maximum value from the computed values of equations (27) and (28) indicating the start of the full pseudosteady behaviour and finally, t D is the dimensionless time considered to date. Authors of [9] also noted that the early radial flow period can end when the wellbore end effects start affecting the flow at a time expressed in dimensionless form as;

Discussion
Considering a single layer for a centrally located well, the models described in cases one and two are used with Odeh and Babu strategies to identify the approximate integration limits. The possible results are analysed using diagnostic plots from a theoretical mathematical perspective from inception to date.

Infinite-acting flow
The infinite-acting flow period starts shortly after the well is put into production. Considering the model equation for dimensionless pressure derivative during infiniteacting flow given by equation (11), the exponential part, The term on the right hand side of equation (34) is a constant and thus on log-log axes, for a plot of P D ′ , against t D , equation (34) will produce a flat horizontal line such that the flat point can be described by a constant given by expression (35).
This flow is radial in the y-z plane and can be considered as the early radial flow period. Considering the equation for dimensionless pressure during infinite-acting flow given by equation (10), and using the approximation of the exponential integral given by [2], for 0.02 This slope can be used to evaluate the anisotropy ratio in the vertical plane for a given formation thickness.

When the vertical boundary has an effect on the flow
The model equation for dimensionless pressure derivative when the vertical boundary has been felt is considered. Taking logarithms on both sides of the model equation as given by equation (15) and simplifying, equation (40) is obtained.
Considering the limits that this flow period occurs, the error function, erf This shows that a log-log plot of P D ′ , against t D , will be an upward straight line having a half slope. Considering the model equation for dimensionless pressure given by equation (14), its indefinite integral solution is given by equation (42). This implies that a graph of P D against t D will be a straight line having a slope given by expression (44).
This slope can be used to approximate the anisotropic ratio in the horizontal direction perpendicular to the well. The next transition flow suggests a radial flow with the graph of dimensionless pressure derivative expected to flatten. This flow period starts after the flow has started coming from beyond the ends of the wellbore. Considering dimensionless pressure derivative as given by equation (15), during the time that the flow period occurs,  Considering the dimensionless pressure as given by equation (14), during the time that this flow period occurs, t erf x Equation (14) approximates to equation (49).  From this equation it implies that a plot of P D against log(t D ) will give a straight line having a slope given by expression (51). k k k e log .
x y This slope can be used to approximate the anisotropic ratio in the horizontal plane. This transition flow period will end if any of the horizontal boundaries is felt.

When the horizontal boundary has an effect on the flow
Considering the dimensionless pressure derivative where the y-boundary is felt first, with the geometry considered and the time that this flow period occurs, equation (17) approximates to equation (52).
Taking logarithms on both sides of equation (52) and simplifying, equation (53) is obtained.
This indicates that a plot of P D ′ against t D will be a straight line on log-log axes with a half slope. Considering the dimensionless pressure, equation (16), with the geometry considered and the time that this transition flow period occurs, the indefinite integral for equation (16) is given by equation (54).  This slope can be applied in determining the anisotropic ratio in the direction parallel to the well and the dimensionless reservoir width if the anisotropic ratio is already known from previous calculations. On the other hand, if the horizontal boundary perpendicular to the well is felt first, considering dimensionless pressure derivative, from the considered geometry, equation (19) reduces to equation (57). This indicates a straight line with a half slope on log-log axes for dimensionless pressure derivative against dimensionless time. Considering the dimensionless pressure as given by equation (18), the indefinite integral is given by equation (59).
This indicates that a plot of P D against t D will give a straight line whose slope is given by expression (60).
This slope can be used to estimate the anisotropic ratio in the direction perpendicular to the well and if this is determined from previous calculation, then the slope can be applied to estimate the dimensionless reservoir length.

When all the boundaries have an effect on the flow
At that point when all the boundaries of the reservoir have been felt, a pseudosteady state flow starts. Considering dimensionless pressure derivative, from the geometry considered and the time that this flow occurs, equation (21) reduces to equation (61).
P π x y t 2 . This indicates that a plot of P D ′ against t D will be a straight line on log-log axes with a unit slope. Considering the dimensionless pressure, the indefinite integral of equation (21) gives equation (63).
This implies that a plot of P D against t D gives a straight line whose slope is given by expression (64). With any of the two parameters identified in previous calculations, this slope can be used to estimate the other parameters. This flow period will prevail to date.

Conclusion
During any flow period from inception to date, any well test procedure can apply the diagnostic plots as discussed in this study to determine the reservoir parameters. In this study the approximate forms of estimating the required parameters in a well test have been derived and presented. The study provides a method to estimate the reservoir and well parameters. The identifications of specific flow periods, particularly the transition flows, require further analysis since this study considers effects of boundaries. For data that is obtained in a well test, a diagnostic plot can be used together with the analysis of this study to estimate reservoir properties that can improve well design and enhance productivity. Since this study considers anisotropy in all directions, it improves the accuracy of the parameters estimated considering that most studies consider isotropic cases.

Appendix
Case example on computing (effect of dimensionless horizontal well length, L D ) To study how the dimensionless horizontal well length affects the flow periods and dimensionless pressure, using the models derived, dimensionless horizontal well length is varied keeping the other parameters constant. Table A1 shows the theoretical dimensional values considered and the computed dimensionless variables. Table A2 shows the dimensionless flow period time computed using Odeh and Babu strategies as described from equations (23-32). From Table A2, it is observed that the y-boundary is felt first. Using equation (22) dimensionless pressure is computed and dimensional pressure derivative is computed using equation (23).
The dimensionless pressure and dimensionless pressure derivative values computed are shown in Table A3.    Figure A1 shows the plot of P D and P D ′ against t D on log-log axes where the solid lines represent P D against t D , while the dashed lines represent P D ′ against t D . From Figure A1, the infinite-acting flow is identified as the first flow period. This flow period starts shortly after the well is put into production. This flow period is evident with the flattening of the dimensionless pressure derivative plot at early times. This flow period is radial in the y-z plane and can be considered as the early radial flow period. It should be noted that this flow period might end even before the vertical boundary is felt for the cases where the flow coming from the ends of the well starts influencing the pressure response. In such a case, a very short transition flow which is still radial continues until the flow reaches the vertical boundary. When the vertical boundary is felt, it is noted that as the dimensionless horizontal well length increases, an early linear flow period occurs in the y-z plane. This flow period is observed to prevail for a very short time and is evident when the dimensionless pressure derivative plot stops flattening and shows an upward trend. Where the early linear flow period does not occur, a transition flow occurs from the time the vertical boundary is felt until the time the flow starts coming from beyond the ends of the wellbore. This is evident for the first two cases where dimensionless horizontal well length was considered. Where the early linear flow occurs as shown in the last three cases of the dimensionless horizontal well length considered, this flow will end when the flow moves beyond the ends of the wellbore. At this point a transition flow occurs until the flow starts coming from beyond the ends of the wellbore. When the transition flow ends, we identify a second radial flow with the flattening of the graph of dimensionless pressure derivative. This flow period starts after the flow has started coming from beyond the ends of the wellbore and can be considered to be the late pseudoradial flow period with one boundary having been felt. This flow period will end when the y-boundary is felt which is followed by a transition flow.
The transition flow will prevail until the x-boundary is felt at a point where it can be considered that all boundaries to have been felt and a pseudosteady state flow has begun. On the plot, this is identified with the straight upward line at late time. This flow period will prevail to date. For the parameters considered, it is noted that as the dimensionless horizontal well length increases, the pressure response decreases during early time but increases during late time when a pseudosteady state behaviour is observed. Figure A1: Variation in dimensionless pressure and dimensionless pressure derivative with L D .