SPE-171031-MS

History Matching and Rate Forecasting in Unconventional Oil ReservoirsUsing an Approximate Analytical Solution to the Double Porosity Model

B. A. Ogunyomi, T. W. Patzek, and L. W. Lake, The University of Texas at Austin; C. S. Kabir, Hess Corporation

Copyright 2014, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Eastern Regional Meeting held in Charleston, WV, USA, 21–23 October 2014.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations maynot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Production data from most fractured-horizontal wells in gas and liquid-rich unconventional reservoirs plotas straight lines with a one half slope on a log-log plot of rate versus time. This production signature (halfslope) is identical to that expected from a one-dimensional linear flow from reservoir matrix to the fractureface, when production occurs at constant-bottomhole pressure. In addition, microseismic data obtainedaround these fractured wells suggest that an area of enhanced permeability is developed around thehorizontal well, and outside this region is an undisturbed part of the reservoir with low permeability.Based on these observations geoscientists have, in general, adopted the conceptual double-porosity modelin modeling production from fractured horizontal wells in unconventional reservoirs. The analyticalsolution to this mathematical model exists in Laplace space but it cannot be inverted back to real-timespace without using a numerical inversion algorithm. We present a new approximate analytical solutionto the double-porosity model in real-time space and its use in modeling and forecasting production fromunconventional-oil reservoirs.

The first step in developing the approximate solution was to convert the systems of partial differentialequations for the dual-porosity model into a system of ordinary-differential equations. After which wedeveloped a function that gives the relationship between the average pressures in the high-and thelow-permeability regions. Using this relationship, the system of ordinary differential equations was solvedand used to obtain a rate/time function that can be used to predict oil production from unconventionalreservoirs. The approximate solution was validated with numerical reservoir simulation.

We then performed a sensitivity analysis on the model parameters to understand how the modelbehaves. Once the model was validated and tested, we applied it to field production data by partiallyhistory matching and forecasting the expected ultimate recovery. The rate/time function fits productiondata and also yields realistic estimates of ultimate oil recovery. We also investigated the existence of anycorrelation between the model-derived parameters and available reservoir and well completion parame-ters.

IntroductionMany studies have been published that focus on the solution of the double porosity model for flow inhydraulically fractured horizontal wells. Barenblatt and Zheltov (1960) presented the first formulation of

the double-porosity model. Warren and Root (1962) presented the first application of the double-porositymodel to flow problems in the petroleum industry. Since then many authors (de Swaan, 1976; Mayerhofer,2006; Carlson and Mercer, 1989; El-Banbi, 1998; Ozkan et al., 1987) have presented applications of themodel.

All the analytical solutions presented have all been in Laplace space and have had to be numericallytransformed to real time space using some form of inversion algorithm of which the Stehfest algorithm(Stehfest 1970) is the most popular. More recently, Bello and Wattenbarger (2008) presented the solutionto the double porosity model for linear flow in which they were able to obtain closed form analyticalsolutions for certain ranges of time. To do this they broke their Laplace space solution in to smaller bitsusing special properties of the solution which they could invert to real-time space. This piece-wisesolution would have to be applied sequentially. Samandarli et al. (2011) presented the application of thissolution to history matching and forecasting the performance of shale gas wells. Song (2014) presenteda finite-difference solution to this problem and its application to oil production from hydraulicallyfractured wells.

In this study, we present an approximate analytical solution to the double-porosity model in real-timespace that is valid across all time domains, that is, it is a continuous function that is valid during thetransient and late time flow from the fracture and matrix. We validate our solution against numericalsimulation and also show that our solution reproduces the production behavior obtained from the invertedLaplace space solution. We also present example applications of our solution to field data.

Model DevelopmentFig. 1 is a schematic diagram of a hydraulically fractured horizontal well where the fractures areperpendicular to the wellbore. Between successive fractures are low-permeability reservoir matrices. Weassumed that the fracture face is at a constant pressure that is equal to the bottom-hole well pressure ofthe well. The dashed red lines represent the no-flow boundaries created by the interference of flow fromthe matrix into the fracture face.

Figure 1—Schematic diagram of a fractured horizontal well with planar fractures

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We made the following assumptions in the model development:

1. Flow is single phase and slightly compressible,2. Flow occurs in the reservoir isothermally,3. The reservoir is isotropic and homogeneous in each compartment,4. There is no direct communication between the matrix and wellbore,5. There is a large contrast in permeability between the fracture and matrix compartments,6. Neglect secondary effects such as stress dependent permeability (porosity) and desorption.

The system of equations that describes this conceptual model is presented as follows; for the lowpermeability reservoir matrix. The governing partial differential equation, initial condition and boundaryconditions are summarized below as:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Eq. 1 is the diffusivity equation for the reservoir matrix and Eq. 2 is the constant pressure initialcondition. Eq. 3 means that there is a no-flow boundary at the external boundary of the reservoir matrix.Eq. 4 states that flow from the matrix into the fracture face (x � xwf) is equal to the out flow from thefracture face. Eq. 5 states that there is a no-flow boundary at the external boundary of the reservoir matrixin the y-direction and Eq. 6 states that there is no interaction between the reservoir matrix and thewellbore, that is, there is no cross flow from the matrix into the wellbore. Eqs. 7 and 8 are no flowboundary conditions and they model the fact that the reservoir is sealed at the top and bottom boundaries.

For the fracture, we have:

(9)

(10)

(11)

SPE-171031-MS 3

(12)

(13)

(14)

(15)

(16)

Eq. 9 is the diffusivity equation for the fracture for which Eq. 10 is the initial condition. Eq. 11 is theno-flow boundary at the fracture tip. Eq. 12 states that at the wellbore, the fracture pressure is equal tothe wellbore pressure and is constant. Eq. 13 states that there is no flow across the center of the fracture(symmetry) and Eq. 14 is identical to Eq. 4 and they have the same physical meaning. Eqs. 15 and 16 areno flow boundary conditions at the top and bottom of the reservoir, they represent the fact that thereservoir is sealed at the top and bottom boundaries.

Eqs. 1 and 9 form a coupled system of partial differential equations (PDEs) because of the boundarycondition defined by Eqs. 4 and 14. We are interested in developing a rate–time relation for forecastingproduction rate from a system described by these set of equations. To achieve this goal we eliminate thespatial dependence in Eqs. 1 and 9 by integrating over the spatial (x, y and z) domains, respectively, andusing the boundary conditions (Walsh and Lake, 2003) to obtain:

(17)

(18)

The details of this derivation and its solution can be found in Ogunyomi (2014) and Appendix A. InEq. 18, is the average fracture pressure, pf is the net flow rate out of the fracture compartment and qm

is the net matrix flow in to the fracture compartment from the matrix. We have thus transformed thesystem of PDEs in to a system of ordinary differential equations (ODEs). The problem is now atwo-compartment problem.

One advantage of transforming the system of PDEs into a system of ODEs is that it is easier to solvefor the producing rate. Another advantage is that it eliminates the need to know the specific location andgeometry/dimensions of the fracture(s). Nobakht et al. (2013) and Ambrose et al. (2011) presented amethod of forecasting production from a multi-fractured horizontal well that considered planar hydraulicfractures of different lengths. This new model from this study applies to fractures of any arbitrary shapeor geometry (planar or otherwise). Fig. 2 is a simplified representation of the new problem, which is aschematic representation of the reservoir as a two-compartment system in serial flow. The first compart-ment can be thought of as the aggregated volume of all the fractures in the reservoir. It is the onlycompartment connected directly to the wellbore. The average pressure in compartment one is and theflow rate from this compartment into the wellbore is qf. The second compartment is the aggregated volumeof the reservoir matrix. The average pressure in the second compartment is . The matrix compartmentdoes not communicate directly with the wellbore; it only has a cross flow term, qm, into the fracturecompartment.

4 SPE-171031-MS

The next step in the solution is to eliminate the average pressures in Eqs. 17 and 18 by using arelationship between the average reservoir pressure and flow rate. This step is achieved by using ananalytical solution to the one dimensional linear flow problem (Wattenbarger et al., 1998, Bello et al.,2009 and Patzek et al. 2014) with constant pressure at the fracture face; from which the average reservoirpressure, as shown in Appendix B is given by:

(19)

Eq. 19 is the complete solution that includes the transient and late-time solutions. This is an importantpoint because flow in unconventional formations exhibit long periods of transient-linear flow and a usefulmodel must be able to predict production for early and late-time flow. Writing Eq. 19 for the fracture andmatrix compartments respectively, we have:

(20)

(21)

Where:

: Dimensionless production rate for the nth normal mode for the fracture compartment

: Dimensionless production rate for the nth normal mode for the matrix compartment

: Initial production rate from the fracture’s nth normal mode

Figure 2—A simplified representation of the double porosity model as a series model with two compartments (tanks) where the first compartmentrepresents the volume of the fracture and compartment two represents the volume of the reservoir matrix

SPE-171031-MS 5

: Initial production rate from the matrix’s nth normal mode

: Fracture productivity index

Eq. 19 was derived with the assumption that the pressure at the fracture face is constant and equal tothe wellbore pressure pwf. In writing Eq. 21 for the matrix compartment, we have assumed that a constantpwf solution is valid even when it is changing. This assumption is a good approximation when there is alarge contrast in permeability between the two adjacent compartments because the high permeability ofthe fracture compartment ensures a quick pressure equilibration with the wellbore pressure in the fracturecompartment and, hence, the pressure at the interface between the two compartments is approximatelyconstant. This assumption was crucial in attaining the final solution.

Substituting Eqs. 20 and 21 into Eqs. 17 and 18 and after some mathematical manipulations (Ogunyomi2014) we obtain:

(22)

(23)

Where:

: Fracture time constant

: Matrix time constant

Tx : Cross flow transmissibility factor

�f and �m are the fracture and matrix time constants respectively. The parameters in the solution are nowtime constants and transmissibilities, not pore volumes and permeabilities as in the original problemstatement. Cao (2014) presented a detailed discussion of the physical meaning of time constants forimmature and mature waterfloods; in this study these time constants are for primary recovery andphysically they indicate how fast the volumes in each compartment would be drained. The index n in Eqs.22 and 23 are the normal modes (independent solutions). Consequently, we can solve the system of ODEsrepresented by Eqs. 22 and 23 for each mode. We then sum these solutions to obtain the complete solutionto the problem. We rewrite this system of ODEs in matrix-vector form for each normal mode as shownbelow and then solve it with eigenvalue-diagonalization.

(24)

The initial conditions to solve the system represented by Eq. 24 are:

(25)

(26)

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Eq. 25 simply states that at time, t � 0, the production rate from the fracture volume is equal to a finitevalue of qfi. While Eq. 26 states that at time, t � 0, the production rate from the matrix volume is equalto zero, qmi

. This is because at time zero the pressure everywhere in the formation is equal to the initialreservoir pressure and, as a result, there is no pressure gradient for flow from the matrix into the fracturebecause the pressure the fracture/matrix interface is still at the initial reservoir pressure.

We solved Eq. 24 by the eigenvalue-decomposition method, as shown in Appendix C, to obtain thefollowing expression:

(27)

Eq. 27 is the approximate analytical solution to the double porosity model. The negative sign under thesquare root of the eigenvalues in Eq. 27 is necessary because the eigenvalues are always negative. Detailsof the derivation can be found in Ogunyomi (2014). The definition of the model parameters aresummarized below:

(28)

(29)

(30)

(31)

Eqs. 28 and 29 are the eigenvalues of the A matrix of the system of ODEs in Eq. 24. These equationsshow the mathematical relationship between the eigenvalues and the fracture, matrix time constants, thetransmissibility coefficient between the fracture and the matrix compartment and the ratio of theirpermeabilities. Note that � and � are the first components of the eigenvector corresponding to theeigenvalues in Eqs. 28 and 29, and the other components are one. Physically, �1 and �2 are the timeconstants of an equivalent parallel flow model that will yield the same results as the original problem whenappropriately weighted with the eigenvectors. One can regard Eqs. 28 through 31 as expressions forscaling parameters that can be used to transform a two compartment series flow model into a twocompartment parallel-flow model without crossflow. A generalizatioin of this solution to three compart-ments is available in Ogunyomi (2014).

Model ValidationWe validate the approximate analytical solution to the double-porosity model, represented by Eq. 27, witha synthetic case. We developed it with a commercial black oil, finite-difference simulator. The model usedin the synthetic case was two-dimensional (2D) and has two adjoining reservoir compartments in whichthe compartment containing the producing well has a higher permeability than the second compartment.

The compartment with the high permeability can be thought of as the aggregated collection of thefracture volume while the second compartment represents the reservoir matrix with lower permeability.The simulation uses spatially resolved permeability cells, which is the main difference between it and the

SPE-171031-MS 7

approximate analytical solution. The volume of the fracture (high permeability) compartment is equal to25% of the volume of the matrix (low permeability) compartment. All other properties are identical forthe two compartments. Table 1 summarizes the other inputs for the synthetic case.

To validate the approximate analytical model, the production rate obtained from running the syntheticcase is matched to the production rate obtained from the approximate analytical solution. Fig. 4 presentsa comparison of the production rate obtained from the synthetic case and the approximate analyticalsolution.

The production history in Fig. 4 exhibits two time scales; the first time scale initially starts as a straightline with a slope of one-half, which indicates transient-linear flow in one dimension. This flow regime isfollowed by an exponential decline that indicates boundary dominated flow (BDF) from the firstcompartment. Following the dissipation of BDF from the first compartment the second compartment startswith an expected straight line with half slope signature. This transient flow regime is then followed by an

Table 1—Summary of input parameters for the synthetic case as used in the commercial simulator

Parameter value

�x � �y � �z, (ft) 100 � 50 � 50

Number of cells 51 � 11 � 1

Subsurface depth, D (ft) 2000

Thickness, h (ft) 50

Compartment 1 (fracture)

Permeability, kf (md) 70

Porosity, � (fraction) 0.3

Volume, vf (ft3) 8.25�106

Compartment 2 (matrix)

Permeability, km (md) 10

Porosity (fraction) 0.3

Volume, vm (ft3) 3.3 � 107

Figure 3—Reservoir grid for the synthetic case showing the permeability field. The grid blocks in green are the high permeability compartment andthe blue grids are the low permeability compartment

8 SPE-171031-MS

external BDF regime from the second compartment.The agreement shown in Fig. 4 is well within engi-neering accuracy. Table 2 summarizes the fittingparameters for the analytical solution.

Comparison of Approximate AnalyticalSolution with the Laplace-Space Solution

In this section, we present a comparison of theanalytical solution derived in the previous sectionwith the Laplace-space solution to the double po-rosity model. The solution to Eqs. 1 through 14using Laplace transforms is given as:

(32)

where: Inter-porosity transfer function

: Dimensionless pressure

: Dimensionless time

: Dimensionless distance in the x-direction

Figure 4—Comparison of the production rate from the synthetic case and the approximate analytical model. The graph in red represents theproduction rate predicted by the analytical solution while the graph in blue is the production rate from the synthetic case.

Table 2—Model parameters used in the validation case

Parameter Value

qfi (b/d) 4696

�f (day) 56

�m (day) 6087

Tx/Jf (ratio) 3.67�10-2

�1 (day-1) -1.60�10-4

�2 (day-1) -1.85�10-2

Y (unitless) �27

� (unitless) 1

Y/(Y-�) (ratio) 9.64�10�1

�/(Y-�) (ratio) �3.60�10�2

SPE-171031-MS 9

: Dimensionless distance in the y-direction

: Inter-porosity transfer parameter

: Storativity ratio

The solution given by Eq. 32 is the constant-pressure solution; that is, it assumes an instantaneousconstant pressure at the fracture face. The details of the derivation of this solution can be found in Belloet al. (2009). A problem with using this solution is that it cannot be inverted back into the real-time spaceto obtain a closed-form analytical solution; hence, we employ a numerical-inversion algorithm to computepressures and rate from this solution. From this solution, we obtain the production rate at the fracture faceby taking the derivative of Eq. 32 and evaluating its value at the fracture face; that is,

(33)

Bello et al. (2008) provided a detailed sensitivity analysis on Eq. 33 to understand how the modelparameters affect its production characteristics. We summarize the result of this sensitivity analysis next.Fig. 5 is a plot of the dimensionless rate versus dimensionless time where the inter-porosity transferparameter and the storativity ratio have been varied. The physical meaning of the general characteristicobserved on this plot can be explained as follows; at the start of production, flow is predominantly linearwith a slope of one-half, which represents transient flow from the fracture. Thereafter, an exponentialdecline period sets in when the effect of the fracture boundary is felt. Following this flow period, anotherlinear-flow period starts (also characterized by a one-half slope) representing transient flow from thereservoir matrix. After this transient flow period another exponential decline period is observed and thisis the effect of the external boundary of the matrix (Walsh and Lake 2003).

Figure 5—Effect of storativity ratio (�) and inter-porosity transfer parameter (�) on the production rate from the double-porosity model.

10 SPE-171031-MS

We now compare the production rate from the approximate analytical solution, Eq. 27, to theproduction rate from the inverted Laplace-space analytical solution, Eq. 33. For the approximate analyticalsolution to be useful, it should reasonably reproduce the observed characteristics in Fig. 5. The results ofthis comparison for three cases are shown in Fig. 6.

In Fig. 6, case 1 shows the history match for � � 1E-3 and � � 1E-5, case 2 shows the history matchfor � � 1E-1 and � � 1E-5 and case 3 shows the history match for � � 1E-1 and � � 1E-9. Clearly,Fig. 6 suggests that the production rate predicted by the approximate analytical solution provides a goodmatch to the production rate predicted by the Laplace-space solution.

Analysis of model parametersPhysical meaning of the model parametersThe definition of the time constants in the approximate analytical solution is identical in definition to thatdefined in the capacitance resistance model (CRM). As a result, we conclude that it has a similar physicalmeaning. In the CRM, Nguyen et al. (2012) and Cao et al. (2014), following the work of Seborg et al.(2003), have defined the time constant to be the time it takes for 63.2 percent of a pressure pulse inputto be observed as the output. The input pulse for our model would be the pressure difference that isresponsible for flow. In the CRM, it is the injection rate.

Inferring fracture and matrix volume from model parametersIn this section, we investigate the possibility of estimating the size (volume) of the fractures induced bythe hydraulic fracture and the reservoir matrix from the model parameters with the approximate analyticalsolution. To accomplish this task, we took the following steps:

1. Build a numerical model with two compartments where one compartment has a high permeabilityand the second compartment has a low permeability with a commercial reservoir simulator.

2. Perform a history match of the production rate from the numerical model to the approximateanalytical solution to obtain the model parameters.

3. Change the relative volume of each compartment in the numerical simulation model while keeping

Figure 6—Comparison of production rate from the approximate analytical solution to the Laplace-space solution

SPE-171031-MS 11

the total pore volume constant and repeat step 2.4. After obtaining the model parameters for a few cases we make a cross plot of each model

parameter with the volume of each compartment as defined in the numerical simulation model.

The model parameters considered for this analysis are the fracture compartment time constant (�m), thematrix compartment time constant (�f) and the initial fracture production rate (qfi). The numerical modelused for this analysis is identical to that presented in Fig. 3 and the model used is Eq. 27. The result ofthis numerical experiment is summarized in Table 3; it presents a summary of the cases considered andthe model parameters obtained from the history matching exercise.

Fracture time constant Fig. 7a. presents the crossplot of the fracture time constant and the pore volumeof the high permeability compartment, whereas Fig. 7b presents the same for the low-permeabilitycompartment. From Fig. 7a as the pore volume of the high permeability compartment increases thefracture time constant increases, indicating a positive correlation between them. The coefficient ofdetermination is large, R2 � 0.98. Recall that the fracture time constant is defined as , where

vpfis the fracture pore volume. This definition of the fracture time constant suggests that we can infer the

size of the fracture volume from the value of the fracture time constant. In contrast, Fig. 7b suggests thatthe fracture time constant decreases with increasing pore volume of the low-permeability compartment.This figure also has a high coefficient of determination, R2 � 0.98. This observation is because of the factthat the fracture volume shares a boundary with the matrix volume and this shared boundary was heldconstant during this experiment while the other boundaries changed.

Table 3—Summary of model parameters obtained from the numerical experiments performed to investigate the possibility of inferring fractureand matrix volumes by using the approximate analytical solution

Case Simulator Vpf (ft3) Simulator Vpm (ft3) �f (days) �m (days) -1/�1 (days) -1/�2 (days) qfi (STB/D)

1 8.25�106 3.30�107 56 6087 6312 54 4696

2 1.65�107 2.48�107 176 3998 4247 166 2696

3 2.48�107 1.65�107 267 1900 2462 206 2080

4 4.13�106 3.71�107 15 7455 7640 15 8196

5 0.00�100 4.13�107 - 8634 8856 - 7096

Figure 7—Fig. 7a is the cross plot of the fracture time constant and the pore volume of the high permeability compartment in the numerical model.Fig. 7b is the cross plot of fracture time constant and the pore volume of the low permeability compartment in the numerical model

12 SPE-171031-MS

Matrix time constant Fig. 8a is the cross plot ofthe matrix time constant and the pore volume of thehigh permeability compartment, whereas Fig. 8brepresents the same for the low permeability com-partment.

Fig. 8a suggests that, as the pore volume of thehigh permeability compartment increases, the ma-trix time constant decreases. This relationship indi-cates a negative correlation between them. The co-efficient of determination is high, R2 � 1.0,suggesting that there is a relationship between thesize of the fracture volume and the matrix time constant. This transmissibility factor is a function of thefracture dimension. From Fig. 8b, we observe that as the pore volume of the low-permeability compart-ment increases, the matrix time constant increases. This observation is expected because in the definition

Figure 8—Fig. 8a is the cross plot of matrix time constant and the pore volume of the high-permeability compartment in the numerical model. Fig.8b is the cross plot of matrix time constant and the pore volume of the low-permeability compartment in the numerical mode.

Figure 9—Fig. 9a is the cross plot of initial production rate and pore volume of the high permeability compartment in the numerical model Fig. 9bcross plot of initial production rate and the pore volume of the low permeability compartment in the numerical model

Table 4—Well data for field examples 1 and 2

Parameter Example 1 Example 2

Well ID UT-ID67 UT-ID265

Well length (ft3) 8894 9965

Spacing (acres) 1280 1280

Initial pressure, pi (psi) 6083 7657

Porosity (fraction) 0.07 0.07

thickness, h (ft) 53 68

Initial number of stages 1 10

SPE-171031-MS 13

of the matrix time constant, as shown above, the matrix time constant is directly related to the matrix porevolume, vpm

. This cross-plot also has a high coefficient of determination.

Initial production rate Fig. 9a presents a cross plot of the initial production rate and the pore volumeof the high permeability compartment. Fig. 9b is the cross plot of the initial production rate and the porevolume of the low permeability compartment. From Fig. 9a as the pore volume of the high permeabilitycompartment increases the initial production rate from the fracture decreases, indicating a negative

Figure 10—Summary of production profiles for Example 1: (a) Presents raw data on the log-log diagnostic plot, (b and c) show history matching andforecasting results on log-log and Cartesian plots

14 SPE-171031-MS

correlation between them. The coefficient of deter-mination is high, R2 � 0.84, suggesting that we caninfer the size of the fracture volume from the initialproduction rate from the fracture. The definition ofthe initial production rate is given as qfi � (pi –pwf)kf Af/�Lf. We note that Af � hwf and pf � Af

Lf in the numerical simulation model. In the exper-iments conducted, when we increased the fracturepore volume we increased Lf and as this variable isin the denominator of the definition of qfi. From thisdefinition there is an inverse relationship betweenthe initial production rate and the fracture volume,which explains the observation in Fig. 9a.

From Fig. 9b we see that as the pore volume ofthe low permeability (matrix) compartment in-creases the initial production rate from the fracture increases. This observation is consistent with the factthat the total pore volume of the reservoir was kept constant, which implies that by increasing the fracturepore volume we decrease the matrix pore volume, pm � Af(L – Lf) � pT – pf. Therefore the initialproduction rate should increase as the matrix pore volume is increased. Given the good correlation, we canestimate the matrix pore volume from the initial production rate.

Model’s Application to Field DataWe present example applications of the approximate solution to field data and demonstrate how to use itto estimate reserves from hydraulically fractured horizontal wells in liquid rich unconventional forma-tions. The model was fitted to production rate data from 88 hydraulically fractured horizontal wells(Ogunyomi 2014). All the fits obtained were within the limits engineering accuracy. To apply the modelto field data from a well, we fit the model to available historical production rate data from the well toobtain the model parameters by minimizing the squared difference between the model estimates and thefield production data, that is, by changing �f, �m and . After obtaining the model

parameters, we proceed to forecast future production rates and cumulative production until 100,000 days.We present two example applications of the model to this data set.

Example 1For this example, we summarized the well details in Table 4. This well has been on production for 1,136days. Fig. 10a presents both the rate and tubinghead pressure on a log-log plot. on a log-log plot. Thefigure suggests that the production rate is relatively constant until about 90 days after which the productionrate from the well declined exponential until it started declining with a slope of one-half. The tubingheadpressure for this well declined with a slope of one half until about 90 days (transient flow) after which itdeclined with a slope of one indicating BDF until about 100 days when it becomes constant.

If we assume that production during the first 100 days is from the fracture volume and the productionafter 100 days is from the matrix then we can match the model to this data making sure we match theexponential decline and the half-slope portions of the data. Fig. 10b presents the results of the rate historymatch and future performance. This figure contains three plots, the original production data (red markers),the history match (green colored markers) and the forecast (black markers). We summarized the modelparameters obtained from the history match exercise in Table 5. The mismatch at the start of theproduction history is because the well was produced at a variable bottomhole pressure during this periodwhile the model presented is based on the assumption that the wellbore pressure is constant. After

Table 5—Summary of model parameters for field examples 1 and 2

Example 1 Example 2

Fitting parameters

qfi (stb/d) 867.9 436.7

Tx/Jf (ratio) 0.2 0.27

�f (days) 34.0 51.9

�m (days) 955.0 392.8

Computed parameters

�1 (days-1) -8.7�10-04 -1.96�10-03

�2 (days-1) -3.5�10-02 -2.5�10-02

Y (unitless) -4.9 -3.37

� (unitless) 1.0 1.11

Y/(Y-�) (ratio) 0.8 0.8

�/(Y-�) (ratio) -0.2 -0.2

SPE-171031-MS 15

obtaining the model parameters from the history match exercise we use the model to forecast futureproduction rate and reserves until 10,000 days. Fig. 10c presents the performance forecasting results.

Example 2We summarized the well details for this example in Table 4. This well has been on production for 531days. The production rate from this well is shown in Fig. 11a on a log-log plot. This figure suggests thatthe production rate is relatively constant until about 10 days after which the well declined exponentially

Figure 11—Summary of production history profiles for Example 2: (a) Presents raw data on the log-log diagnostic plot, (b and c) show historymatching and forecasting results on log-log and Cartesian plots.

16 SPE-171031-MS

until it started declining with a slope of one-half. The tubinghead pressure for this well started decliningwith a slope of one-half (indicating transient flow) until about 10 days after which it started declineingwith a slope of one (indicating boundary dominated flow). After about 100 days the tubinghead pressurewas relatively constant. Again if we assume that production during the first 100 days is from the fracturevolume and the production after 100 days is from the matrix then we can match the model to this datamaking sure we match the exponential decline and the half-slope portions of the data. The result of theproduction rate history match is shown in Fig. 11b and that of future performance in Fig. 11c.

This figure contains three plots, the original production data (red markers), the history match (greencolored markers) and the forecast (black markers). From this figure we have matched the exponentialdecline portion of the rate data and we also matched the linear decline portion of the rate data. We havesummarized the model parameters obtained from the history match exercise in Table 5. After obtainingthe model parameters from the history match exercise, the model was used to forecast future productionrate and cumulative production until 8,000 days. The result of the forecasting process is shown in Fig. 11c.

The model parameters obtained from these two examples and others not shown here are all functionsof the well and reservoir properties. Consequently, the forecasted results have high degree of confidenceparticularly when the fracture/reservoir interface was observed in the production rate data, which providedan opportunity for defining the geometry of the adjoining compartment.

DiscussionA generally accepted conceptual model for fractured horizontal wells is that a stimulated reservoir volume(SRV) develops around the fractured well and there is a region of un-damaged reservoir beyond the SRV(Miller et al. 2010). The SRV is expected to be comprised of a complex network of fractures of differentgeometries ranging from planar, curved, slanted etc and of different lengths. However, for ease ofsolution, existing “physics” based models assume that the hydraulic fractures are planar and perpendicularto the wellbore. The new solution presented in this work overcomes this limitation of existing modelsbecause the assumption of planar fractures is not necessary.

Most empirical models do not account for the second time scale and the end of transient linear flowmust be determined arbitrarily before switching to a boundary dominated flow model. The new solutionpresented here also eliminates this limitation of empirical models. For cases where there is no productiondata from the second time scale the single porosity solution should be used. This solution is shown below:

(33)

In Eq. 34, the first term accounts for BDF and the second term is the transient solution. Thedimensionless time tD is defined as where � is the time constant of compartment 1 and t is time. The

details of the derivation of Eq. 34 are in Ogunyomi (2014).We have shown that the model parameters derived from the use of the new solution are functions of

the reservoir and well completion properties. Particularly, the model parameters can be used to estimatethe drainage volume of a well; this characteristic of the model could have potential application in in-filldrilling and well spacing optimization studies. Because the model has a closed analytical form, it isespecially suited for optimization studies that account for uncertainties in reservoir properties and theoutcome of well stimulation (hydraulic fracturing) jobs.

ConclusionsThe main goal of this work was to develop a rate–time relationship to predict realistic future performancefrom hydraulically fractured horizontal wells in unconventional formations. We summarize the majorfindings below:

SPE-171031-MS 17

1. A simple rate–time relationship is presented to predict well performance in stimulated unconven-tional reservoirs with a double-porosity model. The model developed is valid for all flow regimes(transient and pseudosteady state, including the transition period) and it is a function of reservoirand well properties.

2. The solution presented, although approximate, was validated with numerical flow simulationresults and was shown to accurately reproduce the production rate history and the expectedultimate recovery.

3. The model identifies different flow regimes observed both in the synthetic and field productiondata, thereby largely finessing limitations of other empirically derived models. One of the model’sattributes is that it always extrapolates to a finite cumulative-production volume.

4. We also demonstrated the utility of the model for practicing engineers by presenting exampleapplications to field production data.

AcknowledgmentThis work is supported by the sponsors of the Center for Petroleum Asset Risk Management (CPARM)at The University of Texas at Austin. We thank Hess Corporation for providing data for this work and theComputer Modeling Group (CMG) for use of their black oil simulator. Larry W. Lake holds the Sharonand Shahid Ullah Chair at the Center for Petroleum and Geosystems Engineering at the University ofTexas at Austin. Tad Patzek holds the Cockrell Family Regents Chair in Engineering and the Lois K. andRichard D. Folger Leadership Chair at the Department of Petroleum and Geosystems Engineering.

Nomenclature

�f � Fracture time constant, day�f � Matrix time constant, dayTx �Transmissibility factor between fracture and matrix compartment, barrel per day per psif(s) �Laplace space inter-porosity transfer function, dimensionlesspD �Dimensionless pressuretD �Dimensionless timexD �Dimensionless distance in the x-directionyD �Dimensionless distance in the y-direction� �Inter-porosity transfer parameter, dimensionless� �Storativity ratiopm �Matrix pressure, psipf �Fracture pressure, psipi �Initial reservoir pressure, psipwf �Bottomhole flowing well pressure, psin �Index for normal modeqfn �Production rate for the nth normal mode for the fracture compartment, dimensionlessqmn �Production rate for the nth normal mode for the matrix compartment, dimensionlessqfi �Initial production rate from the fracture’s nth normal mode, dimensionlessqmi

�Initial production rate from the matrix’s nth normal mode, dimensionlessJf �Fracture productivity index, barrel per day per psip¯f �Average pressure in fracture compartment, psip¯m �Average pressure in matrix compartment, psikf �Effective fracture permeability, mdkm �Effective matrix permeability, md�1,�2 �Eigenvalues of the A matrix for the system of ODEs, day-1

18 SPE-171031-MS

� �First element of the eigen-vector corresponding to the other element is 1, dimensionless� �First element of the eigen-vector corresponding to the other element is 1, dimensionless

ReferencesAmbrose R.J., Clarkson C.R., Youngblood J.E., Adams R., Nguyen P., Nobakht M. and Biseda B.

2011. Life-Cycle Decline Curve Estimation for Tight/Shale Gas Reservoirs. Paper SPE 140519presented at the SPE Hydraulic Fracturing Technology Conference and Exhibition, the Wood-lands, Texs, USA 24 – 26, January. doi: http://dx.doi.org/10.2118/140519-MS

Barenblatt, G. I. and Zheltov, Y. P. 1960. Fundamentals Equations of Filtration of HomogeneousLiquids in Fissured Rocks. Soviet Physics, Doklady Vol. 5 page 522.

Bello, R. O. 2009. Rate Transient Analysis in Shale Gas Reservoirs with Transient Linear Behavior.PhD Dissertation, Texas A&M University, College Station, Texas, USA.

Cao F. 2014. Development of a Coupled Two Phase Flow Capacitance Resistance Model, Unpub-lished PhD Dissertation. The University of Texas at Austin, Austin, Texas.

Cao F., Luo H. and Lake L. W. 2014. Development of a Fully Coupled Two-Phase Flow BasedCapacitance-Resistance Model (CRM). Paper SPE 169485-MS presented at the SPE Improved OilRecovery Symposium, Tulsa, Oklahoma, USA 12–16 April. http://dx.doi.org/10.2118/169485-MS.

Carlson E. S. and Mercer J. C. 1991. Devonian Shale Gas Production: Mechanisms and SimpleModels. J Pet Tech 43 (4): 476–482.

de Swaan-O., A. 1976. Analytic Solutions for Determining Naturally Fractured Reservoir Propertiesby Well Testing. SPEJ 16 (3): 117–122; Trans., AIME, 261.

El-Banbi A. H. 1998. Analysis of Tight Gas Wells. PhD Dissertation. Texas A & M University,College Station, Texas, USA.

Mayerhofer M.J., Lolon E. P., Youngblood J. E., and Heinze J. R. 2006. Integration of MicroseismicFracture Mapping Results with Numerical Fracture Network Production Modeling in the BarnettShale. Paper SPE 102103 presented at the SPE Annual Technical Conference and Exhibition, SanAntonio, Texas, 24–27 September. http://dx.doi.org/10.2118/102103-MS.

Nguyen, A. P. 2012. Capacitance Resistance Modeling for Primary Recovery, Waterflood, andWater/CO2 Flood. PhD Dissertation. The University of Texas at Austin, Austin, Texas, USA.

Nobakht M., Ambrose R., Clarkson C.R., Youngblood J. and Adams R. 2013. Effect of CompletionHeterogeneity in a Horizontal Well With Multiple Fractures on the Long-Term Forecast inShale-Gas Reservoirs. J. Cdn. Pet. Tech. 52 (06): 417–425. doi: http://dx.doi.org/10.2118/149400-PA

Ogunyomi B.A. 2014. Mechanistic Modeling of Recovery from Unconventional Reservoirs. Unpub-lished PhD Dissertation. The University of Texas at Austin, Austin, Texas, USA.

Ozkan E., Ohaeri U. and Raghavan R. 1987. Unsteady Flow to a Well Produced at a Constant Pressurein a Fractured Reservoir. SPE Form Eval 2 (2): 186–200. http://dx.doi.org/10.2118/9902-PA.

Patzek T.W., Male F., and Marder M. 2014. Gas Production from Barnett Shale Obeys a SimpleScaling Theory. Proceedings of the National Academy of Sciences of the United States ofAmerica. PNAS December 3, 2013 vol. 110 no. 49 19731–19736 doi: 10.1073/pnas.1313380110.

Samandarli, O., Al-Ahmadi H., and Wattenbarger R. A. 2011. A New Method for History Matchingand Forecasting Shale Gas Reservoir Production Performance with a Dual Porosity Model. PaperSPE 144335 presented at the SPE North American Gas Conference and Exhibition, the Wood-lands, Texas, USA, 12 – 16 June. http://dx.doi.org/10.2118/144335-MS.

Seborg, D. E., Edgar, T. F., and Mellichamp, D. A. 2003. Process Dynamic and Control, 2nd edition.Wiley, John & Sons, Inc., New York. ISBN-13: 9780471000778.

SPE-171031-MS 19

Song, D. H. 2014. Using Simple Models to Describe Production from Unconventional Reservoirs.Master’s Thesis. The University of Texas at Austin, Austin, Texas.

Stehfest H. 1970. Numerical Inversion of Laplace Transforms. Communications of the ACM. 13 (1)47–49. doi: http://dx.doi.org/10.1145/361953.361969

Walsh M. P. and Lake L. W. 2003. A Generalized Approach to Primary Hydrocarbon Recovery.Elsevier. ISBN: 978-0-444-50683-2.

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Wattenbarger R. A., El-Banbi A. H, Villegas M.E., and Maggard J. B. 1998. Production Analysis ofLinear Flow into Fractured Tight Gas Wells. Paper SPE 39931 presented at the 1998 SPE RockyMountain Regional/Low permeability Reservoirs Symposium and Exhibition, Denver, Colorado,5–8 April. http://dx.doi.org/10.2118/39931-MS.

20 SPE-171031-MS

Appendix A

Conversion of the Coupled Partial Differential Equations to a System of Coupled Ordinary DifferentialEquations

This appendix presents the details of how the coupled system of partial differential equations was converted into a system ofcoupled ordinary differential equations that is independent of the spatial variables. We eliminate the spatial dependence in Eq.1 by integrating with respect to x, y and z with x varying from xwf to xe, y from ywf to ye and z from z0 to ze

(A-1)

Using the fact that t is independent of position, then we can bring the time derivative on the right side of Eq. A-1 outsideof the integral to obtain

(A-2)

If we define the average pressure in the matrix as shown below

(A-3)

(A-4)

bmis the bulk volume of the reservoir matrix. Substituting Eq. A-4 into Eq. A-2 and carrying out the integral on the left

(A-5)

Using the boundary conditions defined by Eqs. 3, 5, 6, 7 and 8, Eq. A-5 simplifies to

(A-6)

Multiplying both sides of Eq. A-6 by , Eq. A-6 simplifies to

(A-7)

In Eq. A-7, we note that from Darcy’s law and pm � mbm, where pm is the effective matrix

pore volume. Therefore, we can rewrite Eq. A-7 as:

(A-8)

In Eq. A-8, is the average pressure in the reservoir matrix and qm is the net flow rate from the reservoir matrix.

Performing the same integration over the x, y and z domains of the fracture Eq. 7, x from x � 0 to xwf, y from ywf to ye andz from z0 to ze,

SPE-171031-MS 21

(A-9)

Evaluating the integrals in Eq. A-9 we obtain:

(A-10)

In Eq. A-10, we have made use of .

Multiplying Eq. A-10 by and applying the boundary conditions from Eqs. 11 and 13 we obtain:

(A-11)

Noting that and pf � fbf, Eq. A-11 can be written as:

(A-12)

From the boundary condition given by Eq. 14, substituting this into Eq. A-12 and it becomes

(A-13)

By noting that, then Eq. A-13 is rewritten as:

(A-14)

By eliminating the spatial dependence in Eqs. 1 and 9 we have transformed a microscopic mass balance equation to amacroscopic mass balance equation. The microscopic mass balance equation describes mass balance at a point while themacroscopic equation describes the mass balance for a finite system, Walsh and Lake (2003). Therefore the model parametersin the macroscopic equation are the average properties in the finite volume.

22 SPE-171031-MS

Appendix B

Derivation of a Mathematical Relationship between the Average Reservoir Pressure and Flow Rate

The solution to the 1D linear flow problem (1D diffusivity equation) with constant pressure at the fracture face and a no-flowouter boundary is given by (Wattenbarger et al., 1998; Ogunyomi, 2014)

(B-1)

where pD is the dimensionless reservoir pressure, tD is the dimensionless time and xD is the dimensionless distance in thex-direction. From Eq. B-1 we obtian the production rate at the fracture face as , therefore

(B-2)

By inspection all the terms of the series in Eq. B-2 are positive. The exponential term has two coefficients and

(�1)n. For odd values of n, is �1 (positive 1) and (�1)n is -1 (negative 1). As a result the product of these two

coefficients is always negative. When the product of these two coefficients is multiplied by the negative sign outside thesummation sign, we realize that all the terms of this solution are always positive. Using the same reasoning, we conclude thatfor even values of n, all terms of the solution are positive. Therefore we can write Eq. B-2 as

(B-3)

Again using Eq. B-1 we obtain the volume weighted average reservoir pressure as:

(B-4)

In Eq. B-4 is the dimensionless average reservoir pressure. As with the rate equation, the terms of the infinite series

are always negative therefore we can rewrite Eq. B-4 as:

(B-5)

If we write Eq. B-5 in dimensional form we obtain:

(B-6)

Using the definition of productivity index and Eq. B-3 in Eq. B-6, we obtain an expression that relates

the average reservoir pressure to the well rate as:

(B-7)

Eq. B-7 can be written for the fracture and matrix compartments as:

SPE-171031-MS 23

(B-8)

(B-9)

By writing Eq. B-9 for the matrix compartment we have assumed that the solution to the diffusivity equation with a constantpressure inner boundary condition is valid even when the pressure at the inner boundary is varying. This assumption isreasonable when there is big constrast in transmissibility between two adjoining reservoir compartments.

24 SPE-171031-MS

Appendix C

Solution of the System of Ordinary Differential Equations

Using Eqs. B-8 and B-9 in Eqs. A-8 and A-15 we, after some simplification, obtain the following system of ordinary differentialequations:

(C-1)

(C-2)

In Eqs. C-1 and C-2 we have used qfn � qfiqfDn and qmn � qmiqmDn. The solution to this system of differential equationsis obtained by using eigenvalue diagonalization for each index n and then summing these solutions. For index n, the systemis given as:

(C-3)

And the initial conditions are qf(t � 0) � qfi and qm(t � 0). The eigenvalues of the A matrix of Eq. C-3 are given by:

And the corresponding matrix of the eigenvectors is given by:

Therefore the solution to this systemin Eq. C-3 is given as:

(C-4)

(C-5)

Using the initial conditions, the solution simplifies to

(C-6)

(C-7)

Therefore, the complete solution to Eqs. C-1 and C-2 is given by:

(C-8)

SPE-171031-MS 25

(C-9)

To arrive at the final form of the solution we eliminate the infinite sum in Eq. C-8 by converting the discrete sum to anintegral to obtain:

(C-10)

26 SPE-171031-MS